• Share
  • Email
  • Embed
  • Like
  • Save
  • Private Content
Knowledge varsity  dec 2011 study_session_2
 

Knowledge varsity dec 2011 study_session_2

on

  • 476 views

CFA L - I Notes for Quantitative Mathematics

CFA L - I Notes for Quantitative Mathematics

Statistics

Views

Total Views
476
Views on SlideShare
476
Embed Views
0

Actions

Likes
0
Downloads
6
Comments
0

0 Embeds 0

No embeds

Accessibility

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

    Knowledge varsity  dec 2011 study_session_2 Knowledge varsity dec 2011 study_session_2 Document Transcript

    • Concept Notes For Reading 5 - Time value of moneyReading SummaryThis reading is the most important reading for the CFA Level 1 Examination as it covers the basics ofpresent value (PV) of the expected future cash flows. The PV concept is used throughout the curriculumwith major emphasis in Equity Valuation, Fixed Income valuation and corporate finance. A mastery oftime value is paramount to be successful in the examination.The topic covers calculation of present value and future value for a single cash flow, a series of cashflow (known as annuity), computation of effective rate, problems related to mortgages, retirement.Understanding of time line is critical and it is advised that the candidates should draw the timeline whilesolving any problems. For easy problems it might look that drawing time line is waste of time buttimeline is important when you are solving complex problems with multiple cash flows as the timelinewill give a perfect picture of the cash inflows and outflows.Basic Idea and ConceptsTime Value of Money (TVM) links cash flow, interest rate, compounding frequency, time period and thepresent value or the future value. TVM reflects that a $1 amount with a person today is more valuablethan $1 at a future date because $1 can be invested now and interest can be earned on the investment.TVM would require calculation of Future Value (FV), which is result of compounding the investment at acertain interest rate. TVM also entails finding the present value (PV), which is reverse of compoundingand is referred as discounting. Comparison of PV or FV is very useful in analyzing investments and thenmaking an investment decision. As an investor you would be more interested to invest in thoseproducts which will fetch you more money in the future for the same amount invested today.Usage of Financial CalculatorSolving TVM problems becomes very easy when you are using financial calculator. There are 2 types ofcalculator permitted in the CFA Exam, one from Hewlett Packard and another from Texas Instruments.We recommend Texas Instrument’s calculators because they are similar to the calculators that you haveused in the past and are easy to master.In TI BA II Plus calculator you have a set of TVM Calculation Key. The following are the function keysavailable for TVM calculationN = Number of compounding periods FV = Future valueI/Y = Interest rate per compounding period PMT = Annuity payments, or constant periodic cash flowPV = Present value CPT = ComputeWe will provide detail in some of the questions on the operation of the calculator.TimelineTimeline is a representation of the cash flows with respect to time. We follow a notation when drawinga timeline Cash Outflow – Any cash outflow is treated as Negative Cash Inflow – Any cash outflow is treated as Positive T=0 T=1 T=2 T=3 T=4 -$500 $200 $200 $200 $200© Knowledge Varsity – 2011 Page 1
    • In the timeline above, we have an initial investment of $500 being done and hence it is an outflow, thatis why we have taken the amount as -$500 in the timeline. Since the initial investment is done now, thetime has been put as T=0 in the timeline. Likewise we are getting a series of return every period fromthe investment, since the amount of $200 is cash inflow to us, we have taken a positive sign for it. Thereare four payments of $200 each received and hence we are displaying four payments in the time line.Drawing a time line is a good idea to solve any TVM problem.LOS 5.a. Interpret interest rates as required rate of return, discount rate or opportunity costThere are three ways in which we can interpret the interest rates:-1. Required rate of return is the return required by the investors to postpone their currentconsumption to a future period.2. Discount rate is the rate used to discount the future cash flows (that is the money to be received inthe future) to the current period. In essence discount rate and required rate are one and the same. Weuse the term “discount rate” to bring a future cash flow to present period and the term “required rate”to compound a current cash flow to future period.3. Opportunity cost is the benefit that investors would have received if they had invested their moneyin some other investment and they are investing in any other product of the same risk they should beable to meet the opportunity cost. This can also be said as the value that investors forego by choosing aparticular investment. This would be clear from the following example:-Suppose an investor has taken loan at the rate of 10% pa from a bank and is offered an opportunity toinvest in a product A which is promising a yield of 12%. Now the investor’s opportunity cost is 12%, hewould invest in any other product having same risk if the return from the other product is more than12%.LOS 5.b. Explain an interest rate as the sum of a real risk-free rate, expected inflation, and premiumsthat compensate investors for distinct types of risk;When there are no uncertainty and no inflation then the interest rate received by the investor is knownas Real Risk Free Rate. The risk free rate is observed in only the government bonds (also known astreasury securities), however these are not real rate as there is some amount of inflation present. Theinterest rate given by the government securities is Nominal risk free rate.Here we bring the concept of inflation premium. When an inflation premium is added to real risk freerate, the resulting rate is the nominal risk free rate. Following equation will make things clearOr we can approximate the equation to;Since we are in an uncertain world there are risks present in any investment, an investor should becompensated for these risks, as a result the required return need to be increased by what is termed asrisk premium. There are three types of risk premium we will discuss in this topic, these are:- Default Risk Premium is the premium added for the likelihood or the probability that the borrower will not meet interest or principal payment on time and is likely to default on the loan. When there is higher uncertainty the premium would be more. Maturity Risk Premium is the premium that one obtains for parting away with his money for longer period of time. Typically as the no of days of an investment increases, the investor is being deprived of his wealth for longer time hence the investor should be compensated by way of higher return. This higher return is attributed as maturity risk premium. Liquidity Risk Premium is the premium provided to the investor for investing in securities having low liquidity. Liquidity here implies how fast the asset can be sold in the market and money received. Typically real estate investments give higher return because the investment is not liquid, it may take months to sell a property and hence liquidity premium should be more. © Knowledge Varsity – 2011
    • So, for an investor the required return on equity should be the approximately the sum of real risk freerate and the various premiums.Required rate = Real Risk Free Rate + Inflation Premium + Default Risk premium + Liquidity Risk premium + Maturity Risk premiumIn Fixed Income study session 15, reading 61, we will come across other risk premiums that an investorshould consider when investing.LOS 5.c. Calculate and interpret the effective annual rate, given the stated annual interest rate and thefrequency of compoundingCompounding is a concept in which interest accumulate over a period of time. A compounding period isa period in which interest is accrued or accumulated; there can be different compounding period in ayear.There are three ways in which we can quote the interest rate in a year Periodic Interest rate – This is the interest rate that is applicable for a period, the period can be 1 month, 1 day, 1 year or anything. For example if the 6 months periodic rate is 5%, it means that we will get return of 5% in 6 month. Stated annual interest rate – This is also known as quoted interest rate. Here the interest rate is stated annually, so for the example of the periodic rate given above, the stated annual rate will be 5% multiplied by 2 or 10%. Effective annual rate (EAR) takes the effect of compounding within a year. For a stated annual rate, the higher the number of compounding the higher is the effective annual rate. EAR can be stated as per the following formula Where, periodic rate = Stated annual rate / Number of compounding period in a yearContinuous compounding is a concept in which there is infinite number of compounding period in ayear. The interest rate thus is called as continuous compounding interest rate.For a continuous compounding rate, the EAR is given asWhere r is the stated annual rateConcept Builder – Effective Annual Rate Computation - Single Computation1. Find Effective Annual rate for a bank deposit, in which the stated interest rate is 8% per annum and the frequency of compounding is semi-annuallyAnswerRemember whenever we are computing effective annual rate, we need to find first the periodic rate.Periodic rate = stated rate / number of periods in a year  Periodic rate = 8%/2 = 4%  EAR = (1.04)2 -1 = .0816 = 8.16%Concept Builder – Effective Annual Rate Computation – Understanding increasing frequency2. For a bank deposit, the stated interest rate is 12% per annum compounded monthly? Find the Effective Annual Rate (EAR), for the following compounding frequency a) Semi-annually b) Quarterly c) Monthly d) Daily e) ContinuouslyAnswer a) When the compounding is semi-annual => EAR = (1.06)2 – 1 = 12.36% b) When the compounding is quarterly => EAR = (1.03)4 – 1 = 12.5509% Calculator TIP: Press 1.03 ; then press yx function key (present above 9), now press 4  You have computed 1.034 ; now you press -; then press 1; then multiply by 100 to get the rate© Knowledge Varsity – 2011 Page 3
    • c) When the compounding is monthly=> EAR = (1.01)12 – 1 = 12.6825% d) When the compounding is daily=> periodic rate = 12%/365 = 0.032877% EAR = (1.000329)365 – 1 = 12.7475% e) The EAR for a continuously compounding rate is – ; please note that r is the stated rate  EAR = – Calculator TIP: Press 0.12; then Press 2nd; then press LN (present on the left side of 7); this will compute ; now subtract 1; after this multiply by 100.Now, the stated rate was 12%, as we increase the number of compounding, you can see that the EAR isincreasing, this is primarily because as the compounding period increases, the more interest is earnedon the interest, so it results in higher effective rate.The highest possible effective interest rate would be achieved when continuous compounding is done.LOS 5.d. Solve time value of money problems when compounding periods are other than annual;When compounding periods are not annually then we must take into consideration that the number ofperiod will be more than 1. Also we need to adjust the periodic rate to reflect this.As a thumb-rule, if there are m periods in a year and there are n years, then we should divide theinterest rate by m and multiply the number of years by m.For example, if we are required to solve for a problem in which the stated rate is 8% pa, with quarterlycompounding and 5 years, we should have the periodic rate as 2% and the number of periods as 20.Concept Builder – Finding Future Value when compounding period is more than annual3. Canara bank is offering interest rate of 9.15% for fixed deposits for 2 years. The compounding is quarterly, how much amount would you receive at the end of the period, if you deposit $1000.AnswerThere are 2 approach to solve problems of this kind 1. Find EAR and then find the FV using the compounding for multiple years Periodic rate = 9.15%/4 = 2.2875% EAR = (1.022875)4 – 1 = 9.4688% FV = PV( 1 + EAR)N So the amount after 2 years will be equal to = 1000 * 1.0946882 = $1,198.3412 2. Solve directly using the calculator Here FV = ? PV = -$1000 (Note –ve sign) ; PMT = 0 ; I/Y = 9.15/4 = 2.2875; N = 2 * 4 = 8  FV = $1,198.34 Calculator TIP: First Press 2nd ; Press FV; this will clear the previous TVM calculation For PV: Press 1000; Press +|- (placed right of decimal); Press PV => PV = -1000 For PMT: Press 0; Press PMT => PMT = 0 For I/Y : Press 2.2875 ; Press I/Y => I/Y = 2.2875 For N: Press 8; Press N => N = 8 Now to find out FV; Press CPT (top left corner); Press FV => You will get FV = 1198.3412Concept Builder – Finding Present Value when compounding period is more than annual4. How much money should you deposit now in your investment account to get $1000 after 3 years. Assuming that the investment account offers a return of 10%, compounded semi-annually?AnswerAs we have seen in the above example, its easier to use the TVM function of the calculator to solvethese kind of problems, we will use the TVM keys to solve it.First we need to identify the values associated with the TVM Keys. © Knowledge Varsity – 2011
    • Here, FV = 1000; PV = ? ; N = 2 * 3 = 6; I/Y = 10/2 = 5; PMT = 0So we need to input these values in the calculator to find out the answerThe answer would be -$746.21, the negative sign implies that an amount of 746 goes out from you(outflow) in the investment account.Calculator TIP: First Press 2nd ; Press FV; this will clear the previous TVM calculationFor FV: Press 1000; Press FV => FV = 1000For PMT: Press 0; Press PMT => PMT =0For I/Y : Press 10; Press ÷ ; Press 2; Press =; Press I/Y ; Press I/Y => I/Y = 5For N: Press 8; Press N=> N = 8Now to find out PV; Press CPT (top left corner); Press PV => You will get PV = -746.21LOS 5.e. calculate and interpret the future value (FV) and present value (PV) of a single sum of money,an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;Future value is the money to be received at a later date. We have the following formula for the FutureValue for a single cash investment. We assume that the interest generated is also invested at the givenrate.Concept Builder – Future Value of a Single Sum5. How much would my term deposit of $1000 will become in 7 years, if the bank is offering me 10% interest rate compounded annually?AnswerThis is similar question like Concept Builder # 3; only thing here is that the compounding period isannual.We can solve this using the calculator as shown in the #3. But whenever you see a single cash flow, itsmuch simpler to use the formula directly than using calculator.FV = PV (1 + I/Y) N => FV = 1000 * (1.1)7 = $1948.7171So, if you invest $1000 now, you will get $1948.71 after 7 years.Present Value is the money that you can assume to be equivalent of the future cash flows. The presentvalue of the expected single future cash flows is given by• FV = future value at time n• PV = present value• I/Y = interest rate per period• N = number of periodsConcept Builder – Present Value of a Single Sum6. How much would be the present value of an investment which promises to return $1000 in 7 years, the required rate of return is 10%?AnswerThis is similar question like Concept Builder # 4; only thing here is that the compounding period isannual.We can solve this using the calculator as shown in concept builder #4. But whenever you see a singlecash flow, its much simpler to use the formula directly than using calculator.PV = FV/ (1 + I/Y) N => PV = 1000 / (1.1)7 = 513.158Calculator TIP:First Find 1.17 => Press 1.1; Press YX; Press 7; => you will get 1.17; which is 1.948717Now, Press button (above YX) => you will get 0.513158Now, press X; press 1000; press =; You will get $513.158 (which is the present value)© Knowledge Varsity – 2011 Page 5
    • Perpetuity is a concept in which same cash flow is received for an indefinite (infinite) number of periodsin the future. Present value of perpetuity is what the infinite cash flow should be worth now.PV of Perpetuity = Cash flow / Required rate of returnOrPV of perpetuity is used in the valuation of preferred stock.Preferred stock is a financial product which gives a constant dividend till perpetuity. Value of preferredstock is given byConcept Builder – Present Value of a Perpetuity7. Tata motor has issued a preferred stock, which pays an annual dividend of $10. If an investor’s required rate of return is 10%, how much the preferred stock would be valued by that investor?AnswerPrice of preferred stock would be the same as the present value of the dividends that it is paying.Since, preferred stock is a perpetuity, we will be calculating the present value from its formula.PV = D/r  PV = $10/0.1 = $100So, the price at which the preferred stock should sell will be equal to $100.Concept Builder – Required rate of return of a perpetuity when PV is given8. A preferred stock is trading at a price of $120 per share. The preferred stock pays annual dividend of $8. What is the required rate of return for the preferred stock? AnswerIn this problem, we know the price and the dividend that it is paying, the required rate of return can becalculated using the formula of PV of preferred stock.PV = D/r => r = D/PV => required rate = $8/$120 = 6.67%Annuities – Sometimes we do not invest the money in lump-sum only. We invest regularly, so this is theconcept of annuity. The regular period can be annually, quarterly, monthly or daily. Present Value of Annuity is the sum that you can receive now in lieu of the future cash flows. Future Value of Annuity is the sum that you will receive in the future for the investments that you are doing periodically.You need not remember the formula of the perpetuity, we should use financial calculator to solveperpetuity problems. You will be explained in the class and the video as to how to apply the concept.There are 2 types of annuities Ordinary Annuity – In ordinary annuity the investment is done at the end of the period. So if you are buying a product on 1st January 2010 and the product is annual pay then first payment will be done on 31st December 2010. If the product matures in 5 years then you will make the last payment on 31st December 2014 and also receive the maturity value on 31 st December 2014. Annuity due – In annuity due the investment is done at the beginning of the period. So if you are buying a product on 1st January 2010 and the product is annual pay then first payment will be done on 1st January 2010. If the product matures in 5 years then you will make the last payment on 1st January 2014 and receive the maturity value on 31st December 2014. So, in both the case we are receiving maturity value on 31st December 2014, but in ordinary annuity the last payment is not earning any interest, hence the Future Value will be lesser than that of the annuity due. © Knowledge Varsity – 2011
    • Concept Builder – Present Value and Future Value of Ordinary Annuity9. I have invested in an ordinary annuity product; the annual payment is $1000 for a total of 10 payments. First installment will start after 1 year from today, how much is the PV of this? How much it will become at the end of 10 years? Interest rate on this product is 8%.AnswerWe would be solving all the problems in which there are intermediate cash flows using the financialcalculator. You should be able to identify the value of any 4 of the 5 functions of TVM. One unknownvalue can be easily found out from calculator.NOTE: For the Ordinary Annuity; we use the END Mode in the calculator.For Computing PV – PV is unknownHere, PMT = -$1000; N = 10; I/Y = 8; FV = 0; PV =?Why is FV zero here? – Please note that when you are calculating PV, assume that FV is zero and viceversa. The PMT and N is taking care of the amount of cash that is being deposited.Please see the timeline for this belowSo, using financial calculator, we will input the values for the various parameters and then compute FV.CPT -> PV will give the present value; PV = $6,710.08For Computing FV – FV is unknownHere, PMT = -$1000; N = 10; I/Y = 8; PV = 0; FV =?CPT -> FV will give the future value; FV = $14,486.56Concept Builder – Using Annuity Concept to Value a Bond10. A bond having face value of $100 is paying annual coupon, the coupon rate is 10%. The bond has 5 years to maturity. If the current market interest rate (required rate) is 6%, then at what price the bond should sell in the market?AnswerFace value of a bond is the amount that we receive at the maturity of the bond. So face value equals thefuture value that is received from an investment in bond.The coupon payment is calculated on the face value. So a bond having coupon rate of 10% will givencoupon equal to 10% * Face value = 10% * $100 = $10, coupon can be regarded as the PMT.The bond pays the first coupon at the end of the period, so a bond valuation can be thought of asOrdinary Annuity.For an annual pay bond, the years to maturity will be the number of payment made. For bonds withsemi-annual coupon, the number of period will be twice the number of years to maturity and couponwould be half of the annual coupon.The market interest rate can be thought of as required rate or I/Y.So, for this bond valuation, we have the following parametersN = 5; I/Y = 6; PV = ? ; PMT = $10; FV = $100© Knowledge Varsity – 2011 Page 7
    • Please note that as an investor, we will receive the coupon and also the face value, and hence as perour sign convention, both of these should be Positive.CPT -> PV = -$116.849The negative sign means, that you will have to pay $116.849 to buy the bond.This is how the bonds are valued and this would be used in Financial Reporting and Fixed IncomeValuation.Concept Builder – Present Value of Ordinary Annuity, when the first payment is at a later date11. My grandfather will give me $1000 for 6 years when I become 18 years old. I have just completed 15 years. My required rate of return is 10% p.a. How much is the present value of my grand pa’s gift?AnswerPlease see the timeline below, I will receive the first payment of $1000, when I will become 18 years.Since there are total of 6 payments, I will get the payment till age of 23 (and not 24)For ordinary annuity or In END Mode, the following point need to be imprint in your mind 1. The PV is one period before the first payment day 2. The FV is on the last payment dayIf we calculate the present value from the calculator, it will give us the present value at T = 17 and notat T = 15, which is our requirement.So, this problem involves 2 steps. First find out the present value using the calculator, it would come atT =17 and then discount this for 2 more periods to arrive at the present value at T=15.1st Step: N = 6; I/Y =10; PV = ? ; PMT = 1000; FV = 0CPT -> PV = -$4,355.26 ; Please note that this is PV at T =172nd Step: PV (at T = 15) = PV (at T =17) / (1+r)2  PV15 = PV17/(1+r)2 => 4355/1.12 => $3,599.38So, the present value of the payment is $3,599.38Concept Builder – Present and Future Value of Annuity due12. I have invested in annuity due product of installment $1000 per year for 10 payments. First installment started today, how much is the PV of this? How much it will become at the end of 10 years? Interest rate on this product is 8%.AnswerCOMPUTATION OF Present Value.See the timeline of this problem below, pay attention here, the first payment is being done at T=0 andthere is no payment done at T=10, the last payment was done at T= 9.There are 2 approach to solve such type of problems © Knowledge Varsity – 2011
    • A. Using our financial calculator in Begin Mode B. Using our financial calculator in End Mode (Ordinary annuity mode) & then making adjustmentFirst using the Begin Mode.Please change the mode in your calculator by performing the following functionsPress 2nd; Press PMT; Press 2nd ; Press ENTER – You will begin seeing BGN in your calculator, it showsthat your calculator is now in begin mode.In BEGIN Mode, the following point need to be imprint in your mind 1. The PV is on the first payment dayNow, with begin mode set, it is a simple problem. You just need to enter the parameters like the wayyou had entered in concept builder #9Here, PMT = -$1000; N = 10; I/Y = 8; FV = 0; PV =?CPT-> PV => PV = $7246.88So, the present value of the investment is $7246.88Using the other approach : The problem can be solved using the end mode also. Since your calculator isin Begin Mode, you need to change it to END Mode. The steps to do this is same as the way wechanged to begin modePress 2nd; Press PMT; Press 2nd ; Press ENTER – You will begin seeing END in your calculator, it showsthat your calculator is now in END mode.When the calculator is in END mode, we had observed that the present value is 1 period before thefirst payment.So here the PV will come at T =-1. See the timeline below.But we need to find out the value at T =0 and not at T = -1, so what we need to do is to find out the PVat T =0 from the value we obtained at T = -1PV (At T = 0) = PV (At T = -1) * (1+r)Or, we can say that PV(Annuity Due) = PV(Ordinary Annuity) * (1 + r )Here, PMT = -$1000; N = 10; I/Y = 8; FV = 0; PV =?CPT-> PV => PV = $6710.08This PV is at T=-1=> PV0 = PV-1 * (1+r) => PV0 = 6710.08 * 1.08 = $7246.88So, the value of PV is same in both the case.COMPUTATION OF Future Value.Here also, we have the same 2 approach to solve the problem.First we will solve using the BEGIN Mode.In BEGIN Mode, the following point need to be imprint in your mind 1. The FV is one period after the last payment dayWe will change the mode from END To BEGIN, using the steps highlighted earlier.Please change the mode in your calculator by performing the following functionsPress 2nd; Press PMT; Press 2nd ; Press ENTER – You will begin seeing BGN in your calculator, it showsthat your calculator is now in begin mode.© Knowledge Varsity – 2011 Page 9
    • In the Begin mode, the timeline looks like below.So, we will enter the value and find out the future value.Here, PMT = -$1000; N = 10; I/Y = 8; FV = ?; PV =0CPT-> FV => FV = $15,645.48Secondly, we will solve the problem using the END ModeSince your calculator is in Begin Mode, you need to change it to END Mode. The steps to do this is sameas the way we changed to begin modePress 2nd; Press PMT; Press 2nd ; Press ENTER – You will begin seeing END in your calculator, it showsthat your calculator is now in END mode.When the calculator is in END mode, we had observed that the Future value is on the last payment.So here the FV will come at T =9. See the timeline below.But, we want the FV at T=10So, we can arrive at the FV at 10, by compounding the FV at 9FV10 = FV9 * (1+r)So, we have a new rule => FV(Annuity Due) = FV(Ordinary Annuity) * (1 + r )So, we will enter the value and find out the future value.Here, PMT = -$1000; N = 10; I/Y = 8; FV = ?; PV =0CPT-> FV => FV9 = $14,486.56We know, FV10 = FV9 * (1+r) => FV10 = $14,486.56 * 1.08 = $15,645.48So, the answer is the same as it was obtained from the Begin Mode.IMPORTANTSo, whenever we are asked to calculate the PV or FV of annuity due, we can solve using the default ENDMode. We advise you to follow this approach, because in exam, if you forget to change the mode thenit would be problematic for you, since most of the questions would be end mode questions.Things to LearnIn END Mode 1. The PV is one period before the first payment day 2. The FV is on the last payment dayIn BEGIN Mode 1. The PV is on the first payment day 2. The FV is one period after the last payment day © Knowledge Varsity – 2011
    • Concept Builder – Future Value at a later date of Annuity due13. I am making an investment of $100 every year, starting from now for a total 3 payments. How much money I will receive at the end of 6 years? The interest rate of the investment is 10%.AnswerFirst you need to make timeline for this particular problem, which is given belowWe would be using our calculator in END MODEIn the END Mode, if we are computing the FV, it would come at T=2Now, we should compound the value received at T=2 to get the FV at T=6. Since there are 4 periods inbetween, we can writeFV6 = FV2 * (1+r)4Plugging in the values in the financial calculatorHere, PMT = -$100; N = 3; I/Y =10; PV =0; FV = ?CPT-> FV => FV2 = $331FV6 = $331 * (1.1)4 = $484.617So, I will receive $484.6 at the end of 6 years.Concept Builder – Present and Future value of cash flows when the cash flow is not same14. Find the present value and the future value for the following cash flows. Required rate of return is 10%.AnswerSince, the TVM functions assume that the PMT remains the same, we can’t use TVM here. We shoulduse the Cash Flow function here.You will find, a button named CF in the 2nd row, 2nd Column.Whenever we are calculating using CF, we should first clear the memory.To Clear Memory – Press CF; Press 2nd; Press CE|CYou will observe CF0 on the screen, it is asking for the Cash flow at T=0For CF0: Press 1000; Press +|- ; Press ENTER; Press ↓You See CF1: Press 2000; Press +|- ; Press ENTER; Press ↓You see F01 => By default its value is 1 in the calculator. This is the concept of frequency, it is asking youhow many times -2000 is coming consecutively in the problem. Since -2000 is coming consecutivelyonly once, we will leave F01 at 1 only and press the down arrow ↓ to move to C02.You see C02: Press 3000; Press +|- ; Press ENTER; Press ↓↓ (Yes 2 times, as the frequency of -3000 isalso 1)You see C03: Press 2000; Press +|- ; Press ENTER; Press ↓Now, the data is entered, we will calculate the PV.Note that, for CF function, there is a button to calculate NPV.NPV is Net Present Value and is equal to PV of Inflows – Outflows , but for this NPV will be same as thePV.© Knowledge Varsity – 2011 Page 11
    • Press NPV; You will see I in the screen, Press 10; Press ENTER; Press ↓; You will see NPV on the screen;Press CPTNPV = -6800.15For finding out the future value, you can assume that you are investing 6,800 for a period of 4 years.FV = 6800 * 1.14 => FV = $9,956.1LOS 5.f. Draw a time line and solve time value of money applications (for example, mortgages andsavings for college tuition or retirement).As mentioned, we can solve various type of problems by employing time value of money calculation andit is better to draw a timeline before solving the problem. This LOS covers the application of TVMconcepts in real life.We will cover this LOS through examples and understand the various applications.Concept Builder – Calculation of I/Y15. You have deposited $100 in the account today, after 7 years, the amount would become $200, what is the stated annual rate if the compounding is annual?AnswerHere we need to calculate the rate or I/YThe parameters of TVM are :FV = $200; PV = -$100; N = 7; PMT = 0; I/Y = ?CPT -> I/Y => I/Y = 10.4%Concept Builder – Calculation of PMT16. You are required to deposit a fixed amount every year in your investment account starting from the end of this year. If the stated rate is 10% p.a. and you are depositing for a total of 10 years, you will receive an amount of $100,000. How much amount you should deposit every year?AnswerHere we need to calculate the amount deposited every year or PMTNote that there is no money in the account today, it is a case of ordinary annuity since the money isdeposited at the end of the period.The parameters of TVM are :FV = $100,000; PV = $0; N = 10; I/Y =10; PMT = ?CPT -> PMT => PMT = -$6,274.5So, you need to deposit 6,274.5 every year to get $100,000 at the end of 10 years.Concept Builder – Calculation of Number of Periods17. You are shown an investment plan that will require depositing every year $10,000 starting from the end of this year; it is promised that at the end you will get a sum of $115,000. If the interest rate offered by the investment plan is 10.15%, then find out for how many years you will have to deposit the money in the plan?AnswerHere we need to calculate NThe parameters of TVM are :FV = $115,000; PV = $0; PMT = -$10,000; I/Y =10; N= ?CPT -> N => N = 8.03  So, you need to deposit the amount for 8 years. © Knowledge Varsity – 2011
    • Concept Builder – Compounded Annual Growth Rate18. Tata Steel’s EPS was $5 in 2002, In year 2008 the EPS was $14. Find the compounded annual growth rate (CAGR) of the EPS.AnswerCAGR is the rate at which the EPS has grown.Please see the timeline belowEasily we can solve this with the financial calculatorSo, the number of periods from 2002 to 2008 is equal to 6, therefore N = 6The parameters of TVM are :FV = -$14; PV = $5; PMT = 0; N=6; I/Y =?CPT -> I/Y => I/Y = 18.72%So, the EPS has grown at an average rate of 18.72% for the 6 years.Concept Builder – Loan Calculation19. I took a housing of $200,000 for 15 years; the rate of the loan is 10%. Calculate A. How much monthly payment I am doing. B. How much is the principal payment done in 1st month? C. How much is the principal remaining after 60 installments. AnswerThis is another set of problem, where you will find financial calculator handy.Please note that housing loan is an amortizing loan, where both principal and interest is paid in themonthly payment that is being done. At the end of the loan term, there is no money that is required tobe paid to the bank, so the future value is 0.For Part A – It is just asking for the EMI that I am paying and hence we need to find the PMTThe parameters of TVM are :FV = 0; PV = $200,000; I/Y =10/12 = 0.833; N=15 *12 = 180; PMT =?CPT ->PMT => PMT = -2,149.21So, every month, you should pay $2,149.2 towards your housing loanFor Part B –So, for the 1st month, we can calculate the interest that is accrued on $200,000 for 1 monthusing the simple interest formula  Interest for the 1st month = PRT/100 => ($200,000 * 10 * 1 )/(12 *100) = $1,666.67  Out of the EMI of $2,149.21 ; $1667.67 is towards the interest charge  The principal payment = $2,149.21 - $1667.67 = $482.54For Part C – This is a difficult question if you think conventionally, here you need to think Out Of Box, toanswer it in a simple manner. Think, if everything was as planned, you were paying the EMI and now 60months are left. Can I say, that if you approach the bank after 60 months for a loan, they should giveyou an amount which will be paid off by the end of 120 months, So we can say that whatever theprincipal is remaining, should be equal to the loan amount that you would be given after 60 months.To solve, this now your N = 120FV = 0; I/Y =10/12 = 0.833; N=120; PMT =-2,149.21 ; PV = ?CPT-> PV =>PV = $162,633.24Since you have financial, these things are much easier to solve. Your calculator has AMORT Function,which can be accessed by pressing 2nd and then Pressing PV.Once you do that, you will find P1 on the screen.© Knowledge Varsity – 2011 Page 13
    • P1 is the point from which we want to calculate the principal payment/interest payment etc.If you press ↓; you will find P2.P2 is the point till which we want to calculate the principal payment/interest payment etc.For Part B – we can solve using P1 = 1 and P2 = 1 , since we are interest for the principal payment forthe 1st month only.Calculator TIP: Press 2nd; Press PV; You will see P1; Press 1; Press ENTER; Press ↓; You will see P2;Press 1; Press ENTER;Press ↓; You will See BAL => this is the Principal Remaining after P2Press ↓; You will See PRN => this is the Principal paid between P1 and P2 => $482.54Press ↓; You will See INT => this is the Interest paid between P1 and P2For PART C – Using the financial calculator  P1 = 1; P2 = 60 (as we are interested in finding the principal remaining after 60 months)Calculator TIP: Press 2nd; Press PV; You will see P1; Press 1; Press ENTER; Press ↓; You will see P2;Press 60; Press ENTER;Press ↓; You will See BAL => this is the Principal Remaining after P2 = $162,633Press ↓; You will See PRN => this is the Principal paid between P1 and P2 => $37,366.76Press ↓; You will See INT => this is the Interest paid between P1 and P2 => $91,585.85So, in first 60 months, I have paid $91,585.85 as the interest and $37,366.76 as the principal.Concept Builder – Retirement Calculation20. An investor, plan to retire at the age of 60. He expects to live till the age of 90. He is currently aged 25. He invests the first payment in the account today and invest for 35 years (that is total of 35 investment done). The retirement account earns 12% per annum. Assume that he would like to withdraw $30,000 per year starting from the point when he turns 60 for 30 years, find out the amount he should deposit in his retirement account every year? Here assume, that the investor doesn’t leave any money for his heirs. AnswerThis is a complex problem involving 2 series of cash flows. It’s good to draw a timeline to solve theproblemIn these type of problems, we need to come up with a common point.If we take the middle point as common point, that will be the best.So, we can find out the PV of the cash inflows at T = 59; Using that, we can find out the PV at T=60; andthen we can compute the PMT for the series of cash outflows.Lets do step by step calculationStep 1: Find out the PV at T = 59Keep the calculator in the END mode, the 2nd series of cash flows from T=60 to T=89 looks like ordinaryannuity and the PV will come one period before the first cash flow.FV = 0; I/Y =12; N=30; PMT =$30,000; PV = ?CPT-> PV => PV = $241,655.19Step 2: Find out the PMT. Here you have to note that the FV of the investment would be equal to the © Knowledge Varsity – 2011
    • present value that we have found out.FV = PV59 = $241,655.19; PV = $0; N = 35; I/Y =12; PMT = ?CPT -> PMT => PMT = -$559.82So, the investor need to deposit $559.82 every year for 35 years, in order to get $30,000 every yearwhen he retires.Experience, the power of Compounding, the investor is depositing only $560 and is able to withdraw$30,000 per year.If the investor, delays his investment by 5 years, that is the first payment , he does at the age of 30, thenhe would be required to deposit $1,000 to experience the same inflow during retirement.So, it’s always advisable to start the investment early.© Knowledge Varsity – 2011 Page 15
    • Concept Notes For Reading 6 - Discounted Cash Flow ApplicationsReading SummaryThis topic is another one which is relevant across the CFA Curriculum. In recent times, CFA Institute hasstarted asking questions on the concepts in this chapter and it might happen that you may be gettingquestions on some of these topics. The calculation of cash flow using financial calculator will be coveredin this topic, however the same would be used in Corporate Finance and Equity Valuation. You shouldbe thorough with the Cash Flow function of your calculator. The return concepts of Time Weighted andMoney Weighted Returns may be intimidating for some to start with, but these are essential conceptsfor any person in the investment industry. Lastly you are expected to understand some other returnmeasures which are kind of industry convention and are popular because of their simplicity. You shouldsolve all the questions that are given in this book as well as the problems that are there in the institutebook then only a clear-cut understanding of this chapter will emerge.LOS 6.a. Calculate and interpret the net present value (NPV) and the internal rate of return (IRR) of aninvestmentCompanies take up projects as part of their business activity, typically a project requires upfrontinvestment (a cash outflow) and then subsequently companies benefit in the form of cash inflow. For acompany to undertake a project, not only the amount of return it is getting should be more than theinvestment being done but also the company should be able to earn a decent rate of return on theinvestment. To make investment decisions companies evaluate projects on the basis of NPV and IRR.NPV or Net Present Value of an investment is its present value of the expected cash inflows minus thepresent value of its expected cash outflows.IRR or Internal Rate of Return is the discount rate which would make the NPV of a project equal toZero.NPV Calculation Process – NPV of any project involves the following steps 1. Identify all the cash inflows and outflows 2. Determine the discount rate that should be used to discount these cash flows. In this section we need not calculate the discount rate ourselves, when we cover the corporate finance section, we will understand how to compute the rate. 3. Using the discount rate computed in step 2, find out the present value of cash outflows and cash inflows. We as per our convention treat outflows as negative (hence they decrease NPV) and treat inflow as positive (hence they increase NPV) 4. NPV is the summation of all the present value of the cash flows computed above. Please note that since we are abiding by the sign convention, hence we need to just do the addition. NPV Formula:CFt : Cash flow in time period tN : the number of periods for which investment is mader : the discount rate used for finding out the present value © Knowledge Varsity – 2011
    • Concept Builder – NPV and IRR Computation1. Compute the NPV of a project which requires initial investment of $10 million. The cash flows are expected to be $6 million in year 1, $5 million in year 2, $4 million in year 3 and $3 million in year 4. The discount rate for the investment is 15%.AnswerNPV:The NPV of the project can be calculated by discounting the cash flows.  NPV = -$10 + $5.217 + $3.780 + $2.630 + $1.715  NPV = $3.343 millionIt is better to compute the NPV using financial calculator, the usage of TI BA II Plus will be discussed inthe classroom and also in the Pre-class content.IRR: Computation of IRR is difficult manually and should be done using financial calculator only. There isIRR function in the calculator which will be able to derive the IRR, please note that when discounted atthe IRR rate, the NPV will be equal to Zero.The value of IRR = 32.9783%LOS 6.b. Contrast the NPV rule to the IRR rule, and identify problems associated with the IRR RuleIn any investment the idea is to generate more wealth than what you have invested or produce a returnwhich is more than the cost of capital or required rate of return.Typically the projects can be classified under the following1. Independent Projects – When acceptance or rejection of a project does not affect the acceptance or rejection of other projects then the projects are considered as independent projects. As an example suppose a firm is considering investment in a new printer and investment in a vehicle, these can be considered as two independent projects if the company has fund to buy both and the company is evaluating these two investments as per their cash flow.2. Mutually Exclusive Projects – Typically companies do not have high amount of capital and most of the times it becomes necessary to take up only one project out of two, due to capital constraints. In such case we say that the projects are mutually exclusive. For example if the firm has money either to install printer or to buy the vehicle but not for both then it becomes important for the firm to identify the project which will be more beneficial.To evaluate any investment decision we have the following two rules1. NPV Decision Rule: If a project is yielding positive value to the firm then the project is a good project and it should be accepted. We can summarize this decision rule a. Accept projects which have positive NPV as these projects are adding to the shareholder wealth b. Reject project having negative NPV as these projects are destroying the shareholder wealth c. In case of mutually exclusive projects, select the one which has higher NPV. Mutually exclusive projects are those projects in which only one can be selected, naturally the higher NPV project is increasing the shareholder wealth more and hence we should accept that project2. IRR Decision Rule: The decision rule based on IRR helps in evaluation of how much return is being generated. a. Accept projects whose IRR is more than the required rate of return b. Reject those whose IRR is less than the required rate c. Unlike the NPV decision rule here we can’t say directly that in case of mutually exclusive project we should accept project having higher IRR. This is because of certain problems with the IRR method© Knowledge Varsity – 2011 Page 17
    • Problems Associated with the IRR RuleBefore we take up the problems with the IRR rule, first we need to understand that NPV and IRR wouldgive the same result for the independent projects. This is because If NPV > 0 then IRR should be more than the required rate of return If NPV < 0 then IRR should be less than the required rate of returnHowever for mutually exclusive projects (let’s say projects A &B) there can be conflicting results, that isthe NPV rule may result in selection of project A whereas IRR may result in selection of project B.This conflict happens because of the following problems1. Reinvestment–The calculation of IRR assumes that all the cash flows are reinvested at the IRR rate. For example if a project has IRR of 20%, then it is assumed that the cash inflows which are expected would be re-invested at 20% rate. Now many times it might happen that the firm is not able to find lucrative investments and hence it might not be able to re-invest those cash flows at IRR rate, let’s say that the firm is reinvesting those cash flows at a rate of 15% then the actual return from the investment would be lower than the 20% which was estimated earlier. We will be covering this concept in the Fixed Income in detail.2. Timing – Another reason for conflict is the timing of the cash flows, one project may produce higher cash flows during the early periods and another project may have higher cash flows in the later part of the project.3. Initial Investment –Conflicting results can be observed in cases where one project requires smaller investment and produce smaller amount of cash flows compared to another project which require more investment and produces higher cash flows.You can remember these through the acronym RTI (Right to Information)Concept Builder –NPV and IRR Rule Conflict2.There are two mutually exclusive projects which have the following cash flowsProject A: CF0 = -$10,000; CF1 = $20,000Project B: CF0 = -$40,000; CF1 = $60,000The required rate of return is 10%. Which project should be selected and why?AnswerCompute the NPV and the IRR of each of the project using the financial calculator Project NPV IRR A $8,181 100% B $14,545 50%If these were independent projects then both the project should have been selected because the NPV isgreater than 0 or IRR is more than the required rate of return.However since the projects are mutually exclusive, we should select one of these projects. In case ofmutually exclusive projects we should compare the NPV and the project having higher NPV should beselected, hence project B should be selected.LOS 6.c. Define, calculate, and interpret a holding period return (total return)Holding Period Return (HPR) is the return that is generated over the holding period. The holding periodcan be any period; it can be a minute, several hours, many days or years, please do not confuse holdingperiod as yearly period. Here we are interested in determining the total return which means that thereturn does not only include the change in the investment value but also include any cash flow that isgenerated by the investment within the holding period. These cash flows which are generated arecalled as intermediate cash flows. Stock investment has intermediate cash flow in the form of dividendand bond investment results in coupon payment. © Knowledge Varsity – 2011
    • Where for an equity investmentP1 is the price at the end of the periodP0 is the price at the end of the periodD is the dividend receivedConcept Builder –Holding Period Return3. An investor purchased a share for $20 and sold it at $22 after 6 months, also he received a dividendof $0.5 just before he sold the share. Find his holding period return?AnswerEnding Value of investment = $22Beginning Value of investment = $20Dividend received = $0.5  HPR = (22- 20 + 0.5) / 20 = 2.5/20  HPR = 12.5%Concept Builder –Holding Period Return4. A t bill priced at $98 with face value of $100 and 180 days until maturity. Find the holding periodreturn for the T bill?AnswerEnding Value of investment = $100Beginning Value of investment = $98  HPR = (100- 98) / 98 = 2/98  HPR = 2.0408%LOS 6.d. Calculate, interpret, and distinguish between the money-weighted and time-weighted rates ofreturn of a portfolio, and appraise the performance of portfolios based on these measuresMoney Weighted Return(MWR) – It is actually the internal rate of return for an investment. All thedeposits in the investment account are treated as cash inflow and all the withdrawals are treated ascash outflow. After determining the inflow and outflow, IRR is calculated. This calculation is similar tothe calculation that we had done in Example 1.Time Weighted Return(TWR)– It is actually the Geometric mean return of an investment. Forcalculation of TWR we need to divide the investment horizon into several periods. The period for thereturn calculation is computed whenever there is a major inflow or outflow from the portfolio. Thegeometric mean of all the periodic returns is then calculated to find the time weighted return.If there are N holding periods, the TWR can be calculated as follows (1 + TWR)N = (1+ HPR1)*(1+ HPR2)*(1+ HPR3)* …….. (1+ HPRN)Where HPRi is the Holding Period Return in the ith periodConcept Builder –Money Weighted Return and Time Weighted Return5. An investor purchased one share of Goldman Sachs (GS) at $100 at T = 0, he again purchased onemore share of GS at $120 at T =1. He received total dividend of $2 at T= 1 and a dividend of $3 per shareat T =2, the investor sold the shares at T=2 for a total consideration of $260. Find money weighted andtime weighted return?AnswerImportant assumption : We assume that the dividend is paid just before the period ends.Money Weighted Return Calculation:Step 1 : We should identify the cash flows that are happening at the various intervals. As per our signconvention, money going out from investor should be treated as negative and money coming to an© Knowledge Varsity – 2011 Page 19
    • investor should be treated as positive.At T =0 : Purchase of 1st share was done => Cash outflow of $100 => CF0 = -$100At T =1 : Purchase of 2nd share was done => Cash outflow of $120 Dividend is received => Cash Inflow of $2 Sum of these two results in cash outflow of $118 => CF1 = -$118At T =2 : 2 shares are sold => Cash inflow of $260 Dividend is received => Cash Inflow of $3 * 2 = $6 Sum of these two results in cash inflow of $266 => CF1 = $266Step 2: Plug the cash flow values in the financial calculator to compute the IRR (because MWR is nothingbut IRR)IRR = 14.438% or Money Weighted Return is 14.438%Time Weighted return Calculation:Step 1: We should first find out the number of periods, here significant cash in-flow is happening at T=1and cash outflow is happening at T=2. So we have one period from T=0 to T=1 and another period fromT=1 to T=2Step 2: Compute the portfolio value just before the significant cash inflow or outflow.For Period 1: Beginning Value = $100; Ending Value = $120; Dividend Received = $2For Period 2: Beginning Value = $120 (for the 1st share) + $120 (amount that is invested) = $240 Ending Value = $260; Dividend Received = $6 (for 2 shares)Note: Many candidates have confusion regarding why the beginning value of the portfolio should be$240, why not it should be $220, the point here is that at the beginning of period 2, the first share couldbe sold for $120 and there is an investment of $120 more, hence the portfolio value is $240. Also weshould consider the dividend in this period for both the shares.Step 3: Find out the holding period return of the periodsFor Period 1: HPR = (120 – 100 + 2)/100 = 22%For Period 2: HPR = (260 – 240 + 6)/240 = 10.83%Step 4: Find out the geometric mean return or TWR from the holding period returns (1 + TWR)2 = (1+ HPR1)*(1+ HPR2) => (1 + TWR)2 = (1.22) * (1.1083) = 1.3522 => (1 + TWR) = 1.1628 => TWR = 0.1628 Or Time Weighted return is 16.28%Is it always the cash that Time Weighted return is less than money weighted return?Please note that this is not the case and we got into this situation for this problem only. The nextexample will clear this concept.Concept Builder –Money Weighted Return and Time Weighted Return6. An investor purchased one share of Goldman Sachs (GS) at $100 at T = 0, he again purchased onemore share of GS at $120 at T =1. He received total dividend of $2 at T= 1 and a dividend of $3 per shareat T =2, the investor sold the shares at T=2 for a total consideration of $360. Find money weighted andtime weighted return?AnswerMoney Weighted Return Calculation:Step 1 : Identify the cash flows that are happening at the various intervals.At T =0 : Purchase of 1st share was done => Cash outflow of $100 => CF0 = -$100At T =1 : Purchase of 2nd share was done => Cash outflow of $120 Dividend is received => Cash Inflow of $2 Sum of these two results in cash outflow of $118 => CF1 = -$118 © Knowledge Varsity – 2011
    • At T =2 : 2 shares are sold => Cash inflow of $360 Dividend is received => Cash Inflow of $3 * 2 = $6 Sum of these two results in cash inflow of $366 => CF1 = $366Step 2: Plug the cash flow values in the financial calculator to compute the IRR (because MWR is nothingbut IRR)IRR = 41.2% or Money Weighted Return is 41.2%Time Weighted return Calculation:Step 1: Here significant cash in-flow is happening at T=1 and cash outflow is happening at T=2. So wehave one period from T=0 to T=1 and another period from T=1 to T=2Step 2: Compute the portfolio value just before the significant cash inflow or outflow.For Period 1: Beginning Value = $100; Ending Value = $120; Dividend Received = $2For Period 2: Beginning Value = $120 (for the 1st share) + $120 (amount that is invested) = $240 Ending Value = $360; Dividend Received = $6 (for 2 shares)Step 3: Find out the holding period return of the periodsFor Period 1: HPR = (120 – 100 + 2)/100 = 22%For Period 2: HPR = (360 – 240 + 6)/240 = 52.5%Step 4: Find out the geometric mean return or TWR from the holding period returns (1 + TWR)2 = (1+ HPR1)*(1+ HPR2) => (1 + TWR)2 = (1.22) * (1.525) = 1.8605 => (1 + TWR) = 1.3640 => TWR = 0.364 Or Time Weighted return is 36.4%So for this example the MWR is more than that of the TWR. So which return is more or less depends onthe amount and the timing of the cash flows.What can we observe from the above 2 examples, either TWR or MWR can be more or less, it dependson how the investment is done and the periodic return that is generated.Practical Understanding of the differences in the two return measures1. Money Weighted Return gives weightage to the money that is deposited in the account. If the portfolio performance is good when more money is in the portfolio then the overall return increases, where as if the portfolio performance is bad with more money the overall return decreases.2. In time weighted return, the return is not affected by the timing of the cash flows and the amount of money invested, it is purely depended on the periodic return generated. For purpose of simplicity we will assume that the holding period will be the same. In reality, for the calculation of mutual fund returns, we define the holding period as 1 day, because typically everyday there are cash inflows and outflows.You need to remember the following table because many problems will not ask you to calculate theMoney weighted or Time weighted return but an understanding of the impact on the return measurewhen one of these is choses. Amount of Money compared to Return compared to Interpretation previous period previous period More Money More Return MWR > TWR More Money Less Return MWR < TWR Less Money More Return MWR < TWR Less Money Less Return MWR > TWR© Knowledge Varsity – 2011 Page 21
    • Important: TWR is the preferred method for return calculation in most of the investments. Only in onecase we prefer the MWR over the time weighted return and that is when the portfolio manager has thecomplete control on the cash flow of the portfolio that is the portfolio manager decides on when toincrease or when to decrease the size of the portfolio.LOS 6.e. Calculate and interpret the bank discount yield, holding period yield, effective annual yield, andmoney market yield for a U.S. Treasury billMoney market instruments are highly liquid, low risk instruments having maturity of less than 1 year.These instruments are of two types:-1. Discounted instruments – These instruments do not give any coupon or interest during the holding period. These are priced at less than the face value. Face value is the amount of money that the bondholder will receive when the bond matures. The difference between the face value and the selling price is called as the bond discount. The investor will earn the discount value if he holds the instrument till maturity. The best example of this type of instrument is a treasury bill. For example a T bill having face value of $100 at maturing after 90 days will be selling now at a price which is less than $100, let’s say the price is $99. Then $100 minus $99 or $1 is the discount.2. Interest Bearing Instruments - These instruments will produce intermediate cash flows in the form of interest. You can think of a money market (or income) mutual fund as an example.Yield Measures of Treasury billThere are various yield measures of Treasury Bills; here we will go through the formula andunderstanding of each of the yield measures. You might fear that how you will remember so manyformulas, in reality there are not so many formulas; the idea is to understand the bank discount yieldreally well.1. Bank Discount Yield – Here the yield is calculated on the basis of the face value and not on the actual investment done. The formula for bank discount yield is Where; rBD : Bank Discount yield D: Discount of the T Bill F: Face value of the T bill t : The time to maturity Please note that Bank discount yield is an annualized yield, however it is not a good measure. Think about Bank discount yield as BaD yield, the problems with it are:- a. It is calculated on face value and not the actual price b. The number of days in one year is treated as 360 days instead of 365 or 366 c. The calculation is based on simple interest concept rather than the compound interest concept2. Money market yield – This is another yield measure, here we are correcting one of the problems of the bank discount yield, which is the return is calculated on the actual investment and not on the face value. The formula for money market yield is given as or So money market yield is also not a good measure, but it is better than bank discount yield. The problems are:- a. The number of days in one year is treated as 360 days instead of 365 or 366 b. The calculation is based on simple interest concept rather than the compound interest concept3. Holding Period Yield: We have covered this concept in earlier LOS, here the yield is calculated on the actual investment but only for the period the investment is made and it is not annualized. For a T bill, the HPY is given as; © Knowledge Varsity – 2011
    • 4. Effective Annual Yield: This is the best yield measure for the Treasury bill. It corrects the problem of money market yield by considering a. Compounding b. Considering one year as 365 days The formula is given by: -1 EAY is the annualized yield of the holding period and can be used to compare across various investmentsBased on the above, we can say that for any periodEffective Annual Yield > Money Market Yield > Bank Discount YieldConcept Builder –Various Yield Measures6. A T bill priced at $98 with face value of $100 and 180 days until maturity.a. Calculate the bank discount yieldb. Find the Holding period yieldc. Find the effective annual yieldd. Find the money market yieldAnswerFirst calculate the discount. Discount = Face value – Price => discount = $100 - $98 = $2 a. Bank Discount Yield (The BaD Yield) = (D/F) * (360/t) = (2/100) * (360/180) = 4% b. Holding Period Yield = D/P = 2/98 = 2.0408% c. Effective Annual Yield = (1+ HPY)365/t – 1 => (1.020408)365/180 -1 => 1.0204082.0278 -1 => EAY = 4.1817% d. Money Market Yield = (D/P) * (360/t) = (2/98) * (360/180) = 4.0816%LOS 6.f. Convert among holding period yields, money market yields, effective annual yields, and bondequivalent yields.There is relationship among all the yield measures, if you have understood the underlying concept ofeach of the yield then you need not remember any formula.1. Conversion from bank discount yield to Money Market yield – There is a very complex formula forthe conversion; we will not follow the formula, but the logic for this conversion.During the conversion we will assume that the face value of the T bill is $100. The process is bestdescribed using the below exampleConcept Builder –Conversion from Bank Discount Yield to Money Market Yield7. A T bill having 120 days to maturity is yielding 3.5% on bank discount basis, how much will be itsmoney market yield?AnswerStep 1: Assume the face value to be $100, we would like to find out the discount at which the T bill istrading3.5 % = (D/100) * (360/120) => 3.5% = (D/100) * 3=> D = 3.5% * 100 / 3 = $1.1667Step 2: Once discount is calculated, Find out the price at which the bond is tradingPrice = F – D => $100 - $1.667 = $98.8333Step 3: Since now we know the discount, the price and the days to maturity, calculate MM yieldMoney Market yield = (D/P) * (360/t) => (1.667/98.333) * (360/120)=>Money Market yield = 3.5413%2. Conversion from holding period yield to Money Market yield to Effective Annual Yield –Please note that the Money market yield is annualized version of HPY based on simple interest and 360 days in a year.© Knowledge Varsity – 2011 Page 23
    • EAY is annualized version of HPY based on compound interest and 365 days in a year. -1Concept Builder –Conversion from Money Market Yield to HPY and EAY8. A T bill having 120 days to maturity has money market yield of 3.5413%, calculate the HPY and EAY?AnswerHolding Period Yield = 3.5413 * 120/360 = 1.1804%EAY = (1.011804)365/120 – 1 = 3.6339%Bond Equivalent Yield (BEY) – In US, most of the bonds pay coupon semi-annually, based on the priceof the bond, the market participants calculate the semi-annual yield of the bond. The bond equivalentyield is based on the simple interest concept of the semi-annual yield, therefore3. Conversion from HPY to Bond Equivalent Yield: For this conversion we first need to find the effectivesemi-annual yield. Please note that when we are referring to semi-annual yield; you should convert thesemi-annual into 6 months and the holding period should also be in month. Please note that here weneed to assume compounding for the 6 months rather than simple interest. You can take 30 days as onemonth period over here. We can see an example of this to understand the concept better.Concept Builder –Conversion from HPY to BEY9. A T bill having 120 days to maturity has HPYof 2%, calculate BEY?AnswerFirst Step: Find the Holding period in months by taking 30 days as 1 month=> holding period = 120/30 = 4 monthsSecond Step: Find the effective semi-annual yield=>effective semi-annual yield = (1.02)1.5 -1 = 3.01495%BEY = 2 * 3.01495% = 6.0299%4. Conversion from EAY to Bond Equivalent Yield: Here the idea is to convert the EAY to effective semi-annual yield and then find the BEY. - 1 => BEY = 2 *{ – 1}Concept Builder –Conversion from EAY to BEY10. A T bill having 120 days to maturity has EAY of 5%, calculate the BEY?AnswerFirst Step: Find the semi-annual yield=> Semi-annual yield = (1.05)0.5 – 1 = 2.4695%Second Step: Multiply the semi-annual yield by 2 to get the BEY=> BEY = 2 * 2.4695% = 4.939% © Knowledge Varsity – 2011
    • Concept Notes For Reading #7-Statistical Concepts and Market ReturnsReading SummaryThis topic introduces the concept of descriptive statistics. The focus in this topic is on Measures ofcentral tendency and dispersion. You need to understand and calculate these measures. Frominvestment point of view, important measure of central tendency is mean and standard deviation is theimportant dispersion measure. Percentiles are definitely a testable topic in the examination. Acandidate should understand the deviation of a data set from what is known as Normal distribution.Some may find Chebysev’s inequality to be a difficult concept, but an understanding of it would makelife easier in Readings 10 and 11. Problems related to Skewness and Kurtosis are favorites of theInstitute. Sharpe Ratio is a concept which is found in portfolio management also.LOS 7.a.Differentiate between descriptive statistics and inferential statistics, between apopulation and a sample, and among the types of measurement scales;Statistics can be used to refer to numerical data (as an example the net profit of a firm over the last 10years). It can also be used to refer to the methods of collecting, classifying, interpreting or analyzing thedata. Statistics are widely used in finance and its applications should be well understood by the financeprofessionals.Following are the two categories of statistics:- Descriptive Statistics –This is the branch of statistics where we describe the important aspects or characteristics of the data set that have been collected. The description can consists of o Measure of Central Tendency -Where the data is centered, like mean o Measure of dispersion - How the data is spread out, like range In this reading we will focus on the descriptive statistics, we would understand the concepts through which we can describe huge amount of data sets, thus trying to make sense of the data. Inferential Statistics –This branch of statistics deals with prediction or inference of the large set of data (population) from small set of data (a sample). As an example suppose we want to know the average height of the men aged between 20 and 25, then it would be impossible to measure the height of each and every man. What we can do is to take a sample of the men across the various cities/town/villages and then find out the average. Using this simple average we can try to estimate the average height of the men population. Please note that since we are trying to predict, there will always be an error associated with the prediction. We will focus on this part of statistics in Reading 10 and Reading 11.We have used the word population and sample in the paragraphs above; let’s see what is meant bythese terms Population –A population can be thought of as a universal set. It consists of the entire data set or has each and every member of the specified group. As an example, all the men of India aged between 20 and 25 would form a population. An important point to note is that the population doesn’t have anything to do with the human. We can have a population of living, non-living or even virtual objects. Sample – A sample is a subset of the population or in other words we can say that a sample contains some of the member of the population. It is impractical to conduct analysis on a population and hence typically we select a sample from the population. The best is to get a random sample from the population but sometimes we may be interested in something more specific. We will cover Sampling later in Reading 10.© Knowledge Varsity – 2011 Page 25
    • Measurement ScalesWhen dealing with data, it becomes important to organize the data. The ways in which data can beorganized are listed below Nominal (Grouping): This is the weakest level of measurement scale. Here we only categorize or group the data. However in no way we would be able to find out which group of data is better as there is no ranking done. Example of Nominal Scale would be categorizing the stocks in Large Cap; Small Cap etc.• Ordinal(Grouping and Ranking): This is better measurement than the nominal scale. In this scale the data is grouped as well as ranked. In Ordinal scale the data is ordered data (or ranked data) which means assigning some rating or value to a group with a view that one group is better than the other based on the value. However it provides no information on differences in performance between groups. Example: Categorizing the stocks based on performance; like assigning 5 star rating to the top 20% stocks. Another example is of bond rating, where the bonds are rated as AAA, AA, A etc., a bond with a rating of AAA will be safer than that of a bond with rating of AA, but we don’t have information on how much better the AAA bond is as compared to AA bond.• Interval(Grouping, Ranking, includes exact distance): This scale is one step further, here we work with ranked data but we make sure that the difference between scale values are equal. Here the values can be added or subtracted. However a value of zero here does not mean the absence of what is being measured, in simpler words this scale lacks “True Zero”. We will try to explain this concept using the below two examples Example 1:Temperature is measured in Celsius scale which is an interval scale. Water freezes at a temperature of 0 but Zero 0Cdoes not mean absence of temperature because we have temperature value in negative also. In reality the true absence of temperature is observed at -273 0C which is also Zero Kelvin. Now in the absence of true zero, we can’t say that 50 0C is twice that of 25 0C, so a meaningful comparison is not possible in this scale. Example 2:In GMAT examination, it is not possible for a candidate to score zero marks. The candidates are assigned marks in multiples of 10. So a person scoring 800 in GMAT has not score two times that person who has scored 400 in GMAT.• Ratio: Ratio scale is the strongest measurement scale, it has all the qualities of interval scale plus it also provides a true zero. Since we have a true zero, meaningful comparison can be made among values. This is the most important scale, here the datasets can be evaluated on the basis of ratio. Example 1: The amount of money a person has can be considered as Ratio scale, a person having $10,000 has twice as much money as a person having $5,000. Example 2: In a typical traditional examination, having 100 marks, a person can score zero marks also, here we can say that a person scoring 100 marks has scored 2 times of a person scoring 50 marks.Note: For remembering the measurement scale you can remember NOIR (meaning black in French).Concept Builder – Identification of the scales1. In the cases given below, identify what type of scale it represents A. Ranking of mutual funds by Morning Star B. Runs scored by a player in a cricket match C. Measurement of temperature in Fahrenheit scale D. Classification of students in a class on the basis of sexExplanation A. Morning Star, assigns a one star to five star ranking. Five star ranking is given to the best funds as determined by Morning Star’s evaluation criterion. Here we can say that a 5 star mutual fund © Knowledge Varsity – 2011
    • is better than a 2 star mutual fund, but we can’t say by exactly how much, hence it is an example of ordinal scale. B. The run scored can’t be negative. A batsman who scores 100 runs has scored twice as much a batsman who has scored 50 runs. Hence, it is an example of Ratio Scale. C. Fahrenheit scale like Celsius scale doesn’t have a true zero, but we can identify the difference between the temperatures and hence it is a measure of interval scale. D. Classification on the basis of sex into male or female is just grouping and hence it is an example of Nominal Scale.LOS 7.b. Define a parameter, a sample statistic, and a frequency distributionParameter – A parameter is a numerical quantity measuring some characteristics of the population. Instatistics we have to deal with many parameters, but from our point of view, we are mainly interestedin mean standard deviation of the returns. Parameters are usually denoted by Greek letters. Forexample, population mean is denoted by µ (pronounced as mu) and population standard deviation isdenoted by σ (called as sigma).Sample Statistics – A sample statistics is a numerical quantity measuring the characteristics of thesample. A sample mean is denoted by , whereas sample standard deviation is denoted by s. Asmentioned earlier, most of the times population parameters are not known and they have to beestimated from sample, so we find out sample statistics to make prediction of the populationparameter.Mostly we deal with large data sets and it becomes difficult to deal with individual member of the set,frequency distribution is a simpler way of dealing with the data.The process of creation of Frequency distribution is outlined in the following steps Step 1 - We first create an interval (also known as class), which is basically a group. Following points are important A. An observation has to fall in one of the interval. B. The intervals should not overlap (that is they are mutually exclusive) C. Also an interval has a lower limit and an upper limit. Please note that an interval need not be a range always, it can be classified as one of the scale measure. For example we can have a Class as 1. Just grouping (Nominal) – Type of bond, Govt., private etc. 2. Ranking (Ordinal) - AAA bonds, AA bonds etc. 3. Interval scale – Have a range of return from these bonds 4. Ratio Scale – Have the bond grouped as per the coupon rate Step 2 –Assign the observation in the data set to the class that you have identified in Step 1.This process is called as tallying the data. Step 3 – Count the number of observation in each of the class, this number is called as the class frequency or frequency or absolute frequency.Note: Please note that in the examination, you will NOT be asked to construct a frequency distributiontable, the example below is an illustration of the process. However there can be a question (very lowprobability) where you would be asked to find the frequency of a class interval.© Knowledge Varsity – 2011 Page 27
    • Concept Builder – Frequency Distribution2. Given the following returns data for a stock in the month of January 20X1, construct a frequency distribution table -3.60 3.23 5.32 -5.84 -17.99 -7.14 2.26 6.42 6.70 17.51 -9.26 7.89 0.89 -2.85 -10.50 3.17 -3.25 2.09 -13.98 6.23 3.87 -6.70 1.20 -18.22 18.57ExplanationFor each observation, identify which class interval it should belong to and draw a line in the Tallycolumn. Once done, count the number to identify the absolute frequency. Class Interval Tally Absolute Frequency -20% ≤ R < 15% // 2 -15% ≤ R < 10% // 2 -10% ≤ R < 5% /// 3 -5% ≤ R < 0% //// 4 0% ≤ R < 5% //// // 7 5% ≤ R < 10% //// 5 10% ≤ R < 15% 0 15% ≤ R < 20% // 2 Total 25LOS 7.c. Calculate and interpret relative frequencies and cumulative relative frequencies, given afrequency distributionLets now define the following types of frequency that we observe in a frequency distribution Absolute Frequency – This is the actual number of frequency of a given interval. The minimum value of absolute frequency is ZERO, It can’t be negative. Relative Frequency –This measures how much time one of the class has occurred as relative to the entire class. Numerically, Cumulative Absolute Frequency and Cumulative Relative Frequency – Starting from the first class and moving down the classes, we can add the absolute and relative frequencies to get the cumulative absolute and cumulative relative frequency. Since the frequency will always be greater than or equal to ZERO, the cumulative frequencies are always increasing or remaining the same.The below example will build upon the example #2 to explain the concepts of this LOS. © Knowledge Varsity – 2011
    • Concept Builder – Relative and Cumulative Frequencies3. Given the following returns data for a stock in the month of January 20X1, construct a frequency distribution tableExplanationWe will use the formula given in the text above to calculate each of the measure. Cumulative Cumulative Absolute Relative Absolute Relative Class Interval Frequency Frequency Frequency Frequency -20% ≤ R < 15% 2 0.08 2 0.08 -15% ≤ R < 10% 2 0.08 4 0.16 -10% ≤ R < 5% 3 0.12 7 0.28 -5% ≤ R < 0% 4 0.16 11 0.44 0% ≤ R < 5% 7 0.28 18 0.72 5% ≤ R < 10% 5 0.2 23 0.92 10% ≤ R < 15% 0 0 23 0.92 15% ≤ R < 20% 2 0.08 25 1 Total 25 1LOS 7.d. describe the properties of a data set presented as a histogram or a frequency polygonHistogram – It is a bar chart (without space between the bars) that is constructed using the frequencydistribution. On the X (horizontal) Axis we plot the Class and on the Y Axis (Vertical Axis) we plot thefrequency. Looking at the histogram, you can easily identify some distribution characteristic, that iswhich class has the highest, lowest frequency, how the data is dispersed etc. Hence histogram is a veryuseful tool for an analyst.Frequency Polygon – This is a similar representation like the histogram, but here we don’t display thebars. The mid-points of the intervals are joined with a line to create a polygon structure.© Knowledge Varsity – 2011 Page 29
    • LOS 7.e. define, calculate, and interpret measures of central tendency, including the population mean,sample mean, arithmetic mean, weighted average or mean (including a portfolio return viewed as aweighted mean), geometric mean,harmonic mean, median, and modeMeasures of Central Tendency - Central tendency refers to the presence of center in a data set. Thereare various measures through which we try to identify the central (or middle) point of a population or asample. Please note that practically it is expensive to find out the central tendency measures for apopulation.Population Mean is the average value of a population, there is only one mean for one population. Theformula for a population mean is given asWhere are the individual members or data point of a population. N is the population size (that is thenumber of data point in the population).Sample Mean is the average value of a sample, there is only one mean foronesample. But from apopulation we can create many samples; hence we can have many sample means for one population.We compute sample mean and estimate the population mean from the sample means.The formula for a population mean is given asNote that we are using N (big) to denote the population size and n (small) to denote the sample size.Arithmetic Mean is the simple average for any data set. The population and the sample mean which wehave explained above fall in the category of the arithmetic mean. Arithmetic mean is the most popularmeasure of central tendency because of its calculation simplicity.Properties of Arithmetic Mean:1. Arithmetic mean is unique2. Arithmetic mean is only applicable to interval and ratio scale. Arithmetic mean is meaningless for nominal and ordinal scale.3. Needless to say that arithmetic mean can only be computed by taking the sum of all the data points in the set, its value will be inaccurate if you exclude some data purposely.4. Sum of the deviations from the mean is ZERO. First let’s define what deviation is – A deviation is the distance between the mean and one data point.Concept Builder – Deviation from the Mean4. As an example consider the following data set - 10, 20, 25, 5 The mean of this data set is (10+20+25+5)/4 = 15. Please note that there is no data point which has value of 15, so it’s not necessary that mean would be equal to one of the data point. Now let’s find the deviation: Data Point Mean Deviation 10 15 (10-15) = -5 20 15 (20-15) =5 25 15 (25-15) = 10 5 15 (5 -15) = -10 Sum Of Deviation -5 + 5 + 10 + -10 = 0 Deviation is important information for a data set, as it is a measure of risk. We will cover this point when we are covering the dispersion topic.Advantages of Arithmetic Mean:1. Easy to calculate2. Mean uses all the information about the size and magnitude of the observation (or data points)Disadvantages of Arithmetic Mean: © Knowledge Varsity – 2011
    • 1. Mean is highly sensitive to the extreme values, a higher value will be able to attract the meantowards its side. For example in MBA colleges, average salary is not a good measure because somestudents get very high salary and as a result, the mean is high. For the following data set - 10, 20, 25,965, the mean is 250. See the impact on the mean because of the presence of a high number (ascompared to the previous example). In such cases median is a better measure.2. Mean cannot be calculated for a data set which is not finite, that is the number of data points is notknown.Median: Median divides a data set into two equal parts; it means that median is the Mid-Point of asorted data set. The data set can be sorted either in ascending or descending order.For an odd number of data set, median is easier to identify as we have a data point which is the middlevalue. For odd number, the median is {(n+1)/2}th data pointConcept Builder – Calculation of median for odd number of observation5. A stock has generated the following returns over the past 5 days -2%, 5%, 4%, 2%, 8% Compute the median of the stock return?ExplanationFor computing the median, first we need to arrange the data in either ascending or descending order.Lets arrange in ascending order-2%, 2%, 4%, 5%, 8%There are total of 5 data points, since this is an odd number the median will be {(5+1)/2}th number or(6/2)th or 3rd number.The 3rd number here is 4%. Hence the median is 4%.For an even number of data set, the median needs to be calculated. Median is the average of the(n/2)th and ((n/2) + 1)th data point (or observation)Concept Builder – Calculation of median for odd number of observation6. A stock has generated the following returns over the past 4 days -2%, 5%, 4%, 2% Compute the median of the stock return?ExplanationFor computing the median, first we need to arrange the data in either ascending or descending order.Lets arrange in ascending order-2%, 2%, 4%, 5%There are total of 4 data points, since this is an even number the median will be the average of the(4/)2th and {(4/2)+1}th number or Average of 2 nd and 3rd number.2nd Number is 2%, 3rd number is 4%. Hence the average is (2% + 4%)/2 = 3%Hence the median is 3%.Mode: The mode is the most frequently occurring data point (or observation) in a data set (populationor a sample).Unlike Mean and Median, It is not necessary that a data set should have a mode. If all the observation ina data set is different, then it is said to have No Mode.Unimodal – When a data set has only one observation which has the highest frequency (or mostoccurrences) then the data set is said to have One Mode (or Unimodal)Bimodal -When a data set has only two observations having the highest frequency (or mostoccurrences) then the data set is said to have Two mode (or Bimodal)© Knowledge Varsity – 2011 Page 31
    • Modal Interval – In a frequency distribution with class intervals, we can’t classify as single data pointand hence finding mode is not possible, However, we can find out the class interval which has thehighest frequency, that class interval is said to be the modal interval.Concept Builder – Mode7. Find the mode for the following data set. a. 5%, 7%, 9%, 11%, 13% b. 5%, 7%, 9%, 5%, 11%, 13%, c. 5%, 7%, 9%, 5%, 11%, 13%,13%Explanation a. There is no observation which is occurring more than once, and hence no mode exist for this data set b. The observation 5% is coming twice in the data set, and hence the mode is 5% c. The observations 5% and 13% are coming twice in the data set, so this is an example of bimodal distributionOther Concepts of MeanEarlier we have covered arithmetic mean which is an important concept, but in investment we useother concepts of mean also, and it’s worthwhile to cover those concepts.The Weighted MeanThe concept of weighted mean is used in portfolio analysis. The idea behind weighted mean is that notall observation has the same weight and hence if we take a simple average, then the resulting answerwould not be accurate. Before moving to this concept, let’s see the example given belowConcept Builder – Why We Need Weighted Mean8. Consider that you have made investment in 3 stocks at T=0 and after one year (i.e. at T=1) you sell the stock. You would want to know the return that has been generated. The purchase price (at T=0) and the price at T=1 (the price at which you sold the stock) is given in the table below. Assuming that you have purchased one share of each stock. Stock Price at T=0 Price at T=1 Return A $100 $120 20% B $60 $66 10% C $40 $80 100%What should be the return as per your understanding, Should it be the arithmetic mean of the return?Let’s compute the arithmetic meanIt would be equal to (20%+10%+100%)/3 = 43.33%Now, let’s think in term of the money that is invested at T=0 and the money that you got when you soldthe stocks at T=1.Money Invested = $100 + $60 + $40 = $200Money Received from selling the stocks = $120 + $66 + $80 = $266So the return that you generated = ($266 - $200)/$200 = 33%Now, compare the return that your portfolio has generated with the arithmetic mean return, they arenot same. Hence there should be a way in which we should be able to find out the mean return.Weighted mean is the concept through which we would be able to find out that.In weighted mean calculation, we assign a weight to each observation. For stocks in the portfolio, theweight is the amount of money invested in them. © Knowledge Varsity – 2011
    • Where, X1 , X2 , …. Xn are the observationsW1 , W2 , …. Wn are the weights assigned to these observationsPlease note that, it is must to have the sum of the weights equal to 1, that isIn case of a portfolio of stocks, the Weight is given asNow let’s solve the example above using the weighted mean formula, we should be getting the sameanswerConcept Builder –Weighted Mean9. Consider that you have made investment in 3 stocks at T=0 and after one year (i.e. at T=1) you sell the stock. You would want to know the return that has been generated. The purchase price (at T=0) and the price at T=1 (the price at which you sold the stock) is given in the table below. Assuming that you have purchased one share of each stock. Stock Price at T=0 Price at T=1 Return A $100 $120 20% B $60 $66 10% C $40 $80 100%ExplanationFirst We need to calculate the weight of each of the stocks in the portfolioWA = Amount invested in A / Total amount invested => $100/$200 = 0.5WB = $60/$200 = 0.3WC = $40/$200 = 0.2Please check that the sum of the weight is equal to 1 => 0.5+0.3+0.2 = 1Now, apply the weighted mean formula on the returns that we are seeingWeighted Mean Return = WA RA+ WB RB+ Wc Rc = 0.5*20% + 0.3 * 10% + 0.2 * 100% = 10% + 3% + 20%  Weighted Mean Return =33%This is the same return which we got in the previous example.Hence, for portfolio return, weighted mean is the appropriate return measure.The Geometric MeanGeometric mean is used to calculate investment returns over multiple time periods. It is also used tocompute the average growth rate over a period, for example, the average sales growth of a firm overlast 5 years. When calculating the average growth rate it is called as Compounded Annual Growth Rateof CAGR.The formula for Geometric mean is given below Given that Xi ≥ 0 for i=1 , 2, 3 …… nWhere, GM is the Geometric Mean.X1 ,X2,…… Xn are the observation and n is total number of observationHowever this formula can’t be used to calculate the Geometric mean of the returns, because thereturns can be negative and we can’t have negative number inside the root.So, a solution for that is to replace each Xi with (1 + Return in decimal form). Now for any traditionalinvestment the return will not be less than -100% (that is you can’t lose more than what you haveinvested), so we will not have any negative value inside the root.So, for returns the Geometric Mean is given by the following formula© Knowledge Varsity – 2011 Page 33
    • Where, RG is the Geometric Mean.R1 ,R2,…… Rn are the returns and n is total number of returnsConcept Builder –Geometric Mean Return10. Consider that you have made investment in a stock at T=0, it provided a return of 10% in the first year, 20% in the second year and -30% in the third year. How much is the return you have received from the stock over the past 3 years?ExplanationR1 = 10% or 0.1; R2 = 0.2; R3 = -0.3  (1+R1) = 1.1 ; (1+R2) = 1.2 and (1+R3) = 0.7  (1+ RG) = => (1+RG) = => (1+ RG) = 0.974  RG = -0.026 or -2.6% So, this geometric mean return means that you have lost your money at a rate of 2.6% every year. And if you had invested $100 at T=0, it would have become $92.4 (Think How!!)This is the most appropriate place to cover the last LOS of this reading. The LOS is given belowLOS 7.l. Discuss the use of arithmetic mean or geometric mean when determining investment returnsNow to calculate the performance of any portfolio over a time period, should we use Arithmetic Meanreturn or Geometric Mean Return?Let’s take an example to explain this scenario.Concept Builder –Geometric Mean Return Versus Arithmetic Mean Return11. Consider that you have made investment in a stock and it provided a return of 100% in the first year and -50% in the second year. How much is the return you have received from the stock?ExplanationRather than getting into complication, let’s work from basic reasoning. Let’s assume that you hadinvested $100 in the stock at T=0.Since the first year return is 100%, the value of your stock will be $200 at T=1.Now, in the 2nd year, the return is -50%, therefore the value of your stock will be $100 at T=2.So, you started with $100 and have $100 in the end.It implies that over the 2 years, the return is 0%, so it means that the average return should also be 0%.Suppose, we use Arithmetic Mean return to explain the return over the 2 year period. The arithmeticmean return will be (100% + -50%)/2 = 25%But, we see that the return is 0% over the 2 years. This implies that the Arithmetic mean return is not acorrect measure to find the average stock return.Now Let’s calculate the Geometric mean return over the 2 years.R1 = 100% or 1; R2 = -0.5  (1+R1) = 2and (1+R2) = 0.5  (1+ RG) = => (1+RG) = => (1+ RG) = 1  RG = 0 or 0%So, the geometric mean return is 0%, which is the actual average return that we have calculated.So, from the above example it is clear that the Geometric Mean return is the correct measure of theaverage return over a period of Time. © Knowledge Varsity – 2011
    • The Harmonic MeanAs compared to the arithmetic and geometric mean, harmonic mean is used lesser in the investmentindustry. One scenario, where harmonic mean is used is to find out the average purchase price of shareswhen somebody is depositing equal dollar amount in the stock every period. BTW when somebody isdepositing equal amount every month or every period, the concept is called as Dollar Cost Averagingand in India, we also know this by the term Systematic Investment Plan.Let’s first try to find out the average purchase price of the stock when dollar cost averaging scheme isadopted and in the process we will derive the formula of the harmonic mean.Concept Builder –Harmonic Mean Return12. Suppose you as an investor have invested in a Systematic Investment Plan of HDFC Tax Saver Fund. You are making contribution of $1000 every month in the fund. The below table outlines the price of the fund at which you have made the investment. What is the average purchase price at which you bought the fund? Note: You will learn in the later chapters that for Mutual funds, the price at which we buy is not called as price but it is known as NAV (Net Asset Value) Time Investment Price (NAV) T=0 $1000 $100 T=1 $1000 $125 T=2 $1000 $80ExplanationWill the answer be the arithmetic mean of the price for the 3 periods?Let’s find the arithmetic mean, which will be ($100+$125+$80) = $305/3 = $101.67Let’s try to build the following table, where we identify the number of units of the mutual fund that wehave bought. Time Investment Price (NAV) Number of Units T=0 $1000 $100 =$1000/$100 = 10 T=1 $1000 $125 =$1000/$125 = 8 T=2 $1000 $80 =$1000/$80 = 12.5So, the total investment is 3 * $1000 = $3000Total Number of units that is bought is 10 + 8 + 12.5 = 30.5We know that logically, the average purchase price should be equal to total investment divided by thetotal number of units  Average price = $3000 / 30.5 = $98.3606So, it implies that the arithmetic mean is not the correct approach to find out the average purchaseprice in Dollar Cost Averaging method.Now, what exactly we have calculated which we are saying as the correct answer?We have calculated the harmonic mean of the purchase price, and that is the correct mean.Let’s try to rewrite the Average Price Equation, which we have in the aboveNow, $1000 will get cancelled in the numerator and the denominatorOr,This is the formula of the Harmonic Mean, which is given in detail in the text below© Knowledge Varsity – 2011 Page 35
    • Harmonic mean for a set of observation X1 ,X2 …… Xnis given by the following formulaOr in more general formatLOS 7.f. Describe, calculate, and interpret quartiles, quintiles, deciles, and percentilesOther Measures of Location – QuantilesMedian divides a distribution into half. The concept of Quantile divides the distribution into smallersizes. Median and quantiles are known as measure of location. Following are the examples of Quantiles Percentiles – The distribution is divided into 100 parts. Deciles – The distribution is divided into 10 parts. (Think Deca) Quintiles – The distribution is divided into 5 parts. Quartiles – The distribution is divided into 4 parts. (Think Quarter)Any quantile can be expressed as a percentile. So we use a standard formula to represent the quantiles.As an example, 3rd quintile (or 3/5th) is same as 60 percentile. This is because when we mention 3rdquintile, it means that 60% of the data are below the particular value or observation.For a data set having n members and arranged in ascending order, the formula for finding out theposition for a given percentile y isHere Ly is the position that we will get in the sorted data set.Y is the given percentile.So, if we are asked to find out the 2nd quartile, then y will be 50. If we are asked to find 6thdecile then ywill be 60. If we are asked to find out 90 percentile then y will be 90.Also note that when we are calculating using the formula, we are essentially finding out the position (orlevel) at which y percent of the data set will be below that particular position.100 percentile means that 100% of the data is below the particular observation, which is not possible.Hence the formula has (n+1) in it.Concept Builder –Quantiles13. Suppose there are 9 stocks in a portfolio, whose returns are given below. Find out the 4th quintile of the stock returns. -5%, 4%, 3%, 10%, 2%, 8%, -10%, 7% and 12%ExplanationFirst arrange the returns in ascending order-10%, -5%, 2%, 3%, 4%, 7%, 8%, 10%, 12%Now we need to find the position at which we will get the 4th quintile.4th quintile is same as 80 percentile. => Ly = 8Therefore, the 4th quintile is 8th data from the left.The 8th data is 10%, and hence the 4th quintile is 10%. Or we can say that 80% of the distribution isbelow 10%.Now, let’s see what happens when we do not get a whole number as the position. Whenever thishappens, we need to apply interpolation to get the value. This is a very simple concept related tounitary method, it would be clear from the following example. © Knowledge Varsity – 2011
    • Concept Builder – Quantiles14. Suppose you have added one more stock in the portfolio and now there are 10 stocks in the portfolio, whose returns are given below. Find out the 4th quintile of the stock returns. -5%, 4%, 3%, 10%, 2%, 8%, -10%, 7% , 12% and 18%ExplanationFirst arrange the returns in ascending order-10%, -5%, 2%, 3%, 4%, 7%, 8%, 10%, 12%, 18%Now we need to find the position at which we will get the 4th quintile.4th quintile is same as 80 percentile. => Ly = 8.8Therefore, the 4 quintile is 8.8th data from the left. Now 9th data is 12% and 8th data is 10%, this thmeans that the value would be between 10% and 12%. What we can say is that the value will be 10% +0.8 times the difference between 12% and 10%.This is because for 1 unit, the difference is 12% minus 10% or 2%, therefore for 0.8 units, the differencewill be 0.8 times 2%, this concept is Interpolation.So the value will be = 8th Value + 0.8 * (9th Value – 8th Value)  10% + 0.8 * (12% - 10%) => 10% + 0.8* 2% = 11.6%So the 4th quintile is 11.6%, so we can say that 80% of the distribution is below 11.6%IMPORTANT: We will apply interpolation only when the data set is population. When the data set is asample, then interpolation should not be used. So, if we had mentioned that in this example, it was asample then the interpolation would not have been used. The Ly value was 8.8, so we would just takethe integer part of it, which in this case is 8, so the value would be the observation in the 8 th position.Hence it would still be 10%.LOS 7.g. Define, calculate, and interpret 1) a range and a mean absolute deviation and 2) the varianceand standard deviation of a population and of a sampleMeasures of DispersionThis LOS covers the measures of dispersion, that is, how the data is dispersed around the centraltendency or simply mean. We can think of mean as the reward for investing and dispersion as the riskassociated with investing.There are two types of dispersion Absolute Dispersion – Absolute dispersion is the amount of variability present in a distribution without considering any benchmark or any reference point. In this LOS we will discuss about Absolute dispersion. The measures of absolute dispersion are range, mean absolute deviation, variance and standard deviation. Relative Dispersion - Relative dispersion isthe amount of variability present in a distribution relative to a benchmark or a reference value. Coefficient of variation is an example of relative dispersion and is covered in LOS i.Range is the simplest measure of dispersion. Here we are just interested in finding out the distancebetween the maximum and the minimum value in a distribution. It is useful when we are comparingtwo data set (or distribution) and it is very simple to compute.Range has the following disadvantages It uses only two values from the distribution It is very sensitive to the extreme values. Does not give any idea about the shape of the distributionAs a result, range is not used on a stand-alone basis but is used to supplement other measures ofdispersion.© Knowledge Varsity – 2011 Page 37
    • Mean Absolute Deviation – While covering arithmetic mean, we calculated the mean deviation and wefound that it was always ZERO. Using the deviation we can get information about the dispersion but weneed to address the problem of the sum coming zero, a solution is to take only the absolute value of thedeviation, that is, when the deviation is negative, we ignore the negative sign and take the value. HenceMean absolute deviation is the average of the absolute deviation around the mean. The formula isHere is the sample mean, is the observation and n is the total number of observations.MAD is a better measure than the range, but mathematically it has been found that it’s not the mostsuperior measure of dispersion and hence we move on to the next measure which is Variance.Variance – In MAD, we had taken absolute measure to get the positive deviation. Another way to getpositive deviation is to square the deviations. Mathematically this is a better measure. By taking theaverage of the squared deviations we get Variance. There are differences in the variance calculationwhen we are calculating for population and when it is calculated for sample. The formula for PopulationVariance is given belowHere, is the population mean and is the size of the populationSample Variance –Statistically, it has been found that if we use the population variance formula to findout the sample variance, the variance thus obtained was lesser than what is actually observed for asample. To remove this bias, it has been suggested to divide the squared deviations by (n-1) rather thann. The sample variance thus obtained was closer to what was actually observed. The formula for samplevariance is given belowHere, is the sample mean and n is the size of the sampleStandard Deviation - Variance is the best measure for dispersion, however if you observe, variance is asquared number and hence its unit is also squared. For example if we are measuring the variance ofheight of a group of people, the unit of variance will be height squared. We as human would like tothink in linear measures and also all the calculation that we perform on the distributions are linear andhence it became important to have a measure which is linear along with a superior measure ofdispersion. The solution is to take the square root of the variance. The value obtained after taking theroot is known as standard deviation. So, variance is essentially square of the standard deviation.The population standard deviation is denoted by σ (called as sigma), therefore the population varianceis denoted by σ2. So, the formula for population standard deviation will beThe sample standard deviation is denoted by s, therefore the sample variance is denoted by s2. Theformula for sample standard deviation is given as © Knowledge Varsity – 2011
    • Concept Builder – Measures of Dispersion15. A stock has the following returns over the past 7 years 7%, 6%, 9%, 10%,11%,2%, 4% You are interested in finding out the measures of dispersion. a. What is the range? b. What is the Mean Absolute Deviation (MAD)? c. What is the sample and population variance? d. What is the sample and population standard deviation?ExplanationRange : Range is the difference between the highest and the lowest return values  Range = 11% - 2% = 9%MAD: Before we find MAD, we will have to find the mean of the returnsMean = (7+6+9+10+11+2+4)/7 = 7%  MAD = 2.57%For the rest of the calculations, it is better to use the financial calculator.Using our TI BA II Plus Financial Calculator, we can easily find the variance and the deviation.Press 2nd ; Press 7 ; It means you are pressing DATA.Press 2nd Press CE|C ; It means you are pressing Clear WorkYou will find in your calculator X01 , the calculator is asking you to input the value here.Please note that since we have only one set of data, we will work with only the X data series.Now Press 7 and then Press ENTER. You will see a down arrow, you will have to Press it Twice ↓↓You will get X02, press 6 and then Press ENTER, again press ↓↓.The below has the subsequent steps that you need to performPress 9 Press ENTER Press ↓↓Press 10 Press ENTER Press ↓↓Press 11 Press ENTER Press ↓↓Press 2 Press ENTER Press ↓↓Press 4 ENTERSo, you have entered total of 7 data.Now we need to go to the Statistics function in the calculatorPress 2nd ; Press 8, it means that you are pressing STAT.You will see LIN in the display, it means that you are working on linear data.Now Press ↓=> You will see n = 7, double check that your total data points were 7Now Press ↓=> You will see , = 7, which is the mean of the data (we calculated the same earlier)Now Press ↓=> You will see ,Sx = 3.26598, which is the sample standard deviationNow Press ↓=> You will see , σx = 3.023716, which is the population standard deviationFor this exercise, these were the only calculation that you had to perform in the calculatorPopulation Variance can be found out by squaring the population standard deviation.Population Variance = 9.14% 2Sample Variance can be found out by squaring the sample standard deviation.Sample Variance = 10.67%2Before moving to other LOS, we will visit LOS i as it is closely tied with the concepts which we havediscussed in this LOS.© Knowledge Varsity – 2011 Page 39
    • LOS 7.i. Define, calculate, and interpret the coefficient of variation and the Sharpe ratioWhen we are comparing two distributions which have large differences in the mean and the variance, itbecomes difficult to compare with the absolute dispersion measure, In such cases we employ relativedispersion measure.Coefficient of Variation (CV) is a relative dispersion measure and is used to standardize the absolutedispersion measure. It is given by the following formulaFor portfolios, CV measures the amount of Risk (deviation) per unit of return of the portfolio. We wouldlike to have lesser risk per unit of return and hence the lower the coefficient of variation, the better itis.Concept Builder – Coefficient Of Variation16. You are Fund Of Fund Investment manager and would like to invest some money in one of the fund. You are evaluating two fund managers and would like to invest in one of them. The following table has the return and the standard deviation data for the two fund managers. Based on the concept of Coefficient of Variation, with which fund manager would you invest and why? Fund Manager Return Standard Deviation (Risk) A 10% 20% B 15% 25%ExplanationYou might think it is better to invest with A because he has lower variability of the return. But note thathere we are seeing that the funds have different return and risk measure, and hence we are unable tomake a decision based on the absolute measure. We will have to employ a relative measure to find outthe best manager.Coefficient of Variation (CV) which measures the risk per unit of return would be ideal here. So we willcompute the CV for both the manager. Computation is shown below Fund Manager Return Standard Deviation (Risk) CV A 10% 20% =20/10 = 2 B 15% 25% = 25/15 = 1.67Since the fund manager B has lower Coefficient of Variation (risk per unit of return) we should invest inB’s fund.Sharpe Ratio – This ratio is attributed to Prof. William Sharpe and he received Nobel Prize for coming upwith this concept. This ratio is basically a reward to risk ratio and has prominent place in theinvestment industry. Using this ratio we are able to compare the portfolios on risk and returncharacteristics. You may find it similar to the coefficient of variation, but this ratio is far more reachingthat coefficient of variation because it measures the excess return over the risk free rate. The idea onwhich it is built is that, Since there is no risk when one invest in risk free asset, one should not look atthe absolute return from any investment but should look at the excess return that the investment isgenerating over the risk free rate and then compare investments on the excess return versus the risk.Sharpe ratio is opposite to the coefficient of variation, here we measure the excess return generatedper unit of risk taken. It would become clearer from its formulaHere, is the portfolio return, RFR is the risk free rate of return, is the standard deviation (or risk)of the portfolio.Since higher excess return per unit of risk is better, hence higher the Sharpe ratio, the better it is. © Knowledge Varsity – 2011
    • Concept Builder –Sharpe Ratio17. The data is same as that of the previous problem. Additionally risk free rate of return is given as 4%. Based on Sharpe Ratio, with which fund manager would you invest and why?ExplanationThe following table has the calculation of Sharpe Ratio in the fourth column. Fund Manager Return Standard Deviation (Risk) Sharpe Ratio A 10% 20% =(10-4)/20 = 0.3 B 15% 25% = (15-4)/25 = 0.44Since the fund manager B has higher Sharpe Ratio (excess return per unit of risk) than fund manager A,we should invest in B’s fund.IMPORTANT: Compare, this example with the previous example, in both the case we have got the sameanswer, which is to invest in B’s fund. Is it always the case that the result in which portfolio to invest issame in Sharpe Ratio and Coefficient of Variation?The example, here are peculiar and it is by chance that we are getting the same return, It is not alwaysthe case. Let’s see the next example.Concept Builder – Coefficient of Variation Versus Sharpe Ratio18. You are Fund Of Fund Investment manager and would like to invest some money in one of the fund. You are evaluating two fund managers and would like to invest in one of them. The following table has the return and the standard deviation data for the two fund managers. The risk free rate of return is 7%. Based on the concept of Coefficient of Variation and Sharpe Ratio, find out with which fund manager would you invest and why? Fund Manager Return Standard Deviation (Risk) A 12% 15% B 14% 20%ExplanationThe following table has the calculation of CV and the Sharpe Ratio. Fund Manager Return Standard Deviation CV Sharpe Ratio A 12% 15% =15/12 = 1.25 =(12-7)/15 = 0.33 B 14% 20% = 20/14 = 1.43 = (14-7)/20 = 0.35As per the CV, Fund manager A is better, as CV is lower for A.As per the Sharpe Ratio, Fund manager B is better, as Sharpe Ratio is higher for B.So, we can have conflicting result from Sharpe Ratio and CV comparison. Since Sharpe Ratio, measuresthe excess return relative to Risk, Sharpe Ratio is Preferred over CV.LOS 7.h. Calculate and interpret the proportion of observations falling within a specified number ofstandard deviations of the mean using Chebyshev’s inequalityIdeally, this LOS should have been covered in reading no -9, where we are covering the distribution.Covering the concept over here would entail some unnecessary hard-ship on you. Leaving this LOS, atthis point of time will not impact the flow of study. So, we will cover this LOS in Reading 9.LOS 7.j. Define and interpret Skewness, explain the meaning of a positively or negatively skewed returndistribution, and describe the relative locations of the mean, median, and mode for a nonsymmetricaldistributionSkewness -Ideally the distribution should be symmetrical around the mean, that is if we put a mirroralong the Y Axis on the mean, the reflection of the left side should be identical to the shape of the rightside. However in real life, we usually do not find symmetrical distribution.© Knowledge Varsity – 2011 Page 41
    • In everyday language, the terms “skewed” and “askew” are used to refer to something that is out of lineor distorted on one side. When referring to the shape of frequency or probability distributions,“skewness” refers to asymmetry of the distribution.Skewness is of two types Positively Skewed –In this case the distribution is skewed to the right side and hence it is also called as Right Skewed. If you see the diagram given below, you would see that in case of positive skewed distribution there is a long tail in the right side. The long tail denotes that there are outliers in the positive side of the distribution, that is, there are some observations which have large positive value and hence they are able to shift the mean to the right side. As an example consider that in a class, there is a brilliant student and he scores 100 out of 100 in the examination, there are 4 other student in the class, who are average and they score 60 marks each in the examination. The mean marks of the students in the class would be 68, but the mode and the median are equal to 60. Hence due to the presence of above average student the mean has shifted to the right. For this kind of distribution, you will find that the mean is more than median and median is more than the mode. Negatively Skewed –In this case the distribution is skewed to the left side and hence it is also called as Left Skewed. If you see the diagram given below, you would see that in case of negative skewed distribution there is a long tail in the left side. The long tail denotes that there are outliers in the negative side of the distribution, that is, there are some observations which have large negative value and hence they are able to shift the mean to the left side. An example, here could be the returns in a stock market, many times we observe that there are some returns which are highly negative and there are more returns which are positive , because of large negative returns the mean return shift to the left and hence we can see a left skewed distribution. Mnemonic: As per dictionary, the order of the words is Mean, Median and Mode. Hence Negative skewed distribution follows the dictionary order that is, mean is less than median which in turn is less than mode. Please note that median will always be in the middle.Important Exam Points for Skeweness Mean always get shifted in the direction of skewness, for positively skewed distribution, mean is shifted to the positive side (or right side) For a symmetrical distribution, the distance from the mean to the highest observation would be same as that of distance from the mean to the lowest observation.Concept Builder – Skeweness19. Suppose the mean is given as 50% and the highest return value is given as 100% and lowest return value is given as 0%. State whether the distribution is symmetrical or not?Explanation Since, here the distance from the mean to the highest value and the distance to the lowest value is same, we would say that the distribution is symmetrical. © Knowledge Varsity – 2011
    • 20. Suppose the mean is given as 50% and the highest return value is given as 80% and lowest return value is given as 0%. State whether the distribution is symmetrical or not? If it is not symmetrical then state whether it is positively skewed or negatively skewed?ExplanationSince, here the distance from the mean to the highest value is 30 and the distance to the lowest value is50, then we would say that the distribution is skewed to the left as we have long tail there.LOS 7.k. Define and interpret measures of sample Skewness and kurtosisThis LOS asks us to define the measure of sample skeweness and kurtosis. In the CFA text book, youwould find that formula for sample skeweness and kurtosis are given, but you don’t have to rememberthe formula, since the LOS doesn’t ask you to calculate. However you require interpretation of these.Skeweness is measure of 3rd moment, that is, we are taking the cube of the deviation from the meanand summing them up. This result in the following interpretation o Skewness can range from minus infinity to positive infinity. o For a symmetrical distribution, skeweness = 0 o For positively skewed distribution, skeweness>0 o For negatively skewed distribution, skeweness< 0From examination point of view, you just need to know the above and nothing more.Kurtosis - Kurtosis is a measure of the Peakness of a symmetric distribution as compared to a normaldistribution of the same variance. Here, before covering anything lets cover normal distribution.Normal Distribution – As of now, just understand that this is one type of distribution which issymmetrical in nature. It has a kurtosis of 3. We would cover this distribution in detail in the ReadingNumber 9.Kurtosis is of three types:- Leptokurtic o This type of distribution is more peaked than the normal distribution. o The kurtosis of Leptokurtic distribution is more than 3 o In this type of distribution, there is large number of observation which is near the mean and also large number of observation away from the mean. o It has fatter and long tails o Mnemonic -It has a shape like that of small L and L stands for Lepto Mesokurtic o A mesokurtic distribution has kurtosis of 3 which is same as normal distribution and hence mesokurtic distribution is actually a normal distribution. We compare kurtosis with respect to mesokurtic distribution only. Platykurtic o This type of distribution is less peaked than the normal distribution. o Platykurtic distribution is Flat. o The kurtosis of Platykurtic distribution is less than 3 o It has thinner and shorter tails o Mnemonic – Platy can be thought of as “FLAT”, add P in FLAT to get PLAT© Knowledge Varsity – 2011 Page 43
    • Excess Kurtosis – Excess kurtosis tries to find out the kurtosis which is more than that of a mesokurtic(normal) distribution. Since the Kurtosis for a normal distribution is equal to 3. The excess kurtosisformula is given by Excess Kurtosis = Kurtosis - 3Therefore, Excess Kurtosis for Leptokurtic distribution is more than 0 Mesokurtic distribution is 0. Platykurtic distribution is less than 0Kurtosis Measure: We don’t need to know the formula of Kurtosis; we just need to know that Kurtosisis the 4th moment. That is kurtosis is calculated by taking the sum of the fourth power of thedeviations. As a result Kurtosis is always positive. © Knowledge Varsity – 2011
    • Concept Notes For Reading 8- Probability ConceptsReading SummaryThis reading covers important probability concepts. Here we will discuss about random variables,properties of probability. We will also understand how probability theory is used in betting. Fromexamination point of view, this is an important reading, because the concepts would be used inportfolio management and also in the next readings where without understanding of probability itwould be difficult to understand the distributions. Quite a few questions from this reading are expectedin the examination.From our experience, we have seen that to do well in this topic, you should understand how tostructure a probability problem using a tree diagram. Bayes theorem is a very important topic whichyou need to master. However permutation and combination topic can be taken lightly.LOS 8.a.Define a random variable, an outcome, an event, mutually exclusive events, and exhaustiveeventsA random variable is a quantity whose observed value is uncertain.Outcome is the observed value of a random variable.Event is a specified set of outcomes. It can be a single outcome also.Mutually exclusive events are those events in which only one of the events would happen at aparticular time. So, in mutually exclusive event it is not possible to observe all the events at one time.Exhaustive means that the outcome of the events covers the entire possible scenario.An example of mutually exclusive and exhaustive event would be the pattern of stock price. At anypoint of time, the share price would Increase Decrease Stay sameSo, the three states given above are mutually exclusive but note that they are collectively exhaustivebecause, these are the only states possible. You can’t have a stock’s price changing to some other state.Mnemonic: MECE – Mutually Exclusive and Collectively ExhaustiveLOS 8.b.Explain the two defining properties of probability and distinguish among empirical, subjective,and a priori probabilitiesThe two defining properties of probability are as follows:-1. The probability of any event (E) is a number between 0 and 1, that is,2. The sum of the probabilities of any set of mutually exclusive and collectively exhaustive event shouldbe always equal to 1. That is,Going back to the stock price example of last LOS. It’s not possible to have the probability of any of thestate as negative or more than 1. Let’s say that the probability of increase is 0.4, decrease is 0.5 andremaining the same is 0.1.Here we are seeing that the sum of the probabilities of these mutually exclusive and exhaustive eventsshould be equal to 1.How do we determine the probability of any set of events?Estimation of Probability: There are 3 approaches in which we can estimate the probability. Theapproach is divided into Objective and Subjective. Objective is further divided into Empirical and PrioriProbability. The following diagram will make the concept clearer.© Knowledge Varsity – 2011 Page 45
    • Probability Estimation Objective Subjective Empirical A PrioriLet’s cover what these probability mean Empirical Probability – Empirical probability is estimated using the past data. Example – Suppose we have seen that over the past 250 trading days. The number of days the stock price increased was 100. The number of days the stock price decreased was 125 and the number of days the stock price remained the same was 25. Now probability is also defined as So with the historical observation, we can say that the probability that the stock price will increase will be equal to 100 (no of favorable days) divided by 250 (total number of days), which will be 0.4. Likewise we can estimate the probability of decrease and remaining same also. A Priori Probability–A priori probability is estimation of probability using formal reasoning or logical method. Example – Suppose, we estimate that the inflation of the economy is high and it is expected that the central bank will increase the interest rate. In this scenario with formal reasoning we expect that the probability that the stock price will decrease, we do some analysis and come up with a probability value of 0.7, and similarly other probabilities. Subjective Probability–Subjective probability is estimation of probability based on gut feeling. It is the least formal method and involves lot of subjectivity by the estimator. In investment, its usage is frequent in nature. Example – A person may say that in his personal opinion the probability of stock rising the next day is 0.7.LOS 8.c.State the probability of an event in terms of odds for or against the eventThis LOS covers the concept related to betting. In betting we use the term odds for a team winning etc.Terminologies Odds for an event–Out of the total how much is in favor of a particular event. Example : Odds for India winning against Pakistan is 5 to 3, it implies that out of (5+3 = 8) times, India will win the match 5 times and Pakistan will win the match 3 times. Odds against an event - Out of the total how much is against a particular event. Example : Odds against India winning in a match with Pakistan is 3 to 5, it implies that out of (5+3 = 8) times, Pakistan will match 3 times and India will win the match 5 times .Note: Odds for an event is just the reciprocal of the odds against the eventIn examination, there are 2 types of questions that would be askeda) Find out the probability, given odds for an event or against an event. a. If odds for an event is given as A:B then the probability that event A will occur is b. If odds against an event is given as A:B then the probability that event A will occur isb) Given probability for an event, find out the odds for or against the event © Knowledge Varsity – 2011
    • a. If Probability of an event is given as P(E) then the odds for the event is b. If Probability of an event is given as P(E) then the odds against the event isEven though we have given the formula, dealing these concepts in formula is not the correct approach,rather than that you should understand the underlying concept. It would become clear by seeing thebelow example.Concept Builder – Finding probability when Odds For and Odds Against is given1. The bookies are quoting the following for the final match between Liverpool and ManU Odds For Liverpool : 7 to 11 What is the probability of Liverpool winning the match? What is the probability of ManU winning the match?ExplanationThe probability that Liverpool will win the match is its favorable case divided by the total case  P(Liverpool winning) = 7/18 = 0.389The probability that ManU will win the match is its favorable case divided by the total case  P(ManU winning) = 11/18 = 0.611 Since, This could also be obtained by subtracting P(Liverpool winning) from 1P(ManU winning) = 1 - P(Liverpool winning) = 1 – 0.389 = 0.611 Needless to say, we are ManU fans!!Concept Builder – Finding Odds For and Odds Against when probability is given2. Suppose you determine that the probability of Chennai Super Kings winning a match against Mumbai Indian is 0.6, then what would be the a. Odds that Chennai Super Kings will win the match b. Odds that Chennai Super Kings will lose the match (Note that here we are asking odds against in an indirect way)Explanationa. For finding the odds for; without following the formula; we should think of odds for as Probability the event will occur : Probability the event will NOT occur  Probability that Chennai will win : Probability that Chennai will lose  0.6 : 0.4 Or 6:4 Or Simplifying it a bit, 3:2  So the odds that Chennai Super Kings will win the match is 3 : 2b. For finding the odds against; without following the formula; we should think of odds against as Probability the event will NOT occur : Probability the event will occur  Probability that Chennai will lose : Probability that Chennai will win  0.4 : 0.6 Or 4:6 Or Simplifying it a bit, 2:3  So the odds that Chennai Super Kings will lose the match is 2 : 3LOS 8.d.Distinguish between unconditional and conditional probabilitiesUnconditional Probability: It is also referred as marginal probability. It is the probability that an eventwill occur regardless of any past or future occurrence of any other event.Conditional Probability: The probability that an event will occur, given that one or more other eventshave already occurred. More precisely the probability that B will occur given that A has occurred.The symbol P (B|A) represents Conditional Probability and is read as Probability of B given A hasoccurred.© Knowledge Varsity – 2011 Page 47
    • LOS 8.g.Distinguish between dependent and independent eventsWe will be covering this LOS before we cover LOS 8.e and LOS 8.f1. Independent Events – These are events in which the occurrence of one event has no impact on the occurrence of the other events. As an example, your score in an exam is independent of the score of your friend. In terms of conditional probabilities, two events are independent, if and only if P(A | B ) = P(A) or P(B | A ) = P(B)If the condition above is not satisfied then the events are dependent events2. Dependent Events – These are the events in which occurrence of one event has an impact on other events. As an example, your score in an exam is dependent on the amount of effort you have put in to study for the exam.Here in this section, we will be covering both LOS 8.e and LOS 8.fLOS 8.e.Define and explain the multiplication, addition, and total probability rulesLOS 8.f.Calculate and interpret 1) the joint probability of two events, 2) the probability that at least oneof two events will occur, given the probability of each and the joint probability of the two events, and 3)a joint probability of any number of independent eventsa) Multiplication Rule of probability – It is used to find out the Joint Probability of 2 events. Joint probability is the probability that 2 events would happen together (or simultaneously).We would see the following type of joint probability in the curriculum. 1. Joint Probability of conditional events 2. Joint Probability of independent eventsLet’s cover these two concepts 1. Joint Probability of conditional events –The joint probability of conditional event is given as Where, is the conditional probability that A will happen, given that B has happened. And is the unconditional probability of B. 2. Joint Probability of independent events – In case of independent events A and B, the joint probability is given as So, the joint probability is the multiplication of the individual probabilities and the multiplication rule is used to denote the joint probability. This concept can be extended for any number of independent events Let’s say that A1, A2, ……, AN are independent events, the probability that all of these events occur simultaneously is given by Note that when you see AND , this means that you have to use multiplication.b) Addition Rule of probability – It is used to find out the probability that at least one of the event will occur. In case of two events A and B. Addition rule will say that Event A OR B will occur. Note that when you see OR, this means that you have to use addition.You can extend the addition rule concept to determine the probability that at least one of the eventwould occur. If you notice the below formula, here we are saying that Either A occurs or B occurs, itimplies that sometimes A occurs, sometimes B occurs or sometime both will occur, it will not happenanytime that none of the event is occurring.You can understand this better from the Venn diagram, which is shown below. The shaded regionrepresent that both A and B will occur together. Note that when we add P(A) and P(B), the shaded © Knowledge Varsity – 2011
    • region will come twice, hence we need to subtract it from the addition to come up with the trueprobability that at least A or B will occur.c) Total Probability is the probability of occurrence of an event, given conditional probability. Let’s say there are N mutually exclusive events denoted as A1, A2, ……, AN. Each of these events are conditional on mutually exclusive and exhaustive events B1, B2, ……, BN.We can obtain the unconditional probability of event A (which we say is the sum of the probabilities ofeach event Ai)Since A1 is conditional on B1, We can writeReplacing this for all Ai’s we haveThe probability of A that we have obtained here is un-conditional probability, that is, it doesn’t dependon whether B1 is occurring or B2 is occurring.Hence we can say that Total Probability is used to find out the unconditional probability of an eventwhen the conditional probabilities are given.2 Scenario Case: The total probability rule is stated below for a special case involving 2 scenarios. If wehave an event denoted by B, then we will denote the event which is NOT B (or complement of B) with (spelt as B bar). Here B and are mutually exclusive and collectively exhaustive event.Let’s say we have an event A, which will occur when B occurs and also when B doesn’t occur. The totalprobability for this two scenario case can then be written asBelow is a representation of the total probability concept using tree diagram. Tree diagram isconsidered in a separate LOS, but this is ideal place to introduce the concept of tree diagram.© Knowledge Varsity – 2011 Page 49
    • So, from the above diagram, it becomes clear that Node 1 and Node 3, are representative of occurrenceof A, whereas node 2 and node 4 are representative of occurrence of NOT A (or A bar).If you add the value obtained in Node 1 and Node 3, you will get the probability of occurrence of A.If you add the value obtained in Node 2 and Node 4, you will get the probability of occurrence of A bar(or Not A).The tree diagram will become clearer as and when we solve more problems. We would suggest thinkingabout tree diagram when you are solving problems which involve conditional probability because itgives a better understanding.Concept Builder – Joint Probability Of Conditional Events and Addition Rule3. Suppose probability that RBI will decrease the interest rate is 0.6 and the probability that recession will happen if the interest rate is decreased is 0.3. A. Find the joint probability of decrease in interest rate and recession? B. If the probability that either interest rate will decrease or recession will occur is 0.82, find the unconditional probability of recessionExplanation A. In notation term we can write ; P(Decrease Interest) = 0.6; P(Recession | Decrease Interest) = 0.3 The joint probability is P(Decrease Interest AND Recession); as per the joint probability for conditional probabilities we have P(Decrease Interest AND Recession) = P(Recession | Decrease Interest) * P(Decrease Interest)  P(Decrease Interest AND Recession) = 0.6 *0.3 = 0.18 B. Unconditional probability of recession = P(Recession) Here we have OR condition => P(Recession Or Decrease Interest) = P(Recession) + P(Decrease Interest) – P(Recession AND Decrease Interest) => 0.82 = P(Recession) + 0.6 – 0.18 => P(Recession) = 0.82 -0.6 +0.18 = 0.4 Therefore the unconditional probability that recession will happen is 0.4Concept Builder – Joint Probability Of Independent Events4. Suppose that you are tossing a coin 5 times in a row. What is the probability that Head will come in each of the time?ExplanationHere , the tossing of coins are independent events . For independent events, we have the followingP(A and B) = P(A) * P(B)Probability of Head coming in 1 toss = ½ = 0.5Probability of Head coming in 5 tosses = ½ * ½ * ½* ½* ½ = 0.0312LOS 8.h.Calculate and interpret, using the total probability rule, an unconditional probabilityAs we had stated in the last LOS that total probability is also known as unconditional probability. ThisLOS specifically asks us to calculate the unconditional probability using the total probability rule.Many of us are not able to appreciate fact that the total probability is the unconditional probability.Let’s solve one example to calculate the total probability and understand the concept of unconditionalprobability.Concept Builder – Calculating the Total Probability5. If you are following the recent events, Reserve Bank of India is increasing the interest rates to contain the inflation. One of the major drawback of raising interest rate is that the chances that the economy will go into recession increases. However, it’s not the case that an economy can’t be in recession if the interest rates are decreased (see USA, even though the interest rates are decreased in US, the state of the economy is not good). Coming to the question. Let’s say the probability that the Central Bank would increase the interest rate is 0.7 and the conditional probability of Recession happening when the interest rate is © Knowledge Varsity – 2011
    • increased is 0.9. Also, the conditional probability that the recession will happen if the interest rate is decreased is 0.2. We are interested in finding out a) The unconditional probability that the Recession will happen b) The unconditional probability that the Recession will NOT happenExplanationYou will find that in book, there is notation to the unconditional probabilities. We will go through thisproblem by following those notations, and then try to solve the problem through the formula of thetotal probability. In the next step we will try to solve the problem using the TREE DIAGRAM. We advisethe candidates to follow the tree diagram. Probability of Increase in Interest Rate = = 0.7 Probability of decrease in Interest Rate = = 1 – 0.7 = 0.3 Probability of Recession happening Given Interest Rate Increases = Probability of Recession happening Given Interest Rate Decreases = = 0.2 Probability of Recession NOT happening Given Interest Rate Increases = =1- = 1 -0.9 = 0.1 Probability of Recession NOT happening Given Interest Rate Decreases = = 1- 0.2 = 0.8The unconditional probability that the recession will happen is same as the total probability that therecession will happen.As per the total probability formula, we can write => P(R) = 0.9 *0.7 + 0.2 * 0.3 =0.63 +0.06 = 0.69So, the unconditional probability that recession will happen is 0.69The probabilities that you had seen earlier or were conditional probabilities, that is, theywere dependent on whether the interest rate is increased or decreased. But the total probability is notdepended on increase/decrease of interest rate, it doesn’t matter whether interest rate increases ordecreases, if the interest rate increases, then the probability of recession will increase and if theinterest rate decreases then the probability of recession will decreases, but overall the probability willbe the same and hence un-conditional.We can find the un-conditional probability of recession NOT happening – it will simply be 1 minus theun-conditional probability of Recession happening.The unconditional probability that recession will NOT happen = 1 - 0.69 = 0.31We can also find out the above value using the total probability formula => 0.1 *0.7 + 0.8 * 0.3 = 0.07+0.24 = 0.31Let’s see, how we can solve this using Tree Diagram, You will find it intuitive© Knowledge Varsity – 2011 Page 51
    • While constructing tree diagram,1st STEP - We start with first the un-conditional probability, which we have here for the Interest rate. Sothe first branch will be for the interest rate increase and the other branch is for the interest rate NOTIncreasing.2nd STEP - Then we start with the conditional probabilities and draw branches for them. Throughout theprocess, we will keep in mind that the branches are Mutually exclusive and Collectively Exhaustive.3rd STEP – We highlight the node is related to the respective probabilities. Here Node 1 and Node 3 arethe nodes for Recession happening and Node 2 and Node 4 are the nodes for Recession NOThappening.4th STEP - To find out the Total Probability of recession, we add up the value at the nodes wererecession happens. Therefore Recession probability = 0.63 + 0.06 = 0.69To find out the Total Probability of recession NOT happening, we add up the value at the nodes wererecession does not happen. Therefore Recession probability = 0.07 + 0.24 = 0.31This is the way we will be drawing the TREE Diagram and hence finding out the total or un-conditionalprobabilities.LOS 8.i.Explain the use of conditional expectation in investment applicationsIn previous reading we had covered the concept of Weighted Mean return in a portfolio, which wasbased on the weight of the individual asset. We had obtained the weighted mean return by multiplyingthe weight of asset with the return of the particular asset. If you think about the Total Probability, it canalso be considered like a weighted mean; here the weight is the conditional probability. We are going tointroduce the concept of Expected value which can be viewed like weighted mean.Expected Value –Expected value of a random variable is the probability weighted average of thepossible outcomes of that random variable.Expected value is denoted by E(X) where X is the random variable.Expected value is a futuristic value and is based on our expectation about the future, it is notnecessary that whatever we are expecting about the future will happen.The formula for the expected value can be written asWhere, is the outcome of the random variable with a probability of © Knowledge Varsity – 2011
    • Concept Builder – Finding Expected Value for throw of Dice6. Find the Value that you would expect when you throw a Six faced dice having numbers from 1 to 6ExplanationWhen we throw a dice, we can get any number from 1 to 6. By Expected value we imply what is thevalue on an average we will get when we throw a dice.The random variable can take values from 1 to 6.The probability that either 1,2,3,4,5 or 6 would come when we throw a dice is 1/6Hence Expected Value is calculated as,Therefore we should expect that in one throw we should get on an average 3.5But 3.5 is not a number that you will observe, we get sometimes 1 and sometimes 6, hence there isvariance in our observation.We will take up the variance concept after next example.Concept Builder – Finding Expected Earnings Per Share (EPS) in an investment Setting7. You are analyzing, Infosys Technologies (NASDAQ: INFY) stock and based on your evaluation you have come up with the following estimates of EPS for the year 20X2 and the probabilities that the firm would be able to achieve the EPS. See the table below for the estimates Probability EPS 0.35 $2.8 0.3 $2.9 0.25 $3.1 0.1 $4What is the expected EPS of INFY in 20X2?ExplanationExpected Value is probability weightedHence the expected value of EPS is $3.03, so you have estimated that INFY will have EPS of $3.03 in theyear 20X2.Since, expected value can be considered as mean; hence we will also get variance because there aretimes when the value thus obtained will not be equal to the expected value.Variance –The variance of a random variable is the probability weighted average of the squareddeviations from the expected value. The formula for the variance is given belowStandard deviation is square root of variance.Concept Builder – Finding Variance and Standard Deviation of an Expected value8. You are analyzing, Infosys Technologies (NASDAQ: INFY) stock and based on your evaluation you have come up with the following estimates of EPS for the year 20X2 and the probabilities that the firm would be able to achieve the EPS. See the table below for the estimates Probability EPS 0.35 $2.8 0.3 $2.9 0.25 $3.1 0.1 $4© Knowledge Varsity – 2011 Page 53
    • What is the variance and standard deviation of the expected EPS of INFY in 20X2?ExplanationWe have calculated the expected value in the previous example as $3.025For simplicity, we will calculate the variance using the following table Probability EPS EPS – Expected Value {EPS – E(X)}2 P * {EPS – E(X)}2 0.35 $2.8 (2.8 – 3.025) = -0.225 0.050625 0.01771875 0.3 $2.9 (2.9 – 3.025) = -0.125 0.015625 0.0046875 0.25 $3.1 (3.1 – 3.025) = 0.075 0.005625 0.00140625 0.1 $4 (4 – 3.025) = 0.975 0.950625 0.0950625 Sum 0.118875So, the variance (σ2)is 0.118875 ($2)The standard deviation (σ )= 0.1188750.5 = $0.34478This is a simple problem and there is a high probability that this would be coming in the exam.There are lots of pitfalls here, please avoid them1. Rather than first subtracting the expected value from the observation and then multiplying, manystudents they first multiply and then they subtract2. Many students do not multiply with probability but take the square of the difference and then divideby the number of occurrenceConditional Expectation – In investments, many times we expect outcome which is based on someevent (hence conditional), the expected value based on conditional probability is called as conditionalexpectation. Let’s say that the expected value of a random variable X is based on a scenario S, then theexpected value is denoted by E(X|S) , that is, X is conditional on S.Let’s say that X takes on values X1, X2, ……….Xn then the Expected Value for X is given byIf you see, the above formula looks equivalent to the Total Probability Rule. So, the above can beviewed as the Total Probability rule for Expected Value, when there are N scenarios.LOS 8.k.Diagram an investment problem using a tree diagramAn investment problem is when we are calculating the EPS, return or any other measure related toinvestment. We associate probability to those measures. So, essentially in an investment problem weare interested in finding out the expected value. Tree diagram is used to represent these problems as itbecomes very easy to solve the problems. © Knowledge Varsity – 2011
    • Concept Builder – Tree diagram for Investment Problem9. There is a probability of 0.6 for decrease in interest rate, the probability of stable interest rate is 0.3 and probability of increase in interest rate is 0.1. A firm may perform well or poorly in all these scenarios. The probability that the firm will do well when interest rate increases is 0.3 and the EPS is $2.5. The probability that the firm will do poorly when interest rate decreases is 0.7 and the EPS is $1.5 The probability that the firm will do well when interest rate is stable is 0.5 and the EPS is $3. The probability that the firm will do poorly when interest rate decreases is 0.5 and the EPS is $2 The probability that the firm will do well when interest rate decreases is 0.8 and the EPS is $4. The probability that the firm will do poorly when interest rate decreases is 0.2 and the EPS is $3.A. Draw a tree diagram to represent the above problem.B. Find out the expected EPS conditional on the firm doing wellC. Find out the expected EPS conditional on the firm doing poorD. Find out the expected EPS (that is unconditional EPS)ExplanationB. The Node 1, 3 and 5 represent the Expected EPS conditional on the firm doing well .The Expected EPS will be the probability weighted of the EPSC. The Node 2, 4 and 6 represent the Expected EPS conditional on the firm doing well .The Expected EPS will be the probability weighted of the EPSD. The expected EPS or the unconditional EPS is given asLOS 8.n.Calculate and interpret an updated probability using Bayes’ formulaWe will cover this LOS before we cover the other LOSs.Till now we have covered probabilities which are based on our expectation or historical data and werestatic. In reality, the probabilities keep on changing as and when new information becomes available.Bayes’ formula addresses the concept of updated probability. Following is the formula given for theupdated probability© Knowledge Varsity – 2011 Page 55
    • The following is the probability notationThe updated probability is also known as posterior probability.Rather than focusing on the formula, we will be covering the Bayes’ theorem using the tree diagram.Concept Builder – Bayes’ Theorem10. Going back to one of the previous problem. The probability that the Central Bank would increase the interest rate is 0.7 and the conditional probability of Recession happening when the interest rate is 0.9. Also, the conditional probability that the recession will happen if the interest rate is decreased is 0.2. You want to update your Prior probabilities with some new information a) Given that recession has happened, what is the updated probability that interest rate was increased? b) Given that recession has NOT happened, what is the updated probability that interest rate was increased? c) Given that recession has happened, what is the updated probability that interest rate was decreased? d) Given that recession has NOT happened, what is the updated probability that interest rate was decreased?ExplanationThe Tree Diagram is given below.a) Here, we know that the recession has happened; it means that either Node 1 or Node 3 hashappened. Now since we are interested in finding out the updated probability of increase in interestrate given that recession has happened. So, if you see the tree diagram, our favorable case is Node 1and the total case is Node 1 and Node 3.The probability is given by : favorable case / total case=> Updated probability = 0.63/(0.63+0.06) = 0.9130You will appreciate the fact that recession is mainly happening due to increase in interest rate andhence the updated probability is very high as compared to the prior probability.b) Here our favorable case is node 2 and total case is node 2 and node 4 © Knowledge Varsity – 2011
    • => Updated probability = 0.07/(0.07+0.24) = 0.2258c) Here our favorable case is node 3 and total case is node 1 and node 3=> Updated probability = 0.06/(0.63+0.06) = 0.087d) Here our favorable case is node 4 and total case is node 2 and node 4=> Updated probability = 0.24/(0.07+0.24) = 0.7742If you take the sum of probabilities found in a) and c) it would come to 1. This implies that theprobability of recession happening is 1 (As we already know that recession has happened)LOS 8.k.Calculate and interpret covariance and correlationVariance measures the dispersion of a single random variable. Many times during investment we areinterested in knowing how two variables move with respect to each other. For example, we might beinterested in knowing if the S&P 500 is increasing what would be the behavior of Microsoft stock,whether it would increase or decrease.To understand the movement of 2 variables we have Covariance and Correlation as the measures.Covariance – Covariance measures how 2 variables move with respect to each other. The formula forthe covariance is given belowSo, covariance is the expected value of the product of deviation of the two random variables from theirrespective expected value (can be thought of as mean).Properties of Covariance Covariance values ranges from negative infinity to positive infinity The covariance of a variable X with itself is the variance of X If there is no relation between the assets, then the covariance is 0. The unit of covariance is squared unitsDrawback – Covariance is not able to measure the strength of the relationship. Higher Covariance doesnot necessarily means that the strength of the relationship is high. As a result we have another measureknown as Correlation.Correlation Coefficient – It measures the strength of the relationship. Higher correlation value implieshigher strength. The formula for correlation coefficient is given byProperties of Correlation Coefficient Correlation coefficient measures only strength of Linear Relationship Correlation coefficient is unit less Correlation ranges between -1 and +1 If there is no linear relation between the assets, then the correlation coefficient is 0. If the correlation coefficient is +1, then the variables are Perfectly Positively Correlated If the correlation coefficient is +1, then the variables are Perfectly Negatively CorrelatedMost of the time we would be using covariance and correlation from investment point of view, so thereturns are used as the random variable.© Knowledge Varsity – 2011 Page 57
    • Concept Builder – Covariance and Correlation11. You are analyzing two stocks and are trying to find out the expected return. You are predicting that the economy can be in either good state or bad state. Following table illustrate the return and the probabilities that you are expecting from the two stocks. State of the economy Probability Return of A Return of B Good 0.7 15% 20% Bad 0.3 10% 5%ExplanationFirst we will have to find out the expected returns of A and BE(RA) = 0.7 *15% + 0.3*10% = 13.5%E(RB) = 0.7 *20% + 0.3*5% = 15.5%The following table has the calculation for Covariance Probability Return of A Return of B RA – E(RA) RB – E(RB) P * {RA-E(RA)} * {RB-E(RB)} 0.7 15% 20% 1.5% 4.5% 4.725 0.3 10% 5% -3.5% -10.5% 11.025 Sum 15.75So, the covariance is 15.75 %2 or 0.001575For finding out the Correlation we need to find the standard deviation of A and BThe standard deviation is computed in the following tables Probability Return of A RA – E(RA) P * {RA-E(RA)} * {RA-E(RA)} 0.7 15% 1.5% 1.575 0.3 10% -3.5% 3.675 Sum 5.25So, the variance of A = 5.25%2 => standard deviation = 2.2913 % Probability Return of B RB – E(RB) P * {RB-E(RB)} * {RB-E(RB)} 0.7 20% 4.5% 14.175 0.3 5% -10.5% 33.075 Sum 47.25So, the variance of B = 47.25%2 => standard deviation = 6.8739 %So, the correlation coefficient will be given bySo, here we see that the correlation coefficient is the maximum.LOS 8.l.Calculate and interpret the expected value, variance, and standard deviation of a randomvariable and of returns on a portfolioA portfolio of assets will also have expected return and variance similar to the assets itself. The onlything is that we observe the benefit of diversification in case of portfolio. By diversification we meanthat we are including more assets in the portfolio. If there are 2 assets in a portfolio and the correlationcoefficient between them is less than 1, then the standard deviation of the portfolio is lesser than theweighted standard deviation of the individual assets.Asset Weight in a portfolio - The weight of any asset in the portfolio is given by the following formulaExpected Return of Portfolio – The expected return of the portfolio is the weighted average of theexpected return of the individual asset. © Knowledge Varsity – 2011
    • Variance of the Portfolio – There is a complex formula for the variance of a portfolio having n assets.For level 1 we are required to remember the variance of 2 asset portfolio.The variance of a 2 asset portfolio is given byIn terms of covariance, the formula can be written asConcept Builder – Portfolio Variance12. A portfolio is made up of 2 assets A and B. The current market value of investment in A is $600 and current market value of investment in A is $400. Correlation between A & B is 0.3. The standard deviation of A is 20% and standard deviation of B is 30%. Find out the portfolio standard deviation?ExplanationFirst find out the weight of each asset in the portfolio.Weight of asset A = 600/(600+400) = 0.6 => Weight of B = 0.4We will use the 2 asset, standard deviation formula.=> = 374.4 2The variance of the portfolio is 374.4 %The standard deviation will be square root of the variance = 19.3494%If, you calculate the weighted average standard deviation it will be 24%So, you are seeing that the standard deviation of the portfolio is lesser than the weighted average, thisis the benefit of diversification, which we had discussed.LOS 8.m.Calculate and interpret covariance given a joint probability functionJoint Probability Table: This is a table in which we mention the joint probabilities of the two variablesfor a specified set of outcomes. The Joint probability is very neat way to denote the probabilities ofoccurrence of return of 2 investments.Below is an example of a Joint Probability table Asset RB = x% RB = y% RB = z% RA = a% A1 A2 A3 RA = b% A4 A5 A6 RA = c% A7 A8 A9Some Observations from the joint probability table Here P1 means the probability of A returning a% and B returning x% The total probability of A returning a% will be the some across the row which will be A1 + A2 + A3 The total probability of B returning x% will be the some across the column which will be A1 + A4 + A7 The sum of all the Ai should be equal to 1; that comes from the probability property.Correlation Matrix – In a correlation matrix the correlation between the variables is mentioned. Theusage of matrix is a very clear representation of the correlation and is widely used.Below is an example of correlation matrix Asset A B C A 1 B 1 C 1© Knowledge Varsity – 2011 Page 59
    • You will find that the lower part of the matrix is redundant, because the correlation coefficient betweenA and B is same as the correlation coefficient between B and A.So, in many places you will find that only the top part is mentioned.Covariance Matrix – In covariance matrix, the covariance between the random variables is mentioned.Below is an example of covariance matrix. Asset A B C A Cov (RA, RA) = Var(A) Cov (RA, RB) Cov (RA, RC) B Cov (RB, RA) = Cov (RA, RB) Cov (RB , RB) = Var(B) Cov (RB, RC) C Cov (RC, RA) Cov (RC , RB) Cov (RC, RC) = Var(C)Like correlation matrix, here also you will find that the lower part of the matrix is redundant.LOS 8.o.Identify the most appropriate method to solve a particular counting problem, and solvecounting problems using the factorial, combination, and permutation notationsThere is a very low probability of this section coming in the examinationWe will be covering 3 type of formula here1. Labeling Formula – The number of ways in which n object can be labeled with k different labels, with n1 of first type, n2 of second type and so on. n1 + n2 + ….. +nk = 1 The formula is given by2. Selection Or Combination formula – This is used when we are selecting from a group. It is a special case of labeling formula. Here the order of the selection doesn’t matter. For example if we are selecting 6 players from 9 players, it doesn’t matter who gets selected first. The formula is given as or = Where n is the total number of object and we need to select r object from it. So, in the example above, n = 6 , r = 9, so the number of ways = 843. Permutation – This formula is used when arrangement of the items selected is also important. For example in a cricket match, let’s say that the player who gets selected first will be the first one to bat, hence here the order of selection is important. The formula is given as = As an example, if there are 9 players and we need to select 6 players and the order of selection is important, then the number of ways in which it can happen = 60,480Concept Builder – Counting Problem13. Your portfolio has 10 mutual funds, a. You met an investment advisor and he advised to hold 6 mutual funds and sell 4 mutual funds. In how many ways this can be achieved? b. In how many ways you can select 3 mutual funds from the 10 funds c. In how many ways you can sell 3 mutual funds, where the order of selling is importantExplanationa. The total number of ways in which we can arrange 10 mutual funds is 10!. The total no of ways in which we can arrange 6 and 4 mutual fund is 6! And 4! respectively. Here the sequence doesn’t matter, so we are doing double counting by arranging the 6 and 4 mutual funds, which have to be removed. © Knowledge Varsity – 2011
    • Hence, the number of ways = 10!/(6! * 4!) = 210 Alternatively, we can directly use the labeling formula, because we are labeling 4 mutual funds as sell and 6 funds as hold.  the number of ways = 10!/(6! * 4!) = 210b. This is a selection problem, the number of ways in which we can select 3 mutual funds is given by 10C3 = 120c. Here the order is important. So we will use permutation No of ways = 10P3 = 720© Knowledge Varsity – 2011 Page 61