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# Financial maths

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Introduction to financial maths. 6th Years please use in conjunction with the questions given on Friday.

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• Draw attention to the compounding period and the number of periods After 12 months F= P(1+i) n F= 100(1+.00287) 12 € 103. 50 Hence the annual equivalent rate which takes compounding into account is 3.5% 0.287% x12= 3.44%. This is the APR ( we have not specified any charges in this case). It is less than the AER as it does not include compounding. Instead of leaving the money on deposit for 1 year – how much would it earn if left on deposit for 15 months? After 15 months F= P(1+i) n F= 100(1+.00287) 15 € 104.39
• Same as (1.035) ^1.25
• ### Financial maths

1. 1. Financial Maths <ul><li>Leaving Cert. Hon </li></ul><ul><li>Ms. Carter </li></ul>
3. 3. What do you notice? <ul><li>Different methods of calculating interest. </li></ul><ul><li>Government have rules about what information must be provided in advertisements for financial products and in the agreements that businesses make with their customers. </li></ul>
4. 4. Getting Money! <ul><li>Where can we get money? </li></ul><ul><li>What is it called when we borrow money? </li></ul>CREDIT!
5. 5. Financial Institutions Banks Credit Unions Financial Companies Pawn Brokers Charge Accounts Illegal Lenders
6. 6. Types of Credit <ul><li>Overdrafts </li></ul><ul><li>Credit Cards </li></ul><ul><li>Personal Loans </li></ul><ul><li>Credit Union Loans </li></ul><ul><li>Hire Purchase </li></ul><ul><li>Credit Sale Agreements </li></ul><ul><li>Top-up Mortgages </li></ul><ul><li>Moneylending </li></ul><ul><li>Store Cards </li></ul>- its a way of borrowing on your bank account. - allows you to borrow money on a monthly basis. - these loans are suitable for medium to longer term needs. - you have to be living or working in the area. - non flexible, ‘ballon payment’, exspensive. - immediately own item, pay over time. - extending your mortgage to consolidate debts. - companies or individuals who lend at a rate of 23% or higher. Repayments geberally collected in cash each week. - treated like a credit card. Limit of use in store.
7. 7. Irish Credit Bureau <ul><li>Irelands biggest credit referencing agency. </li></ul><ul><li>Not a state body - its owned and ran by ICB members, which are mainly financial institutions. </li></ul><ul><li>Electronic library/database that contains information on the performance of credit agreements between financial institutions and borrowers (the citizen). </li></ul><ul><li>Lending institutions register information with the ICB generally on a monthly basis. </li></ul><ul><li>ICB give you a Credit Score based on credit history. </li></ul><ul><li>All records remain on the database for 5 years.* </li></ul>
8. 8. Comparing Credit Cards. AIB mc2 Student MasterCard Bank of Ireland Student Credit Card
9. 9. Introduction <ul><li>Financial services use different terms for the interest you are charged or earn on their financial products. </li></ul><ul><li>The four most common terms are: </li></ul>- Annual Percentage Rate (APR) - Equivalent Annual Rate (EAR) - Annual Equivalent Rate (AER) - Compound Annual Return (CAR) When borrow When save
10. 10. APR - Annual Percentage Rate <ul><li>Real cost of borrowing to the consumer. </li></ul><ul><li>Defined as “ being the total cost of credit to the consumer expressed as an annual percentage of the amount of credit granted”. </li></ul><ul><li>In case of loans and other forms of credit, there is a legal obligation to display the APR prominently. </li></ul><ul><li>Clear rules in legislation on how APR is calculated ( Section 21 of the Consumer Credit Act, 1995) . </li></ul><ul><li>Term is not used the same way in all countries. </li></ul>
11. 11. <ul><li>Concerned with only Irish meaning, where its governed by Irish and European law. </li></ul><ul><li>The APR is calculated each year on the declining principal (amount outstanding) of a loan. </li></ul><ul><li>The interest rates are set out by the European Central Bank and can change daily. </li></ul><ul><li>APR is calculated each year on the declining principal of a loan. That is the amount you still owe, not the original amount you borrowed. </li></ul>
12. 12. 3 Key Features 1. All the money that the customer has to pay must be included in the calculation - the loan repayments themselves, along with any set-up charges, additional unavoidable fees,etc. 2. The definition states that the APR is the annual interest rate (expressed as a percentage to at least one decimal place) that makes the present value of all of these repayments equal to the present value of the loan. 3. In calculating these present values, time must be measured in years from the date the loan is made.
13. 13. Additional Notes: <ul><li>If a credit rate is not APR, then it may be referred to as “nominal rate” or “headline rate”. </li></ul><ul><li>Less relevant now as illegal not to say APR in ad’s for a loan or credit. </li></ul>Nevertheless lets look at an example! If a loan or overdraft facility is governed by a charge of 1% per month calculated on the outstanding balance for that month, that might have been considered to be “nominally” a 12% annual rate, calculated monthly. However, it is actually an APR of 12.68%, (since €1 owed at the start of a year would become (1.01)¹²=€1.1268 by the end of the year).
14. 14. <ul><li>APR is reserved for use when the customer is borrowing from the service provider so in the opposite case, where the customer is saving or investing money, the comparable term is the EQUIVALENT ANNUAL RATE (EAR). EAR applies to deposits aswell as overdrafts. EAR calculates the interest as if it is paid once a year, even if it is paid twice or three times per year. </li></ul><ul><li>Often referred to as Annual Equivalent Rate (AER) or Compound Annual Return/Compound Annual Rate (CAR). </li></ul><ul><li>In Ireland all these terms mean the same thing. </li></ul><ul><li>The rules governing their use in advertising are not as clearly specified as in the law governing the use of APR. </li></ul><ul><li>In the case of investments that do not have a guaranteed return, the calculation of EAR often involves estimates of future growth. </li></ul><ul><li>Despite the differences the method of calculation is the exact same as is the case with the APR. </li></ul>Savings and Investments
15. 15. Example. <ul><li>If a financial institution quotes an interest rate of 4% per year compounded every 6 months. We call this 4% the 'Nominal Rate'. This means that the financial institution pays 2% compound interest every 6 months. The interest paid at the end of 6 months, actually earns interest for the second 6 months of the year. For this reason, 4% compounded every 6 months, is not the same as 4% compounded annually. </li></ul><ul><li>You invest €500 with your financial institution at a rate of 4% each year, compounded every 6 months. </li></ul><ul><ul><li>Time Period Interest Accumulated Value </li></ul></ul><ul><ul><li>After 6 months €10 €510 </li></ul></ul><ul><ul><li>After 12 months €10.20 €520.20 </li></ul></ul><ul><li>The 410 interest for the first 6 months is simply €2% of 500. This is then added to the initial investment to give a running total of €510. The interest for the second 6 months of the year €10.20 is 2% of €510. The effective annual interest rate is therefore 20.20 / 500 x 100 = 4.04%. </li></ul>
16. 16. Examples.
17. 17. <ul><li>Bank of Ireland is currently offering a 9 month fixed term reward account paying 2.55% on maturity, for new funds from €10,000 to €500,000. (That is, you get your money back in 9 months time, along with 2.55% interest.) Confirm that this is, as advertised, an EAR of 3.41% </li></ul>Question 1
18. 18. Solution <ul><li>For every euro you put in you are getting €1.0255 in ¾ of a year’s time. At 3.41%, the present value of this return is €1.0255/(1.0341⁰.⁷⁵) = €1, which is as it should be. (Alternatively, just confirm that 1.0341^¾ = 1.0255, or that 1.0255^(4/3) = 1.0341. </li></ul>
19. 19. Example 2 <ul><li>The Government’s National Solidarity Bond offers 50% gross return after 10 years. Calculate the EAR for the bond. </li></ul>
20. 20. Solution <ul><li>(1 + i)^10 = 1.5 => i = 0.041379...=> EAR = 4.1% (as advertised) </li></ul>
21. 21. When do we use Geometric Sequences? <ul><li>for Regular Repayments or Savings over time. </li></ul><ul><li>E.g. term loan or mortgage. </li></ul><ul><li>-> Calculations involving such regular payment schedules, when they are considered in terms of the present values of the payments, (or the future values,) will involve the summation of a geometric series. </li></ul>
22. 22. Annuity <ul><li>Any regular stream of fixed payments over a specified period of time is technically referred to as an annuity . </li></ul><ul><li>Can be used with slightly different meaning, such as a regular pension payment that lasts as long as the person is alive. </li></ul><ul><li>Annuity Mortgage far most popular, involves paying a fixed amount and some interest over a long period of time. </li></ul>
23. 23. Amortization <ul><li>A loan that involves paying back a fixed amount at regular intervals over a fixed period of time is called an amortized loan . </li></ul><ul><li>When the regular payments are being used to pay off a loan , then we are usually interested in calculating their present values, because this is the basis upon which the loan repayments and/or the APR are calculated. </li></ul><ul><li>When the regular payments are being used for investments, we may instead be interested in their future values, this tells us we can expect to have when the investment matures. </li></ul>
24. 24. Regular Savings A/C & Similar Investments <ul><li>any regular payment over time will give rise to a geometric series, irrespective of whether its purpose is to repay a loan or to generate savings for the future. </li></ul><ul><li>In the case of savings and investments, we are generally interested in the future value rather than the present value. </li></ul>
25. 25. Annuities <ul><li>An annuity is a form of investment involving a series of periodic equal contributions made by an individual to an account for a specified term. </li></ul><ul><li>Interest may be compounded at the end or the beginning of each period. </li></ul><ul><li>Can also be used for case of regular payments paid to an individual, such as a pension. </li></ul><ul><li>When receiving payments from an annuity the present value of the annuity is the lump sum that must be invested now in order to provide those regular payments over the term. </li></ul><ul><li>Comes from annuity meaning yearly! </li></ul>
26. 26. Examples of annuities: <ul><li>Monthly rent payments </li></ul><ul><li>Regular deposits in a savings account </li></ul><ul><li>Social welfare benefits </li></ul><ul><li>Annual premiums for a life insurance policy </li></ul><ul><li>Periodic payments to a retired person from a pension fund </li></ul><ul><li>Dividend payments on stocks and shares </li></ul><ul><li>Loan repayments </li></ul>
27. 27. <ul><li>The future value of an annuity is the total value of the investment at the end of the specified term. This includes all payments deposited as well as the interest earned. </li></ul><ul><li>Note: a BOND is a certificate issued by a government or a public company promising to repay borrowed money at a fixed rate of interest at a specified time. </li></ul>
28. 28. Amortization formula (pg 31 tables) associated terms Present Value is the value on a given date of a future payment or series of future payments discounted to reflect the time value of money and other factors such as investment risk. An annuity is a series of equal payments or receipts that occur at evenly spaced if intervals. The payment of receipts occurs at the end of each period for an ordinary annuity. An amortized loan is a loan for which the loan amount plus interest is paid off in a series of regular payments. (2 types – add on interest loan and a simple interest amortized loan for which the payments are smaller than for the former) A simple interest amortized loan is an ordinary annuity whose future value is the same as the loan amount’s future value, under compound interest. A simple interest amortized loan’s payments are used to pay off a loan where other annuities payments are used to generate savings as for example retirement funds.
29. 29. We can think of the situation in two ways which give the same end result: 1) The sum of the present values of all the annual repayment amounts = sum borrowed. (This principle is enshrined in European Law) 2) Future value of loan amount = Future value of the annual repayment amounts(i.e. future value of the annuity )
30. 31. Red = Principal Blue = Interest No. of Repayments Monthly Repayment
31. 32. SEC Sample Paper 2011 Q. 6 Q. 6 Padraig is 25 years old and is planning for his pension. He intends to retire in forty years’ time, when he is 65. First, he calculates how much he wants to have in his pension fund when he retires. Then, he calculates how much he needs to invest in order to achieve this. He assumes that, in the long run, money can be invested at an inflation adjusted annual rate of 3%. Your answers throughout this question should therefore be based on a 3% annual growth rate. (a) Write down the present value of a future investment of €20000 in one years’ time. Log Tables: Which formula is to do with preset value? A : P = 20000/1.03= €19,417.48 (b) Write down in terms of t, the present value of a future payment of €20000 in t years’ time. A : P=20000/(1.03) t
32. 33. (c) Padraig wants to have a fund that could, from the date of his retirement, give him a payment of €20,000 at the start of each year for 25 years. Show how to use the sum of a geometric series to calculate the value on the date of retirement of the fund required. A: The amount of money in the fund on the date of retirement = sum of the present values of all the payments on the date of retirement retirement. Present value of the first payment = 20000 Present value of the second payment i.e. €20000 due in 1 year’s time = 20000/(1.03) Present value of the third payment i.e. €20000 due in 3 years = 20,000/(1.03) 2 Present value of the last payment = 20000/(1.03) 24 The value on the date of retirement of the fund required = 20000+20000/(1.03)+ 20,000/(1.03) 2 +............................................20000/(1.03) 24 Fund =20000(1+1/1.03+1/(1.03) 2 +1/(1.03) 2 +.....................................1/(1.03) 24 ) Fund =20000(Sn of a geometric series with n =25, r = 1/1.03, a = 1) = 20000(a (1- r n )/(1-r))= 20000(1(1-1/1.03) 25 )/(1-1/1.03) Fund = €358,710.84 (note- less than 20000*25 which could be an initial rough estimate)
33. 34. 6(d)(i) Padraig plans to invest a fixed amount of money every month in order to generate the fund calculated in part(c). His retirement is (40x12)480 months away. Find correct to four significant figures the rate of interest per month that would, if paid and compounded monthly, be equivalent to an effective annual rate of 3%. A: (1+i) 12 =1.03 (1+i) = 1.03 1/12 = 1.002466 i= rate of interest per month = 0.002466 = 0.2466% 6(d)(ii) Write down in terms of n and P, the value on the retirement date of a payment of €P made n months before retirement date. A: FV of payment of €P paid n months before retirement date is P(1+.002466) n
34. 35. 6(d)(iii) If Padraig makes 480 equal monthly payments of €P from now until his retirement, what value of P will give the fund he requires? A: The FV(annuity) = 358,710.84 Let i = .002466 €P = monthly contribution € 358,710.84 = P(1+i) 480 .................................... + P(1+i) 2 + P(1+i) = P(1+i)+ + P(1+i) 2 .................................... P(1+i) 480 (reversing the order) = P( (1+i) +(1+i) 2 +..........................................(1+i) 480 ) =P ( S n of a geometric series with n=480, a = 1+i,r = (1+i)) [S n = (a(r n -1))/(r-1)] =P (1.002466((1.002466) 480 -1)/ (0.002466)) =P (919.38) Monthly contribution P = €358,710.84/€919.38 = €390.17
35. 36. 6(e) If Padraig waits for ten years before starting his pension investments, how much will he then have to pay each month in order to generate the same pension fund? The FV(annuity) = 358,710.84 = P (1.002466((1.002466) 360 -1)/ (0.002466)) 358,710.84 = P(580.11) P = €618.35 NOTE: We are assuming above as in part (d) that the payment is made at the beginning of each payment period. If the payment was made at the end of each payment period, the answer would be as below. A: The FV(annuity) = 358,710.84 = P((1+0.002466) 360 -1))/(0.002466) 358,710.84 = P(578.68)
36. 37. Financial Mathematics <ul><li>€ 100 earns 0.287% per month compound interest. </li></ul><ul><li>What is its final value after 1 year? </li></ul><ul><li>If interest was added annually what is the annual equivalent rate? </li></ul>Your turn! <ul><li>Solution </li></ul><ul><li>(Note: Match the compounding period with the number of periods) </li></ul><ul><li>After 12 months F = P(1+i) n </li></ul><ul><li>F = 100(1+.00287) 12 = 100(1.034988) =100 (1.035) </li></ul><ul><li>= €103. 50 </li></ul><ul><li>Hence the annual equivalent rate (AER) = 3.5% ( correct to 1 d.p.) </li></ul>
37. 38. <ul><li>The €100 is left on deposit for 15 months at 0.287% per month compound interest. </li></ul><ul><li>Calculate the final value. Give the answer to the nearest 10 c. </li></ul><ul><li>What is the interest rate for the 15 months? </li></ul><ul><li>What is this interest rate called? </li></ul><ul><li>Solution: </li></ul><ul><li>After 15 months F = P(1+i) n </li></ul><ul><li>F = 100(1+.00287) 15 </li></ul><ul><li> = €104.39 </li></ul><ul><li>This means the interest earned is 4.4 % ( correct to 1 dp) </li></ul><ul><li>This is the Gross interest rate . </li></ul><ul><li>It is the percentage the total interest is of the initial investment. </li></ul>Financial Mathematics Your turn!
38. 39. Using amortized loans. <ul><li>Sean borrows €10,000 at an APR of 6%. He wants to repay it in ten equal instalments over ten years, with the first repayment one year after he takes out the loan. How much should each repayment be? </li></ul>
39. 40. Solution: <ul><li>Let each repayment equal A. Then the present value of the first repayment is A/1.06, the present value of the second repayment is A/10.06², and so on. Therefore, the total of the present values of all repayments is </li></ul>