Chapter 5-time valueofmoney (1)
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Time Value of Money.
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Chapter 5-time valueofmoney (1)
- 1. Financial Management FIN 3300 Weeks 2 and 3
- 2. Corporate Finance FIN 3300 Time Value of Money 2
- 3. Review What are the purposes of following types of ratios? • Market value • Profitability • Leverage • Liquidity • Efficiency FIN 3300 Time Value of Money 3
- 4. Time Value of Money Future values – compounding Present values – discounting Multiple cash flows Perpetuities Annuities Interest rates FIN 3300 4
- 5. Future Value Equation: FV = PV ( 1 + r ) t • FV = future value • PV = present value • r = interest rate, discount rate or cost of capital per period • t = number of time periods FIN 3300 5
- 6. Future Value Examples What will $100 be worth in one year, assuming you can invest at 2% interest per year? What will $100 be worth in five years, assuming you can invest at 2% interest per year (assume interest reinvested - compounding)? FIN 3300 6
- 7. Present Value FV Equation: PV = (1+r) t • FV = future value • PV = present value • r = interest rate, discount rate or cost of capital per period • t = number of time periods FIN 3300 7
- 8. Present Value Examples If you will receive $102 in one year, what is it worth to you today? Assume you can invest now at 2% interest per year (opportunity cost of capital). If you will receive $110.41 in five years, what is it worth to you today assuming 2% interest per year (assume interest reinvested)? FIN 3300 8
- 9. More Problems Implied interest rates: You buy a new recliner and can pay $600 now or $800 in one year. How should you pay if you can get a one year 25% loan? Internal rate of return: (compound annual growth rate - CAGR): What is the internal rate of return if you invest $100 and get back $1,000 in ten years? FIN 3300 9
- 10. More Problems Time needed to save: If you have $1,000 now and want to put it in a savings account to grow to $2,000, how many years will you need to wait assuming you get 2% interest per year? Comparing future cash flows: Which is worth more, $1,500 in 1 year or $2,379 in 5 years? Let r = 12%. FIN 3300 10
- 11. Another Problem The value of free credit: You have the option of paying for a new one-wheeled motorcycle with cash now for $10,000 or pay $12,500 in two years. If a car loan would cost you 12% interest per year, should you pay now or in two years? FIN 3300 11
- 12. Present Value of Multiple Cash Flows C1 C2 Equation: PV = ( 1 +r ) 1 + ( 1 +r ) 2 + .... • PV = present value • C = future cash flow in one period 1 • C = future cash flow in two periods 2 • r = interest rate, discount rate or cost of capital per period FIN 3300 12
- 13. Present Value of Multiple Cash Flows Example Draw time-lines to organize cash flows Discount each cash flow separately Find the combined present value of getting $1,000 in one year and $1,500 in two years if r = 10%? FIN 3300 13
- 14. Multiple Cash Flows Problem Choose the less expensive option to buy a car if your cost of money is 8%: • Pay $15,500 cash now • Pay $8,000 now and $4,000 at the end of each of the next two years FIN 3300 14
- 15. Perpetuity A stream of level cash payments that starts one period into the future and never ends. Equation: C PV = r • PV = present value • C = periodic cash payment • r = discount rate or interest rate per period FIN 3300 15
- 16. Perpetuity Example To create an endowment for a new charity which will pay $100,000 per year forever starting next year, how much money must you invest today if the interest rate will be 10%? FIN 3300 16
- 17. Perpetuity Example (continued) If you need the first perpetuity payment to start today, how much money do you invest now? If you need the first perpetuity payment to start in three years, how much money do you invest now? • This is a delayed perpetuity and is covered further at the end of this presentation. FIN 3300 17
- 18. Annuity A stream of level cash payments that starts one period into the future and continues for t periods. 1 1 Equation: PV = C − r r( 1 + r) t • PV = present value • C = periodic cash payment • r = discount or interest rate per period • t = number of payment periods FIN 3300 18
- 19. Annuity Example You purchase a TV by paying $1,000 per year at the end of the next three years. What is the price you are paying if the interest rate is 10%? FIN 3300 19
- 20. Annuity Problems You plan to save $4,000 every year for 20 years and then retire. Given a 10% interest rate, what will be the value of your savings at retirement? FIN 3300 20
- 21. Annuity Problems You purchase a $200,000 condominium with 100% financing over 30 years at 10% interest per year. If you make annual payments, what will they be? FIN 3300 21
- 22. Interest Rates Simple and compound interest Annual percentage rate (APR) Effective annual interest rate (EAR) Inflation: nominal and real interest rates FIN 3300 22
- 23. Simple and Compound Interest Simple • Interest earned only on original investment • No interest on interest • Invest $100 at 16% simple interest and have $132 after two years Compound • Interest earned on interest by reinvesting • Time value of money method • Invest $100 at 16% compounded interest and have $134.56 after two years FIN 3300 23
- 24. Annual and Effective Interest Rates Many interest rates are expressed as daily or monthly interest rates • To compare rates over one year, we annualize them Annual percentage rate (APR) • Interest rate that is annualized using simple interest • APR = (periodic rate) x (number of periods in a year) Effective annual interest rate(EAR) • Interest rate that is annualized using compound interest • EAR = (1 + periodic rate)(number of periods in a year) - 1 FIN 3300 24
- 25. APR and EAR Examples Find the APR and EAR for a 2% monthly interest rate. What is the EAR for a car loan requiring quarterly payments at an 8% APR? FIN 3300 25
- 26. Interest Rates and Inflation Inflation rate • Rate at which prices of goods increase • Consumer price index (CPI) Nominal interest rate • Rate at which an investment grows Real interest rate • Rate at which the purchasing power of an investment grows Equation: (1 + real interest rate) = (1 + nominal interest rate) (1 + inflation rate) real interest rate ≈ nominal interest rate – inflation rate FIN 3300 26
- 27. Interest Rate Example If the interest rate on one year government bonds is 5.9% and the inflation rate is 3.3%, what is the real interest rate? FIN 3300 27
- 28. Inflation History 100 Years of Inflation 20 15 Annual Inflation % 10 5 0 1900 1920 1940 1960 1980 2000 -5 -10 -15 FIN 3300 28
- 29. Nominal versus Real for Time Value of Money Problems Normally use nominal cash flows with nominal interest rates If you need to use real data, use real cash flows with real interest rates Both should give the same answer if done properly FIN 3300 29
- 30. Delayed Perpetuity A stream of level cash payments that starts “t” periods into the future and never ends. Equation: C 1 PV = (1 + r ) n −1 r • PV = present value • C = periodic cash payment • r = discount rate or interest rate per period • n = number of periods until first payment FIN 3300 30
- 31. Delayed Perpetuity Example To create an endowment for a new charity which will pay $100,000 per year forever starting in three years, how much money must you invest today if the interest rate will be 10%? FIN 3300 31
- 32. Delayed Annuity A stream of level cash payments that starts “n” periods into the future and continues for t periods. 1 1 1 Equation: PV = C − r t n −1 r (1 + r ) (1 + r ) • PV = present value • C = periodic cash payment • r = discount or interest rate per period • t = number of payment periods • n = number of periods until first payment FIN 3300 32
- 33. Delayed Annuity Example Starting in 3 years, you will $4,000 every year for 17 years. Given a 10% interest rate, what is the present value? FIN 3300 33
Editor's Notes
- Today we will briefly review ratio analysis and then introduce the time value of money.
- Ratio analysis does not necessarily tell us if a company is good or bad. Its real value comes in using the ratios to compare companies within the same industry. Then we can ask why one company has better ratios than the other. Market value – compare how expensive different companies are to one another Profitability – compare how profitable different companies are to one another Leverage – how much debt a company has Liquidity – how much cash a company has or how easily it can pay debt in near future Efficiency – how well does a company manage its cash cycle – how quickly it pays its bills and how long its customers pay them
- Future values are value some time in the future of a dollar today after it is invested and all interest is reinvested – this is compounding The present value is the value you must invest today to get a certain amount of money in the future assuming compound interest – this is discounting Given multiple future cash flows, we can find one combined present value for all of them Perpetuities are a stream of the same cash flow every period that lasts forever Annuities are a stream of the same cash flow every period that last for certain amount of time Interest rate are the percentage income we receive by investing our money
- This is the fundamental equation for future value Think of it as pv equals your initial investment, while r is your interest rate in decimal form and t is the amount of time you invest the money. At the end of your investment you are left with a given amount of money called fv. The equation assumes that the interest you earned every period was reinvested. Note that for these problems that the interest is per periods – like 5% per year, and that the number of periods must be in the same unit of measurement – years if our interest is compounded per year.
- For the first example: pv =100, t = 1, r = .02 Fv=100(1+.02)^1 = 102 Second example: pv=100, t=5, r=.02 Fv = 100(1+.02)^ = 110.41
- Think of this equation answering the question – how much do I invest today to get a given future value
- First example: fv = 102, t=1, r = .02 Pv = (102)/[(1+.02)^1] = 100 Next example: fv=110.41, t=5, r=.02 Pv = (110.41)/[(1+.02)^5] = 100
- We can also use the fv and pv value equations to solve for r or t. Example 1: you are given two options to pay for the recliner. The problem is that the dollar options are given in different time periods. To answer the question we need to make an apples to apples comparison of the two options – we can put them into present value terms to do this. option 1: 600 today is already in present value terms options 2: 800 in one year needs to be put into present value fv=800, t=1, r=.25 pv=800/[(1+.25)^1] = 640 Now we can directly compare the two options and can see that option 1 at 600 is cheaper than option 2 of 640 Example 2: In this example we solve for r Pv = 100, fv=1000, t=10 Fv = pv(1+r)^t Using algebra we can solve for t first divide both sides by pv: Fv/pv = pv(1+r)^t/pv => fv/pv = (1+r)^t so the pv’s cancels out on the right side next take both sides to the 1/t power: (fv/pv)^(1/t) = [(1+r)^t]^(1/t) => (fv/pv)^(1/t) = 1+r so the exponents cancel out on the right side now just solve for r r = CAGR = (fv/pv)^(1/t) – 1 r = (1000/100)^(1/10) – 1 = 1.259 – 1 = .259 = 25.9%
- Example 1: we can also solve for t in the fv equation Fv = pv(1+r)^t first divide both sides by pv: fv/pv = (1+r)^t next take the natural log of both sides: ln(fv/pv) = ln[(1+r)^t] by the natural log rules we can convert this to: ln(fv/pv) = t x ln(1+r) now we solve for t: t = ln(fv/pv) / ln(1+r) = ln(2) / ln(1.02) = 35 Example 2: to compare different future values, let’s put them both into present value terms: 1500 in 1 year: pv = 1500/(1.12) = 1250 2379 in 5 years: pv = 2379/[(1.12)^5] = 1349.9 >>>> this is worth more
- We did a similar example to this already, let’s try the previous method again. Put both payment options into present value so we can directly compare them. Option 1: 10,000 today is already in pv Option 2: fv=12500, t=2, r=.12, thus pv = 12500/(1.12^2) = 9965 >>>>> thus option 2 is cheaper
- We can take the present value of multiple future values and combine them all together
- A great strategy to solve multiple cash flow problems is to draw each cash flow separately on a timeline that starts at 0 which is the present value 0 1 2 3 years |------------|--------------|----------------|------------ 1000 1500 We can see that the 1000 must be discounted back 1 year to get to year 0 which is present value 1500 must be discounted back 2 years to get to present value This method will be especially useful when future cash flows become more complicated Thus pv = 1000/1.1 + 1500/(1.1^2) = 909.09 + 1239.67 = 2148.76
- To solve for the pv of the second option the timeline of cash flows is: 0 1 2 3 years |------------|--------------|----------------|------------ 8000 4000 4000 The 8000 is already in present value terms, so we do not have to discount it back. Thus pv = 8000 + 4000/1.08 + 4000/(1.08^2) = 8000 + 3703.7 + 3429.35 = 15,133 Thus choose the second option since 15133 < 15500
- How does this look on the timeline? 0 1 2 3 years |------------|--------------|----------------|------------ 100,000 100,000 100,000 -> payments go on forever Use the perpetuity formula to solve this problem Pv = C/r = 100,000/.1 = 1,000,000
- For example 1, how does the timeline look? 0 1 2 3 years |------------|--------------|----------------|------------ 100,000 100,000 100,000 100,000 -> payments go on forever Notice how the first payment is already in present value terms, so we do not have to do anything to it. The other payments fit the definition of a perpetuity, so we can use the perpetuity formula on them Pv = 100,000 + C/r = 100,000/.1 = 100,000 + 1,000,000 = 1,100,000 For example 2, the timeline is: 0 1 2 3 years |------------|--------------|----------------|------------ 100,000 -> payments go on forever The difficult part of this problem is that the payments start in 3 years. Our perpetuity formula assumes that the payments start in 1 year. How do we deal with this? We can use a little time value of money trick. If we first use the perpetuity formula it will give us a value of the perpetuity one year before the first payment. Thus it will give us a perpetuity value at year 2. Remember the perpetuity formula just assumes the payments start in one period, so no matter where the payment start, if we use the formula on them, the value of the perpetuity will be at one period before the first payment. Value of perpetuity at one period before first payment = C/r = 100,000/.1 = 1,000,000 0 1 2 3 years |------------|--------------|----------------|------------ 1,000,000 So how do we get this value to the present? Just take present value of 1,000,000 in 2 years Pv = 1000000/(1.1^2) = 826,446
- In finance, we calculate the present value of cash flows, because we want to know what something is worth. The price you pay should be equal to or less than what it is worth. If you pay less than what it is worth you will make extra profit. To calculate what something is worth and what we are willing to pay today, we calculate its present value. So let’s start with the timeline. 0 1 2 3 4 years |------------|--------------|----------------|------------|-- ---- 1000 1000 1000 Notice how the first payment starts in year one. So when we use the annuity formula, it will give us a value in year 0 – the present value. Pv = 1000{ 1/.1 – 1/[.1(1.1)^3] } = 1000( 10 – 7.51) = 2490
- This problem looks more confusing than it really is. The key with multiple cash flows like this annuity type problem is to get all the cash flows into one value. Here we can use the annuity formula to do that. Then we can use the present value and future value formulas to give us a value at any point in time. If we calculate the present value of the retirement savings, we can use the future value formula to calculate the value at retirement. This is how it works: 0 1 2 3 20 years |------------|--------------|----------------|----------/ / -------|------ 4000 4000 4000 4000 Notice how the first payment starts in year one. So when we use the annuity formula, it will give us a value in year 0 – the present value. Pv = 4000{ 1/.1 – 1/[.1(1.1)^20] } = 34,054 But we are not done yet. We want to know the future value at retirement which is in 20 years: Fv = 34,054 (1.1)^20 = 229,098
- This looks like an annuity, but the difference is that now you need to calculate the annual payment. We are basically given the present value of 200,000 so we can solve for C in the annuity formula. 0 1 2 3 30 years |------------|--------------|----------------|----------/ / -------|------ C C C C Pv = 200,000 = C{ 1/.1 – 1/[.1(1.1)^30] } = C {9.4269} C = 200,000/9.4269 = 21,216
- Simple interest assumes you earn interest on your investment, but you do not reinvest your interest. So you get no compounding. Compound interest assumes you reinvest your interest. So you get interest not only on your original investment, but also the interest you previously reinvested. Many interest rates have periods that are less than 1 year. Your credit card may quote a daily interest rate. APR and EAR are ways of converting interest rates with periods less than 1 year into interest rates with a 1 year period. This way it makes it easier to compare interest rates. Say one credit card offers a .1% daily rate and another offers 24% annual rate, which one is better? That’s is why we use APR and EAR. APR assumes simple interest, while EAR assumes compounded interest. Another interest rate to take into account is inflation. The normal prices and investment interest rates you see everyday are called nominal prices and interest rates. If you want to look at prices and interest rates by taking out the effect of inflation, then we calculate real prices and real interest rates. These have inflation removed.
- In the simple interest example, you earn 16% on $100 the first year to give you $16. The second year, due to simple interest, you earn 16% only on your original $100 investment. So in year 2 you get $16 again. Thus over two years you earned a total of $32 in interest. So you have your $100 original investment plus the $32 in interest for a total of $132. In the compound interest example, you reinvest your $16 in interest so after the first year your total investment is $116. Then the second year interest is 16% of 116, which is $18.56. You are then left with $116 + $18.56 after two years for a total of $134.56. Compound interest will always grow your investment faster (good) or how much you owe someone faster (bad) than simple interest.
- Example 1: APR = (periodic rate)(number of periods per year) = .02(12) = .24 = 24% remember there are 12 months per year EAR = (1+periodic rate)^(number of periods per year) - 1 = 1.02^12 – 1 = .268 = 26.8% Example 2: We must find the EAR given an APR. To find the EAR we need the periodic rate, which we can find using the APR. APR = periodic rate x number of periods .08 = periodic rate x 4 since there are 4 quarters in a year Periodic rate = .08/4 = .02 Now find the EAR = (1+periodic rate)^(number of periods per year) - 1 =1.02^4 – 1 = 1.082 -1 = 8.2%
- Nominal rates are the normal rates we see and use every day. Real interest rates have the effect of inflation removed. This allows us to better compare real interest rates over time. With the equations you can convert a nominal interest rate into a real interest rate and vice versa.
- Using the exact formula: (1+real) = (1+nominal)/(1+inflation) plug in the numbers (1+real) = (1+.059)/(1+.033) = 1.0251 subtract 1 from both sides Real = 1.0251 – 1 = .0251 = 2.51% Using the approximation formula: Real = nominal – inflation = .059 - .033 = .026 = 2.6%
- Inflation is an important factor to take into account. See how it has varied over history. As long as your income goes up with inflation you can still afford to buy everything you bought in the past. Unfortunately, since 2000 incomes have not risen with inflation which mean our income has not increased as much as the costs we pay for things like food and gas. This has lowered our standard of living.
- How does this look on the timeline? 0 1 2 3 years |------------|--------------|----------------|------------ 100,000 -> payments go on forever Use the delayed perpetuity formula to solve this problem Pv = (C/r)(1/(1+r)^n-1) = (100,000/.1) /(1/(1+.1)^3-1)= 751,315
- 0 1 2 3 20 years |------------|--------------|----------------|----------/ / -------|------ 4000 4000 Notice how the first payment starts in year three. So we will use the delayed annuity formula. Pv = 4000{ 1/.1 – 1/[.1(1+.1)^17] }{1/(1+.1)^(3-1)} = 26,518