Time Value of Money (Examples) The sample problems are not in order of topic or difficulty.

Part 1: Example 1 - PV If you invest $15,000 for ten years, you receive $30,000. What is your annual return? A:

Example 2 - The Multi-Period Case Assume that the average college tuition costs $20,000 dollars per annum (paid at the end of the year). For a freshman just starting college, what is the present value of the cost of a four year degree when the interest rate is 10%? If the tuition is paid at the beginning of the year, what is the PV?

Ex. 3 - Computing Payments with APRs Suppose you want to buy a new computer system and the store is willing to sell it to allow you to make monthly payments. The entire computer system costs $3500. The loan period is for 2 years and the interest rate is 16.9% with monthly compounding. What is your monthly payment? An annuity problem... Monthly rate = .169 / 12 = .01408333333 Number of months = 2x(12) = 24 3500 = C[1 – 1 / 1.014083)24] / .014083 C = 172.88 2(12) = 24 N; 16.9 / 12 = 1.408333333 I/Y; 3500 PV; CPT PMT = -172.88

Ex. 4 - Future Values with Monthly Compounding Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years? Monthly rate = .09 / 12 = .0075 Number of months = 35(12) = 420 FV = 50[1.0075420 – 1] / .0075 = 147,089.22 35(12) = 420 N 9 / 12 = .75 I/Y 50 PMT CPT FV = 147,089.22

Ex. 5 - Present Value with Daily Compounding You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an effective annual rate of 5.63% based on daily compounding, how much would you need to deposit? Daily rate = (1.0563)1/365 – 1= 0.00015 Number of days = 3x(365) = 1095 PV = 15,000 / (1.00015)1095 = $12,728 3(365) = 1095 N 5.5 / 365 = .015068493 I/Y 15,000 FV CPT PV = -12,718.56

Example 6 - Pure Discount Loans A T-bill promises to repay $10,000 in 6 months. If the bill sells for $9,600 in the market, what is the annual rate of return (simple and compound)? HPR = (10,000-9,600)/9,600= 0.04167 Simple annual return=2x0.04167= 0.0833 Compound annual return= (1.04167)2 – 1= =1.0851 – 1= 0.0851 Remind students that the value of an investment is the present value of expected future cash flows. 1 N; 10,000 FV; 7 I/Y; CPT PV = -9345.79

Example 7 - Investment C0 = C1/(1 + r) + C2/(1+r)2 Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive two cash flows: $40 next year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment? C0 = C1/(1 + r) + C2/(1+r)2 PV = 40/(1.15) + 75/(1.15)2 = PV = 34.78 + 56.71 = 91.50$ (PV<C0), No – the broker is charging more than you would be willing to pay. You can also use this as an introduction to NPV by having the students put –100 in for CF0. When they compute the NPV, they will get –8.51. You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV.

Ex. 8 - Saving For Retirement You are offered the opportunity to put some money away for retirement now. You will receive ten annual payments of $25,000 each beginning in 30 years. How much would you be willing to invest today if your money can earn an interest rate of 12% per annum over the years.

Ex. 8 - Saving For Retirement Timeline 0 1 2 … 29 30 31 32 ... 39 ? 0 0 … 0 25K 25K 25K 25K 25K Notice that the year 0 cash flow = CF0 = ? The cash flows years 1 – 29 are 0 The cash flows years 30 – 39 are 25,000

Ex. 8 - Saving For Retirement Timeline PVA= 25,000{[1 – 1/(1.2)10]/0.12}= 141,255 PVA due = PVA x (1.12) = 158,206.24 CF0 = FV / (1+r)30 = 158,206.24 / (1.12)30 = = 158,206.24 / 29.96 = $5,280.58

Example 9 - Buying a House Fixed rate mortgage loans are basically annuities. They promise a specified stream of cash payments to a lender. Suppose, you want to take out a 30-year mortgage loan for 500,000$ at an interest rate of 7.5% per annum. A) What would be your monthly payment? B) How much would you owe to the bank after third monthly payment? Show the amortization schedule for the first three months. C) How much would you owe to the bank after the 60th payment?

Example 9 - Buying a House A) 500,000 = PMT * [1/0.625% - 1/(0.625%(1+0.625%)360)] Monthly payment will be 3,496.07$ for 360 months beginning next month. B) C) After the 60th payment, you still have an annuity of 300 months to pay. The answer is the PV of this annuity. Make the computation. Beg Bal Pmnt Int Principal End Bal 500,000 3,496 3,125 371 499,629 3,122.7 373.3 499,255.7 3,120.3 375.7 498,880.0

Ex. 10 - Computing EARs Suppose you can earn 1% per month on $1 invested today. What is the APR? 1(12) = 12% How much are you effectively earning? FV = 1(1.01)12 = 1.1268 Rate = (1.1268 – 1) / 1 = .1268 = 12.68% Suppose you put it in another account, you earn 3% per quarter. What is the APR? 3(4) = 12% FV = 1(1.03)4 = 1.1255 Rate = (1.1255 – 1) / 1 = .1255 = 12.55% Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.

Ex. 10 - Computing EARs Which account would you prefer? Monthly or quarterly compounding? Choice should be based on the EAR, hence monthly compounding account.

Ex. 11 – EAR’s You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? First account: EAR = (1 + .0525/365)365 – 1 = 5.39% Second account: EAR = (1 + .053/2)2 – 1 = 5.37% Which account? Remind students that rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates. Calculator: 2nd I conv 5.25 NOM up arrow 365 C/Y up arrow CPT EFF = 5.39% 5.3 NOM up arrow 2 C/Y up arrow CPT EFF = 5.37%

Ex. 11 – EAR’s Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year? First Account: Daily rate = .0525 / 365 = .00014383562 FV = 100(1.00014383562)365 = 105.39 Second Account: Semiannual rate = .0539 / 2 = .0265 FV = 100(1.0265)2 = 105.37 You have more money in the first account. It is important to point out that the daily rate is NOT .014, it is .014383562 First Account: 365 N; 5.25 / 365 = .014383562 I/Y; 100 PV; CPT FV = 105.39 Second Account: 2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV = 105.37

Example 11/A – EAR’s Your money is promised a return of 100% over 15 years. Is this a lot or a little? (1 + r0,t) = (1 + r) t (1 + r0,t)1/t = (1 + r) => (1 + 1) 1/15 = (1 + r) 1 + r = 1.04729 => r = 0.04729 4.729%, the annualized rate of return is the interest rate r, which, if compounded for 15 years, offers a 100% rate of return. How does this compare to 1% over 3 months? (1 + 0.01)4 = 1.0406 => r = 0.0406

Example 12 - APR Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must the account pay? On the calculator: 2nd I conv down arrow 12 EFF down arrow 12 C/Y down arrow CPT NOM

Example 14 - APR If a 3 month bond has a 8% APR, how much interest will I earn over the life of the bond? Since the APR quote does not include interest on interest and since a 3 month bond can be reinvested 4 times during the year, the bond will earn 2% interest over its life.

Example 15 - EAR If a 3 month bond has a 8% EAR, how much interest will I earn over the life of the bond? Since the EAR quote does include interest on interest and since a 3 month bond can be reinvested 4 times during the year, Why is this rate lower?

Ex. 16 - Finding the Number of Payments Suppose you borrow $2000 at 5% and you are going to make annual payments of $734.42. How long will it take before you pay off the loan? 2000 = 734.42(1 – 1/1.05t) / .05 .136161869 = 1 – 1/1.05t 1/1.05t = .863838131 1.157624287 = 1.05t t = ln(1.157624287) / ln(1.05) = 3 years Sign convention matters!!! 5 I/Y 2000 PV -734.42 PMT CPT N = 3 years

Ex. 17 - Amortized Loan with Fixed Principal Payment Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year. Click on the Excel icon to see the amortization table

Ex. 18 - Amortized Loan with Fixed Payment Each payment covers the interest expense plus reduces principal Consider a 4 year loan with annual payments. The interest rate is 8% and the principal amount is $5000. What is the annual payment? 5000 = PMT (1/0.08 – 1/(0.08 * (1.08)^4)) PMT = 1509.60 Click on the Excel icon to see the amortization table

Example 19 - Perpetuity You made your fortune in the dot-com boom (and got out in time!). As part of your legacy, you want to endow an annual MBA graduation party at your alma matter. You want it to be a memorable, so you budget $30,000 per year for the party. If the university earns 8% per year on its investments, and if the first party is in one year’s time, how much will you need to donate to endow the party?

Ex. 19 - Solution PV = C / r = $30,000 / .08 = $375,000 today. 3 2 1

Example 19 - Cont’d Suppose instead the first party was scheduled to be held 2 years from today. How would this change the amount of the donation required?

Ex 19 - Solution PV = $375,000 / 1.08 = $347,222 today. 3 30,000 2 1 2 1 375,000

Non-standard TVM Problems Sometimes the problem we face is not stated as a typical time value of money problem and so does not exactly fit any formula. In these cases, we often have to “back into” a solution.

Example 24– Deceptive Ads (1) Motown Autos (MA) Advertisement: American Classic Cars! Finance Special! Sprite Conversion! Now Only $15,000! Just $1,000 Down, 0% interest, and 3- years to pay with easy monthly payments! What is the true interest rate in this deal? Can you give an answer yet?

Example 24– Deceptive Ads (2) When purchasing from MA, you are buying a bundle of financing and car Classic Car News has an almost identical car advertised for $9,000, but it needs $3,000 of work to match the condition of the car offered by MA. To un-bundle the package, you use the cost of acquiring the competing car Cash value of car = $9,000 + $3,000 = $12,000

Example – Deceptive Ads (3) Next, determine the cash flows associated with the financed car

Example – Deceptive Ads (4) Next, determine the cash flows and interest rate on an equivalent loan if you took the alternative.

Example – Deceptive Ads (5) APR = 1.3664%p.m.*12 = 1.1639 or 16.39% p.a. EAR = 1.013664^12 -1 = 17.69%p.a. Clearly, the true interest rate on this loan is much higher than that in the advertisement. The trick is to sell something worth of 11,000USD for 14,000USD, and tell that the interest is 0%!

Example 25 $15,000 is financed at seven percent APR You have $30,000 in student loans that call for monthly payments over 10 years. $15,000 is financed at seven percent APR $8,000 is financed at eight percent APR and $7,000 at 15 percent APR What is the interest rate on your portfolio of debt? Hint: don’t even think about doing this: 15,000 + = 30,000 × 7% 8,000 8% 7,000 15%

Example 25 Find the payment on each loan, add the payments to get your total monthly payment: $384.16. Set PV = $30,000 and solve for I/YR = 9.25% N 7 120 15,000 –174.16 8 120 8,000 –97.06 15 120 7,000 –112.93 –384.16 30,000 120 I/Y 9.25 PV PV + + = PMT FV

Example 27 You are considering the purchase of a prepaid tuition plan for your 8-year old daughter. She will start college in exactly 10 years, with the first tuition payment of $12,500 due at the start of the year. Sophomore year tuition will be $15,000; junior year tuition $18,000, and senior year tuition $22,000. How much money will you have to pay today to fully fund her tuition expenses? The discount rate is 14% CF0 = ?

Example 28 You are thinking of buying a new car. You bought your current car exactly 3 years ago for $25,000 and financed it at 7% APR for 60 months. You need to estimate how much you owe on the loan to make sure that you can pay it off when you sell the old car.

Example 29 You have just landed a job and are going to start saving for a down-payment on a house. You want to save 20 percent of the purchase price and then borrow the rest from a bank. You have an investment that pays 10 percent APR. Houses that you like and can afford currently cost $100,000. Real estate has been appreciating in price at 5 percent per year and you expect this trend to continue. How much should you save every month in order to have a down payment saved five years from today?

Example 29 Continued FV= 100,000x(1.05)^5 = 127,628.1563 First we estimate that in 5 years, a house that costs $100,000 today will cost $127,628.16 Next we estimate the monthly payment required to save up that much in 60 months. FV= 100,000x(1.05)^5 = 127,628.1563 20% down= 0.2x 127,628.1563= 25,525.63 Use FV of annuity formula to compute PMT: 25,525.63= PMT x {[1 – 1/(1.00833)^60]/0.00833} PMT= 329.634$

Part 2: Some Problems for practice: Q1) On January 1 you deposit $100 in an account that pays a nominal interest rate of 11.33463%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.) 4/16/2017

( ) ( ) Part 2: Q1: Answer iPer = 11.33463%/365 = 0.031054% per day. 1 2 273 0.031054% 100 FV=? FV = $100 ( 1.00031054 ) 273 273 = $100 ( 1.08846 ) = $108.85. 4/16/2017

Q2) What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 1 2 3 4 5 6 6-mos. periods 5% 100 100 100 4/16/2017

Payments occur annually, but compounding occurs each 6 months. Q2 - Answer Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques. 4/16/2017

1st Method: Compound Each CF 1 2 3 4 5 6 5% 100 100 100.00 110.25 121.55 331.80 FVA3 = $100(1.05)4 + $100(1.05)2 + $100 = $331.80. 4/16/2017

2nd Method: Treat as an Annuity Yes, by following these steps: a. Find the EAR for the quoted rate: EAR = (1 + ) - 1 = 10.25%. 2 0.10 2 b. Use EAR = 10.25% as the annual rate for an annuity for N=3 years. Do the exercise! 4/16/2017

2nd Method: Treat as an Annuity 1 2 3 5% 100 100 100 4/16/2017

You plan to leave the money in the bank if you don’t buy the note. Q3) You are offered a note which pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank which pays a 6.76649% nominal rate, with 365 daily compounding, which is a daily rate of 0.018538% and an EAR of 7.0%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless. Should you buy it? 4/16/2017

1. Greatest future wealth: FV 2. Greatest wealth today: PV iPer = 0.018538% per day. 365 456 days -850 1,000 3 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EFF% 4/16/2017

1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1,000. FVBank = $850(1.00018538)456 = $924.97 in bank. Buy the note: $1,000 > $924.97. 4/16/2017

2. Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV = $1,000/(1.00018538)456 = $918.95. PV of note is greater than its $850 cost, so buy the note. Raises your wealth. 4/16/2017

Now we must solve for i per day. 3. Rate of Return Find the EAR% on note and compare with 7.0% bank pays, which is your opportunity cost of capital: FVn = PV(1 + i)n $1,000 = $850(1 + i)456 Now we must solve for i per day. 4/16/2017

Using interest conversion: İ% = 0.035646 EAR = (1.00035646)365 - 1 = 13.89%. Since 13.89% > 7.0% opportunity cost, buy the note. 4/16/2017

Q4) Future Values FV = 25,000(1.04)40 = 120,025$ Suppose you just started a new job at a current annual salary of $25,000. Average expected inflation rate is 4% for the next 40 years. If you receive annual cost-of-living raises tied to the inflation rate, what would be your ending salary? FV = 25,000(1.04)40 = 120,025$ Also today’s $20,000 car will cost $96,020 under the same assumptions.

Q5) Future Values Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today? FV = 10(1.055)200 = 447,189.84 What is the effect of compounding? Simple interest = 10 + 200(10)(.055) = 120.00 Compounding added $447,069.84 to the value of the investment 5F-55

Q6) Present Values You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today? PV = $150,000 / (1.08)17 = $40,540.34 We can use a negative exponent to avoid having to invert and multiply by the result from (1 + r)^n: Key strokes: 1.08 yx 17 +/- = x 150,000 = Calculator: N = 17; I/Y = 8; FV = 150,000; CPT PV = -40,540.34

Q7) Discount Rate Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest? r = (20,000 / 10,000)1/6 – 1 = 0.12246= 12.25% Calculator: N = 6; FV = 20,000; PV = 10,000; CPT I/Y = 12.25%

Q8) Number of Periods Suppose you want to buy a new house. You currently have $15000 and you figure you need to have a 10% down payment plus an additional 5% of the loan amount for closing costs. Assume the type of house you want will cost about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?

Q8) Number of Periods How much do you need to have in the future? Down payment = 0.1(150,000) = $15,000 Closing costs = 0.05(150,000 – 15,000) = 6750 Total needed = 15,000 + 6750 = 21,750 Compute the number of periods Using the formula: t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years Loan amount = 150,000 – down payment = 150,000 – 15,000 = 135,000