Time Value of Money (5)

MGMT 235 Dr. Sharp Review Fundamentals of Valuation These class notes review this material and also provide some help for a financial calculator. It also has some self-test questions and problems. Class notes are necessarily brief. See any principles of finance book for a more extensive explanation. Eugene F. Brigham, Joel F. Houston Fundamentals of financial management HG 4026 B6693 1998 Ross, Stephen A, Westerfield, and Jordan Fundamentals of corporate finance HG 4026 .R677 1995 PART I : Single Sum. Time Value of Money: Know this terminology and notation FV Future Value (1+i) t Future Value Interest Factor [FVIF] PV Present Value 1/(1+i) t Present Value Interest Factor [PVIF] i Rate per period t # of time periods Question: Why are (1+i) and (1+i) t called interest factors? Answer: 1. Start with simple arithmetic problem on interest: How much will $10,000 placed in a bank account paying 5% per year be worth compounded annually? Answer: Principal + Interest $10,000 + $10,000 x .05 = $10,500 2. Factor out the $10,000. 10,000 x (1.05) = $10,500 3. This leaves (1.05) as the factor . 1. Find the value of $10,000 earning 5% interest per year after two years. Start with the amount after one year and multiply by the factor for each year. [ Amount after one year ] x (1.05) = [ $10,000 x (1.05) ] x (1.05) = $10,000 x (1.05) 2 = $11,025. . © 2004 So (1+i) t = (1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)… ·(1+i) for “t” times

Class Notes A. Future Value Find the value of $10,000 in 10 years. The investment earns 5% per year. FV = $10,000 · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) FV = $10,000 · (1.05) · (1.05) · (1.05) · (1.05) · (1.05) · (1.05) · (1.05) · (1.05) · (1.05) · (1.05) FV = $10,000 x (1.05) 10 = $10,000 x 1.62889 = $16,289 Find the value of $10,000 in 10 years. The investment earns 8% for four years and then earns 4% for the remaining six years. FV = $10,000 · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) FV = $10,000 · (1.08) · (1.08) · (1.08) · (1.08) · (1.04) · (1.04) · (1.04) · (1.04) · (1.04) · (1.04) FV = $10,000 x (1.08) 4 x (1.04) 6 FV = $17,214.53 B. Present Value : Same idea, but begin at the end. Rearrange the Future value equation to look like this: PV = FV÷ [(1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i) · (1+i)] P V = FV ÷ (1+i) t [2] Example: How much do I need to invest at 8% per year, in order to have $10,000 in__. a. One year: PV =10,000 ÷ (1.08) = $9,259.26 b. Two years: PV = $10,000 ÷ (1.08) ÷ (1.08) OR $10,000 ÷ (1.08) 2 = $8,573 c. Ten years PV = $10,000 ÷ (1.08) 10 = $10,000 ÷ 2.1589 = $4,632 C. Rate of Return START WITH SAME RELATIONHSIP: FV = PV x (1+i) t Solve for i. (1+i) t =FV/PV. 1+i = (FV/PV) 1/t i = (FV/PV) 1/t -1. Question: An investor deposits $10,000. Ten years later it is worth $17,910. What rate of return did the investor earn on the investment? Solution: $17,910 = $10,000 x (1+i) 10 (1+i) 10 = $17,910/10,000 = 1.7910 (1+i) = (1.7910) 1/10 = 1.060 i = .060 = 6.0% 2

Class Notes F. Finding the Present Value Find the present value of $10,000 to be received at the end of 10 periods at 8% per period. a. Scientific Calculator Scientific Calculator: Use [y x ] where y = 1.08 and x = -1,-2, or -10. 1. Enter 1.08. 2. Press [y x ] 3. Enter the exponent as a negative number 4. Enter [=]. 5. Multiply result by $10,000. b. Spreadsheet c. Financial calculator. You may need to input something like this. Specific functions vary. Be sure to consult the calculators’ manual!!!!!! n [N] i [I/YR] PV PMT FV c. 10 8 10,000 0 ? The present value will be negative, to indicate the opposite direction of cash flow. KEY RELATIONHSIP: PV = FV ÷ (1+i) t 4

Class Notes G. Finding the [geometric average] rate of return : Scientific Calculator To find i, use [y x ] and [1/x]. 1. Enter 1.7910 , 2. Press [y x ] 3. Enter the exponent 10 then press [1/x] 4. Press [=] . 5. Subtract 1 2. Spreadsheet 3. Financial Calculator. (Your financial calculator may differ. Consult your manual.) n [N] i [I/YR] PV PMT FV 10 ? -10,000 0 17,910 Answer i = 6%  Question: Today your stock is worth $50,000. You invested $5,000 in the stock 18 years ago. What average annual rate of return [i] did you earn on your investment? Answer: 13.646%.  Question: The total percentage return was 45,000÷5000=900%. Why doesn’t the average rate of return equal 50%, since 900%÷18 = 50%? KEY RELATIONHSIP: (1+i) t =FV÷PV (1+i) = (FV÷PV) 1/t 5

Class Notes Alternatively , FV t  m = PV x (1+i/m) t  m . m= periods per year , t= number of years , i = the interest per year [APR] . Example: What will $1,000 be worth at the end of one year when the annual interest rate is 12% [This is the APR .] when interest is compounded: Annually : t=1 i =12% FV 1 = PV x (1+i) 1 = $1,000 x (1.12) 1 = $1,120. Quarterly: t=4 i = 3% FV 4 = PV x (1+i) 4 = $1,000 x (1.03) 4 = $1,125.51. Monthly: t=12 i =1% FV 12 = $1,000 x (1.01) 12 = $1,000 x (1.126825) = $1,126.825. Daily: t=365 i = (12%÷365) = 0.032877% FV 365 = $1,000 x (1.00032877) 365 = $1,000 x (1.12747) = $1,127.47. n [N] i [I/YR] PV PMT FV 1 12 1,000 0 ? 4 3 1,000 0 ? 12 1 1,000 0 ? 365 .032877 1,000 0 ? How about compounding at every instant? E. CONTINUOUS COMPOUNDING: [Used in Black Scholes option pricing model.] t · m lim 1 + __i__ = e i t m  m Example: What is $1,000 worth in one year if compounded at 12% continuously. FV = $1,000 x e .12 = $1,000 x 1.127497 = $1,127.50 This is $.03 more than daily compounding. Try this on your calculator. Find the e x button. e .12 = 1.12749 Present Value Interest Factor = [ e -i t ] Problem: What is the present value of $10,000 to be received 3 years from today compounded continuously at 10%?PV = $10,000 x e -.10 x 3 = $10,000 x 0.74082=$7,408 Try this on your calculator. Find the e x button. e -0.3 = 0.74082 7

Class Notes Practice Quiz Questions: PV and FV of a Single sum. Review Problems 1. How much must you deposit today in a bank account paying interest compounded quarterly: a. if you wish to have $10,000 at the end of 3 months, if the bank pays 5.0% APR? Answer: $9,877 b. if you wish to have $50,000 at the end of 24months, if the bank pays 8.0%APR? Answer: $42,675 c. if you wish to have $6,000 at the end of 12 months, if the bank pays 9.0% APR? Answer: $5,489 2. a. What rate of interest [APR] is the bank charging you if you borrow $77,650 and must repay $80,000 at the end of 2 quarters, if interest is compounded quarterly? Answer: 6.0% APR b. What rate of interest [APR] is the bank charging you if you borrow $49,000 and must repay $50,000 at the end of 3 months, if interest is compounded monthly? Answer: 8.0% APR 3. How much must you deposit today in a bank account paying interest compounded monthly: a. if you wish to have: $10,000 at the end of 1 months, if the bank pays 5.0% APR ? Answer: $9,959 b. if you wish to have: £6,000 at the end of 6 months, if the bank pays 9.0% APR ? Answer: £5,737 c. if you wish to have: $12,000 at the end of 12 months, if the bank pays 6.0% APR ? Answer: $11,303 4. If interest is compounded quarterly , how much will you have in a bank account: a. if you deposit today £8,000 at the end of 3 months, if the bank pays 5.0% APR ? Answer: £8,100 b. if you deposit today $10,000 at the end of 6 months, if the bank pays 9.0% APR ? Answer: $10,455 c. if you deposit today ¥80,000 at the end of 12 months, if the bank pays 8.0% APR ? Answer: ¥86,595 d. if you deposit today $5,000 at the end of 24 months, if the bank pays 5.0% APR ? Answer: $5,522 5. If interest is compounded monthly , how much will you have in a bank account, a. if you deposit today £8,000 at the end of 3 months, if the bank pays 5.0% APR ? Answer: £8,100 b. if you deposit today $10,000 at the end of 6 months, if the bank pays 9.0% APR ? Answer: $10,459 c. if you deposit today ¥80,000 at the end of 12 months, if the bank pays 8.0% APR ? Answer: ¥86,640 d. if you deposit today £5,000 at the end of 24 months, if the bank pays 5.0% APR ? Answer: £5,525 8

Class Notes Review Fundamentals of Valuation Part II Multiple Periods: Uneven and Even (Annuities)  Periodic Uneven Cash Flows What is the value of the following set of cash flows today? The interest rate is 8% for all cash flows. Year and Cash Flow 1: $ 300 2: $ 500 3: $ 700 4: $ 1000  Solution: Find Each Present Value and Add 300 1.08 1 + 500 1.08 2 + 700 1.08 3 + 1000 1.08 4 = 277.78 428.67 555.68 735.03 = 1997.16  Periodic Cash Flow: Even Payments An annuity is a level series of payments. For example, four annual payments, with the first payment occurring exactly one period in the future is an example of an ordinary annuity. $1,000 $1,000 $1,000 $1,000 0 1 2 3 4 A. Present value of an annuity: The present value of each of the cash flows is the value of the annuity. This could be done one at a time, but this might be tedious. Annuity Present Value Interest Factor PVIFA = [1/(1+i) + 1/(1+i) 2 + ... + 1/(1+i) t ] 10

Class Notes Example: What is the present value of a 4-year annuity, if the annual interest is 5%, and the annual payment is $1,000? i = 5%; PMT = $1,000; t =4; PV = ? PV = 1,000 /(1.05) + 1,000/(1.05) 2 + 1,000/(1.05) 3 + 1,000/(1.05) 4 Factor out the single sum interest rate factors: PV = 1,000 x [1/(1.05) + 1/(1.05) 2 +1/(1.05) 3 + 1/(1.05) 4 ] = PV = 1,000 x [PVIFA ( 4,5% )] = Calculate: PVIFA( 4,5% ) = 1-1/(1+i) t = 1- PVIF 4,5% 1- 0.8227 = 3.54595. i 5% .05 PV = 1,000 x [3.5460] = $3,546. Finding the Future Value of an annuity on a: 1. Scientific Calculator. To calculate PVIFA using scientific calculator: FIRST FIND: PVIF 4,5% = 1/(1+i) t = 1/(1.05) 4 = 0.82270 THEN FIND: PVIFA( 4,5% ) = 1-1/(1+i) t = 1- PVIF 1- 0.8227 = 3.54595. i i .05 = 1,000 x [3.5460] = $3,546. 2. Using a spreadsheet. 3. Using a financial calculator, the Present Value of an annuity. n [N] i [I/YR] PV PMT FV 4 5 ? -1000 0 PV= $3,546. Note: Most financial calculators require i [I/YR] to be a percentage. That is enter a 5, not .05. However, Excel requires .05 or 5%. Long way. Short Way 11

Class Notes B. Future value of an annuity :  Annuity Future Value Interest Factor FVIFA = [1+ (1+i) + (1+i) 2 + ... + (1+i) t-1 ]. . Example: What is the future value of a 4-year annuity, if the annual interest is 5%, and the annual payment is $1,000? i = 5%; PMT = $1,000; t =4; FV = ? $1,000x [1+ (1.05) + (1.05) 2 + (1.05) 3 ] = $1,000 x [FVIFA (4,5%)] = $1,000 x [4.3101] = $4,310.1 Finding FVIFA 1. Using scientific calculator: FIRST FIND: FVIF = (1+i) t = (1.05) 4 = 1.2155 THEN: FVIFA(t,i) = (1+i) t -1 = FVIF- 1 i i . FVIFA(t,i) = FVIF 4,5% - 1 1.2155-1 = 4.3101 5% .05 2. Using a Spreadsheet Note: Last Payment earns no interest in ordinary annuity. The interest factor on that payment is 1. The first payment earns interest for t-1 periods, not t periods. Use Short Cut Formula 13

Class Notes 3. Using a financial calculator, the Future Value of an annuity: n [N] i [I/YR] PV PMT FV 4 5 0 -1000 ? FV = $4,310 Question: How much would you need to deposit every month in an account paying 6% a year to accumulate by $1,000,000 by age 65 beginning at age 20? Data: FV = $1,000,000 PMT = ? i = 6%÷12 = 0.5% per month n = (65-20) x 12 = 45 x 12 = 540 months. Answer: PMT = $362.85 C. RATE OF RETURN OF AN ANNUITY  You borrow $60,000 and repay in 8 equal annual installments of $12,935 with the first payment made exactly 1 year later. To the nearest percent, what rate of interest are you paying on your loan? Difficult without financial calculator. Can use table to find answer to the nearest percent. Data: i = ? PV = $60,000 PMT=$12,935 t = 8 years Relationship : PV = PMT x PVIFA(t, i) 1. Solution: (Trial and Error with Table) PVIFA(t, i) = PV/PMT= $60,000/12,935 = 4.6386 Table ..... 14% .... : 8 .. 4.6389 .. : Therefore: PVIFA = 4.6386. S o > i = 14% 2. (Trial and Error using a spreadsheet program) 14

Class Notes 3. (Trial and Error using a financial calculator) n [N] i [I/YR] PV PMT FV 8 ? 60000 -12935 0 i = 14% D. Example of Annuity with quarterly compounding: An investment of $3000 per quarter for 6 years at annual interest rate of 8%, compounded quarterly, will accumulate by the end of year 6 to: Solution: FV = ? PMT = $3,000 t = 24 i = 2% FV = PMT x FVIFA (t, i). FV = $3,000 x [30.422] = $91,266. n [N] i [I/YR] PV PMT FV 24 2 0 -3000 ? $91,266 Review Problems with solutions. 1. This one: is a typical mortgage problem. You borrow $80,000 to be repaid in equal monthly installments for 30 years. The APR is 9%. What is the monthly payment? PV = $80,000 i =0.75%, t=360 PMT = ? $80,000 = PMT x 124.282 PMT = $643.70 n [N] i [I/YR] PV PMT FV 360 .75 -80000 ? 0 2. Try this one. You make equal $400 monthly payments on a loan. The interest rate equals 15% APR, compounded monthly. The loan is for 12 years. What is the amount of the loan? Answer: PV = $26,651 3. Retire with a million: How much would must you deposit monthly in an account paying 6% a year [APR], compounded monthly, to accumulate $1,000,000 by age 65 beginning at age 30? Answer: PMT = $701.90 n [N] i [I/YR] PV PMT FV 420 0.50 0 ? 1000000 4. Using a financial calculator for annuity calculations : Calculate the future value of $60.00 per year at 7% per year for eight years. 16

Class Notes n [N] i [I/YR] PV PMT FV 8 7 0 -60 ? FV = $615.50 5. Calculate the future value of $50.00 per month at 6% APR for 24 months n[N] i [I/YR] PV PMT FV 24 0.5 0 -50 ? FV = $1,217.60 6. Calculate the present value of $500 per year at 6% per year for 5 years (monthly compounding). n[N] i [I/YR] PV PMT FV 5 6 ? -500 0 PV=$2,106 7. You borrow $5,000 and repay the loan with 12 equal monthly payments of $500? Calculate the interest rate per month and the APR. n[N] i [I/YR] PV PMT FV 12 ? 5,000 -500 0 i = 2.92% per month. APR = i x 12 APR = 2.92% x 12 = 35.04 8. Problem on inflation.  You will receive $100,000 dollars when you retire, forty years from today. If inflation averages 3% per year for the next forty years, how much would that amount be worth measured in today's dollars? (Note, this is not a time value of money problem, but it solved with a similar calculation. Such adjustments are necessary to overcome “money illusion”]   Solution: $100,000 ÷ (1.03) 40 =100,000 ÷ 3.26204 = $ 30,655 D. Annuity Due 17

Class Notes Try the following practice questions: Review Problems 1. Suppose you are trying to find the present value of two different cash flows. One is $100 two periods from now, the other a $100 flow three periods from now. Which of the following is/are true about the discount factors used to value the cash flows? a. The factor for the flow three periods away is always less than the factor for the flow that is received two periods from now. b. The factor for the flow three periods away is always more than the factor for the flow that is received two periods from now. c. Whether one factor is larger than the other will depend on the interest rate. d. Since the payments are for the same amount, the factors will yield present values that are the same. e. None of the above statements are true. 2. What is the present value of a stream of $2,500 semiannual payments received at the end of each period for the next 10 years? The APR is 6%. a. 37,194 b. 38,310 c. 35,810 d. 36,885 3. What is the future value in 10 years of $1,500 payments received at the end of each year for the next 10 years? Assume an interest rate of 8%. a. $25,260 b. $23,470 c. $21,730 d. $18,395 e. $15,000 4. You are given the option of receiving $1,000 now or an annuity of $85 per month for 12 months. Which of the following is correct? a. You cannot choose between the two without computing present values. b. You cannot choose between the two without computing future values. c. You will always choose the lump sum payment. d. You will always choose the annuity. e. The choice you would make when comparing the future value of each would be the same as the choice you would make when comparing present values. 5. You open a savings account that pays 4.5% annually. How much must you deposit each year in order to have $50,000 five years from now? a. $8,321 b. $9,629 c. $8,636 d. $9,140 e. $6,569 6. You are considering an investment in a 6-year annuity. At the end of each year for the next six years you will receive cash flows of $90. The initial investment is $414.30. To the nearest percent, what rate of return are you expecting from this investment? (Annual Compounding) a. 8% b. 9% c. 12% d. 21% e. 10% 19

Class Notes 7. You are saving up for a down payment on a house. You will deposit $600 a month for the next 24 months in a money market fund. How much will you have for your down payment in 24 months if the fund earns 10% APR compounded monthly? a. $14,480 b. $15,870 c. $12,930 d. $10,560 e. $ 9,890 8. Your mortgage payment is $600 per month. There is exactly 180 payments remaining on the mortgage. The interest rate s 8.0%, compounded monthly. The first payment is due in exactly one month. What is the balance of the loan? [Balance = PV of remaining payments.] a. $62,784 b. $77,205 c. $63,203 d. $82,502 e. $85,107 9. Your mortgage payment is $755 per month. It is a 30-year mortgage at 9.0% compounded monthly. How much did you borrow? a. $93,800 b. $97,200 c. $92,500 d. $85,100 e. $89,400 Possibly New Problems. 10. What is the value of the following set of cash flows today? The interest rate is 8.5%. Year Cash Flow 0: -$1,000 1: $ 200 2: $ 400 3: $ 600 4: $ 800 a. $ 800 b. $ 571 c. $1072 d. $ 987 e. $ 520 11. The present value interest factor of an annuity due for 3 years at 8% equals: a. 1/(1.08) 3 b. 1/(1.24) c. [1 + 1/(1.08) + 1/(1.08) 2 ] d. [1/(1.08) + 1/(1.08) 2 + 1/(1.08) 3 ] e. None of the above. 12. What is the present value of $2,500 semiannual payments received at the beginning of each period for the next 10 years? The APR is 6%. a. 37,194.70 b. 38,309.50 c. 35,809.50 d. 36,884.80 20