# 1 PRESENT VALUE AND THE OPPORTUNITY COST OF CAPITAL What is the NPV criterion? How do we use the time line and the basic equation to determine present.

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1 PRESENT VALUE AND THE OPPORTUNITY COST OF CAPITAL What is the NPV criterion? How do we use the time line and the basic equation to determine present value? How do we determine present value of a perpetuity and an annuity? How do we determine present value a bond and a stock?

2 The Time Line 0 CF 0 =-\$100 1 i% = 10%, annually CF 1 = ? How much will we have in the bank at the end of one year if we deposit \$100 and the interest rate is 10% each year?

3 The time line – Future value at end of year 1 i% = 10%, annually 0 CF 1 = 100 + 10%(100) = 100 + 10 = 110 CF 1 = 100 (1 +.10) = 110 CF 0 =-\$100CF 1 =\$110 1

4 The time line – Future value at end of year 2 i% = 10%, annually Year 01 100CF 2 = ? 2 110 CF 2 = 110 + 10%(110) = 110 + 11 = 121 CF 2 = 110 (1+.10) = 100 (1+.10)(1+.10) CF 2 = 100 (1+.10) 2 At the end of 2 years, we have \$100 principal, \$20 of simple interest and \$1 of compound interest.

5 The basic equation i% annually Year FV = PV (1 + i) t Note: If compounding frequency is not annual, two changes must be made. Change the interest rate and change the number of periods. 01 PVFV t...

6 Uneven cash flows – present value i% = 10% PV (1 ST CASH FLOW) = \$ 90.91 PV (2 ND CASH FLOW) = \$165.29 PV (3 RD CASH FLOW) = \$225.39 PV = \$ 481.59 3 2 1 100200300 PV=? 0

7 Uneven cash flows – future value 0 i% = 10% 3 2 1 100200300 FV=\$641 100 200300 0 FV=\$705.10 3 2 1 4

8 TIME VALUE OF MONEY PROBLEMS Single lump sum cash flows –Present value You want to set aside some money today in order to present your daughter with a \$3,000 trip upon her graduation from high school in seven years. If you can earn 10% on your money, how much do you need to set aside? Answer: PV = \$1,539.47 –Future value Janet opened a savings account for her new-born niece with \$2,000. The account pays 4% interest a year. How much will the account have in it when the niece turns 20? Answer: FV = \$4,382.25 –Return You purchased a house four years ago for \$130,000.You can sell it today for \$177,000. What effective annual rate of return have you earned? Answer: r = 8.02%

9 Problem Solutions Problem 1: PV FV = \$3,000, PMT = 0 N = 7, I = 10% CPT PV = -\$1,539.47 Problem 2: FV PV = -\$2,000, PMT = 0 N = 20, I = 4% CPT FV = \$4,382.24 Problem 3: Return PV = -\$130,000, PMT = 0 N = 4, FV = \$177,000 CPT I/Y = 8.02%

10 TVM PROBLEMS ….continued Single lump sum cash flow –Time Mary Vane has saved all the money she has received for gifts this year – a total of \$500. She has deposited the money in an account that pays an effective annual rate of 8%. When she has accumulated \$3,000, she plans to treat herself to a face lift. How long will Mary have to wait, assuming annual interest payments? Answer: 23.3 years –Future value using quarterly compounding If you deposit \$1,000 into a savings and loan account that earns 4% interest compounded quarterly, what will be the balance in the account at the end of four years if you make no withdrawals and the savings and loan is not insolvent prior to the end of the fourth year? Answer: FV = \$1,172.58

11 Problem Solutions Problem 1: Time PV = -\$500, PMT = 0, I = 8%, FV = \$3,000 CPT N = 23.3. years Problem 2: FV using quarterly compounding PV = -\$1,000, PMT = 0 N = 4 x 4 =16, I = 4%/4 = 1 CPT FV = \$1,172.58 Problem 3: Interest Rate (Icahn Trust) EAR = (1 + r)^n – 1 = (1 + 0.50)^7.3 – 1 = 18.29%, N = 365/50 = 7.3 and I = 500/1000 = 0.50, or 50%

12 TVM PROBLEMS ….continued Interest rate –A credit card company charges 1.5% on the unpaid balance each month. What is the annual percentage rate associated with this credit card? What is the effective annual rate associated with this credit card? What is the periodic rate associated with this credit card? Answer: r = 1.5%, APR = 18%, EAR = 19.56% –The Icahn Trust Company is willing to lend you \$1,000 for 50 days. After 50 days, you must pay Icahn the \$1,000 you borrowed, plus \$500. What is the effective annual cost of borrowing from Icahn? Answer: EAR = 18.29% –The rate of inflation for consumer goods for January 1990 was reported to be approximately 1.5%. If inflation had continued to grow at this same rate for the entire year, what would have been the effective annual rate of inflation for 1990? Answer: EAR = 19.56%

13 TVM PROBLEMS ….continued Multiple cash flows (annuities and perpetuities) Present value of an ordinary annuity –Joe Peshy plans to hit the campus scene. He expects his tuition for each of the next four years to be \$2,000, payable at the end of each year (He’s cool, not smart). At a 6% annual interest rate, how much would have to be invested today to provide for all his future tuition payments? (The first payment will be due a year from today.) Answer: PV OA = \$6,930.21 Future value of an ordinary annuity –You have decided that you can save \$200 a month by giving up your frequent visits to the vending machines. If you can earn 12%, compounded monthly, on this money, how much will you have in four years, assuming you make the deposits at the end of each month? Answer: FV OA = \$12,244.52

14 Problem Solutions Problem 1: PV of ordinary annuity PMT = \$2,000 N = 4, I = 6%, FV = 0 CPT PV = -\$6,930.21 Problem 2: FV of ordinary annuity PMT = \$200 N = 4 x 12 = 48, I = 12%/12 = 1 PV = 0 CPT FV = \$12,244.52 Problem 3: PV of annuity due PMT = \$900,000 N = 20, I = 10%, FV = 0 CPT PV = -\$8,428,428.08 (use BGN mode)

15 TVM PROBLEMS ….continued Multiple cash flows (…continued) Annuity due –Suppose that you have won a lottery where the prize is stated as \$18 million. The paying agent will pay you the winnings in twenty annual installments, starting immediately (i.e., \$900,000 each year starting now). If your opportunity cost is 10%, what is the value today of these twenty installments? Answer: \$8,428,428 (use BGN mode) Deferred annuity –An insurance agent has recommended you invest in a policy that promises to pay you \$10,000 a year upon retirement. The payments will continue for 15 years after you retire. If you plan to retire in 25 years and can earn 12% on similar investments, what is the maximum you should be willing to pay for this policy? (The first payment will be made at the end of the 25 th year.) Answer: PV OA = \$68,108.64, PV DA = \$4,487.14

16 Problem Solutions Problem 4: PV of deferred annuity Step 1: PV of an ordinary annuity PMT = \$10,000 N = 15, I = 12%, FV = 0 CPT PV = -\$68,108.64 Step 2: PV of a future cash flow FV = \$68,108.64 N = 24, I = 12%, PMT = 0 CPT PV = -\$4,487.14 Problem 5: PV of a perpetuity PV = C/r = 10/0.10 = \$100

17 TVM PROBLEMS ….continued Multiple cash flows (continued) Perpetuity –Joe Blow (whom you trust completely) has agreed to give you \$10 a year forever in exchange for a lump sum of cash from you today. What is the most you would be willing to give him if your opportunity cost of funds is 10%? Answer: PV = \$100 –Keyco Inc. pays a preferred dividend of \$5 a share. If the stock is selling at \$62.50, what is the opportunity cost of holding preferred stock in Keyco Inc.? Answer: r = 8%

18 GROWING PERPETUITY A growing perpetuity is a stream of cash flows that grows at a constant rate forever. Simplification: PV t = C t / (r - g), C t = \$100 ( r = 10%) (g = 2%) \$104.04... \$102\$100 0123 … forever… PV 0 = \$100/(0.10 – 0.02) = \$1250

19 GROWING ANNUITY A Growing Annuity is a stream of cash flows that grows at a constant rate over a fixed number of periods. Simplification: PV 0 = C t  {1/(r-g) - 1/(r-g)  [(1+g)/(1+r)] T }, C t = \$100 0 ( r = 10%) (g = 2%) \$104.04\$102 \$100 123 PV 0 = 100  [1/0.08 - (1/0.08)  (1.02/1.10) 3 ] = \$253.37

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