The time value of money and cash-flow valuation

The time value of money and cash-flow valuation Readings: Ross, Westerfield and Jordan, Essentials of Corporate Finance, Chs. 4 & 5 Ch. 4 problems: 13, 16, 19, 20, 22, 25. Ch. 5 problems: 14, 15, 31, 32, 44, 55, 56 The answers to these problems are posted to the class web page. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 1

Basic Definitions Present Value earlier money on a time line Future Value later money on a time line Interest rate exchange rate between earlier money and later money Discount rate Cost of capital Opportunity cost of capital Required return May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 2

Time Value Terminology 0 1 2 3 t... PV FV The convention is that time 0 is today ; 1 is one year later ( end of the first year ) etc. Problems with a single cash flow involve the four values: PV, FV, r, and N. Given three of them, it is always possible to calculate the fourth. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 3

Simple vs. compound interest Suppose you leave $1,000 on deposit at 5%. How much will you have two years from now? FV with simple interest = 1,000 + 50 + 50 = 1,100 FV with compound interest: First year: 1,000 x 1.05 = 1,050 Second year: 1,050 x 1.05 = 1,102.50 The extra 2.50 comes from the interest of.05(50) = 2.50 earned on the first interest payment May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 4

HP-10b (or 10bii) Calculator Keys FV = future value PV = present value I/YR = interest rate per year N = number of periods P/YR = [compounding] periods per year The 10b comes out of the box with P/YR set to 12 (i.e., monthly). Initially, we ll be using annual compounding. So we need P/YR set to 1. The keys to do this are: 1; Shift [the green or gold key]; P/YR. P/YR is not cleared or reset when the calculator is cleared. It is there until you explicitly change it. To display what the calculator thinks P/YR is: Shift; Clear All. Remember to Clear All before starting a new problem. To change the number of decimal digits displayed to 3, use: Shift; Disp; 3. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 5

Calculators, etc. We normally assume that a period 1 cash flow occurs at the end of period 1. (The end of period 0 is today.) For most problems, this is the appropriate setup. The BEGIN key on a financial calculator causes the calculator to assume that cash flows occur at the beginning of the period. ( Today is the beginning of period 1.) Most calculators show a BEGIN flag in this case (and show END or nothing at all otherwise). BEGIN does not mean Begin a new calculation. On an HP-12c, the interest key is labeled i. This is the interest rate per period, not per year. (They might differ if the problem involves compounding within the year.) May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 6

Future value on a caculator $1000 on deposit for 5 years at 5%. How much would you have? 1; Shift; P/YR (In the notes, I won t always include this step explicitly) Shift; Clear All (I ll assume that we always include this step.) 5 N 5 I/Y 1000 PV FV = 1276.28 Why does the FV appear with a negative sign? Calculators use a + sign for cash inflows and a for cash outflows. From the viewpoint of our bank account: 1000 is coming in today; 1276 is going out (a withdrawal) in 5 years. The magnitudes don t change if all signs are reversed: 5 N; 5 I/YR; 1000 PV; FV 1276.28 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 7

The long-run return on common stock According to Stocks for the Long Run, by Jeremy Siegel, the average annual return on common stocks was 8.4% over the period from 1802-1997. Suppose a distant ancestor of yours had invested $1,000 in a diversified common stock portfolio in 1802. Assuming the portfolio remained untouched, how large would that portfolio be at the end of 1997? May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 8

Present Values Example 1 You want to begin saving for you 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? 17 N 8 I/YR 150,000 FV PV = 40,540.34 (remember the sign convention) May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 9

Present Values Example 2 Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest? 10 N 7 I/YR 19,671.51 FV PV = 10,000 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 10

What interest rate should we use when we compute a present value? I ve just won the Genovia lottery: I ll receive $100,000 (US) in 6 years. The government has offered to give me the alternative of receiving $60,000 today. I can borrow at 10%, but invest at 7%. Should I take the government s offer? May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 11

Solving for the interest rate Example 1 You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest? May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 12

Solving for the interest rate Example 2 Benjamin Franklin died on April 17, 1790. In his will, he gave 1,000 pounds sterling to Massachusetts and the city of Boston. He gave a like amount to Pennsylvania and the city of Philadelphia. Franklin originally specified that the money should be paid out 100 years after his death and used to train young people. Later, it was agreed that the money would be paid out 200 years after Franklin s death in 1990. By that time, the Pennsylvania bequest had grown to about $2 million; the Massachusetts bequest had grown to $4.5 million. Assuming that 1,000 pounds sterling was equivalent to 1,000 dollars, what rate did the two states earn? (Note: the dollar didn t become the official U.S. currency until 1792.) May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 13

Solving for the number of periods Example 1 You want to purchase a new car and you are willing to pay $21,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 14

Cash flow valuation (Ch. 5) Future and Present Values of Multiple Cash Flows Valuing Level Cash Flows: Annuities and Perpetuities Comparing Rates: The Effect of Compounding Periods Loan Types and Loan Amortization May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 15

Multiple cash flows: Example We have $7,000 in a bank account that pays 8%. At the end of each of the next three years, we plan to deposit $4,000. How much will we have at the end of three years? Four years? Time line: 0 1 2 3 4 7,000 4,000 4,000 4,000 Find the value at year 3 of each cash flow and add them together. Today (year 0): FV of 7,000 in 3 years = 8,817.98 Year 1: FV of 4,000 in 2 years = 4,665.60 Year 2: FV of 4,000 in 1 year = 4,320 Year 3: value = 4,000 Total = 8817.98 + 4665.60 + 4320 + 4000 = 21,803.58 Value at year 4 = FV of 21,803.58 in 1 year = 23,547.87 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 16

Valuing multiple cash flows: saving for retirement You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years [at the end of the 40th year]. How much would you be willing to invest today if you desire an interest rate of 12%? Time line: 0 1 39 40 41 44 0 0 0 25,000 25,000 25,000 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 17

Solution I 0 1 2 39 40 44 +25 +25 Find the PV of the annuity as of the end of year 39: 25 PMT; 12 I; 5 N; PV= 90.1194 The annuity has the same value as 90.1194 received at the end of year 39. The present value of 90.1194 to be received in 39 years is [90.1194 FV; 12 I; 39 N] PV=1.0847 ($ thousand). May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 18

Solution II 0 1 2 39 40 44 +25 +25 We can get the desired cash flows by taking a 44-year annuity and subtracting a 39-year annuity: 0 1 2 39 40 44 +25 +25 +25 +25 +25-25 -25-25 PV of the first annuity is [12 I; 44 N; 25 PMT] = 206.9105 PV of the second annuity is [12 I 39 N; 25 PMT] = 205.8257 The difference is 206.9105 205.8257 = 1.0848 ($ thousand). May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 19

Annuities and Perpetuities Defined Annuity finite series of equal payments that occur at regular intervals If the first payment occurs at the end of the period, it is called an ordinary annuity If the first payment occurs at the beginning of the period, it is called an annuity due Perpetuity infinite series of equal payments May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 20

Annuities and the Calculator You can use the PMT key on the calculator for the equal payment The sign convention still holds Ordinary annuity versus annuity due The usual calculator default is that the payment is made at the end of the period (an ordinary annuity). To tell the calculator to assume payments are at the beginning of the period, use [shift] BEG/END (This will toggle back and forth.) May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 21

Annuity Due You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years? May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 23

A WSJ Advertisement Before you buy a CD, read this. Consider an alternative. It's a Single Premium Deferred Annuity, SPDA. Consider the advantages of an SPDA It offers high interest rates that can give you all the impact of a CD. Interest accumulates with no tax liability until you actually receive cash. It offers valuable annuitization options. (A typical $20,000 SPDA purchased at age 45 could provide an income at age 65 of $8,698 every year for life.) SPDA. We think it's a better way. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 25

Analysis Age45 65 66 67 0 20 21 22 20,000 8,698 8,696 At an investment rate of 8%, how much will the seller of the annuity (the insurance company) have to fund the annuity at the end of year 20? How many payments will the insurer be able to make? Redo the analysis assuming an interest rate of 12%. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 26

Perpetuities A perpetuity is an annuity with an infinite horizon. The PV = p/r. Ex: On 1/17/94, a share of GM preferred series D pays a 1.98 annual dividend. the annual yield on a long-term corporate bond was 7.5% PV = 1.98/.075 = 26.40 (Actual closing share price was 26 1/8) May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 27

Intraperiod Compounding. Compounding refers to when the interest gets added to the account. Ex: With annual interest, the FV of 100 at 12% for one year is 112. With semiannual compounding: FV=100(1.06)(1.06)=106(1.06)=112.36 With quarterly compounding, FV=100(1.03) 4 = 112.55 With monthly compounding, FV=100(1.01) 12 = 112.68. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 28

Effective Annual Rate (EAR) This is the actual rate paid (or received) after accounting for compounding that occurs during the year If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison. In the previous problem, we say: An annual percentage rate (APR) of 12% compounded monthly has an effective annual rate (EAR) of 12.68%. I am indifferent between investing at 12% compounded monthly or 12.68% compounded annually. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 29

Annual Percentage Rate This is the annual rate that is quoted by law By definition APR = period rate times the number of periods per year Consequently, to get the period rate we rearrange the APR equation: Period rate = APR / number of periods per year You should NEVER divide the effective rate by the number of periods per year it will NOT give you the period rate May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 30

Computing APRs What is the APR if the monthly rate is.5%?.5(12) = 6% What is the APR if the semiannual rate is.5%?.5(2) = 1% What is the monthly rate if the APR is 12% with monthly compounding? 12 / 12 = 1% May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 31

Things to Remember You ALWAYS need to make sure that the interest rate and the time period match. If you are looking at annual periods, you need an annual rate. If you are looking at monthly periods, you need a monthly rate. If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 32

Computing EARs - Example 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 (EAR = 12.68%) On the HP-10b, the APR is the NOM% rate, and the EAR is the EFF% rate. 12 [shift] P/YR; 12 [shift] NOM%; [shift] EFF% display 12.68 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 33

In another account, you earn 3% per quarter. What is the APR? 3(4) = 12% How much are you effectively earning? 4 [shift] P/YR; 12 [shift] NOM%; [shift] EFF% display 12.55 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 34

Finding the best investment 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? May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 35

Going in the other direction: APR - Example 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 they pay? 12 [shift] P/YR; 12 [shift] EFF%; [shift] NOM% display 11.39% May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 36

Loan amortization Amortization is the payback of the amount borrowed. The payments on a loan will cover the interest owed on the loan and the full amortization of the principal. The amounts and timing may vary depending on the type of the loan. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 37

The usual case: Loans with constant payment size Ex: 10,000 car loan with 3 annual payments, r=10%. 0 1 2 3 p p p 10,000 PV; 3 N; 10 I/YR; PMT display 4,021 Amortization: 1 2 3 Loan balance 10,000 6,979 3,656 (beg of yr) Interest 1,000 698 366 Payment 4,021 4,021 4,021 Amortization 3,021 3,323 3,655 Loan balance 6,979 3,656 1 (end of yr) May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 38

House mortgages $100,000, 30-year mortgage at 10% compounded monthly; monthly payments. 12 [shift] P/YR; 100,000 PV; 360 N; 10 I/YY; PMT display 877.57 For the first payment, 833.33 is interest; 44.24 is principal. After the 12 th payment, the loan balance is 99,444.13 You ve paid the bank 12 x 878 = 10,531. Your loan has only been reduced by 556. Last payment: 7.28 is interest; 870.29 is principal. You can get this using your calculator s amortization function or x + (.10/12)x = 877.57 x = 870.32 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 39

Variations: Deferred payments and subsidized interest Sometimes with a loan Payment might be deferred. Interest accrual might be deferred The interest rate might be subsidized May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 40

Case study: student loans Terms $10,000 loan at 5% 2 years until graduation; Repayment with 5 equal annual payments (first pmt one year after graduation) Timeline: 0 1 2 3 4 5 6 7 today grad n +$10,000 p p p p p May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 41

If no interest accrual until graduation, we will owe $10,000 at graduation. What is the size of the loan payment? Suppose that we can invest the loan proceeds at 8% How much will we have in the bank when we graduate? If we make the loan payments out of the bank account, how much will we have at the end of year 7? What is the net worth of our position at graduation? Should we repay the loan immediately? May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 42

Solving for the number of periods (supplemental) What does N=3.5303 years mean? After 3 years, our 15,000 will have grown to: 15,000 PV; 10 I/YR; 3 N; FV=19,965 Over the next 0.53 years, this will grow to: 19,965 (1.10)0.5303 = 21,000 1.10; Shift; yx; 0.5303; = ; x ( times ); 19965; = or: 19965 PV; 10 I/YR; 0.5303 N; FV: 21,000 This implies that the interest is 21000 19965=1035 Note: This is 0.5303 x 0.10 x 19965 (=1059) Interest does not accumulate proportionately throughout the year. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 44

Effective and nominal rates on the HP-12C The HP-12C doesn t have NOM% and EFF% functions. Instead, you program simple investment problems. An APR of 12% compounded monthly is equivalent to what EAR? Clear FIN 12,, 12, [Figure out the monthly interest rate] i, 12, n, -1, PV, FV [display shows 1.1268] 1, n, i, [display shows 12.68] An EAR of 12.68% is equivalent to what monthly APR? Clear FIN 1.1268, FV, -1, PV, 12, n, I [display shows 1%], 12,, [display shows 12%] May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 45

Variations: Loans with constant amortization The amortization must be 10,000/3 = 3,333/yr Amortization 1 2 3 Loan balance 10,000 6,667 3,333 (beg of yr) Interest 1,000 667 333 Amortization 3,333 3,333 3,333 Payment 4,333 4,000 3,666 Loan balance 6,667 3,333 0 (end of yr) Payment size does not stay fixed. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 46

Variations: Initial loan payments are interest only Amortization schedule 1 2 3 Loan balance 10,000 10,000 10,000 (beg of yr) Interest 1,000 1,000 1,000 Amortization 0 0 10,000 Payment 1,000 1,000 11,000 Loan balance 10,000 10,000 0 (end of yr) May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 47

Variations: loans with a teaser rate Take out a 100,000 30-yr mortgage. Pay 5% the first two years, and 10% for the next 28 years. Payments in first two years: 100,000 PV; 30 N; 5%; PMT display 6,505 Amortization schedule for first two years: 1 2 Beg balance 100,000 98,495 Interest 5,000 4,925 Payment 6,505 6,505 Amortization 1,505 1,580 End balance 98,495 96,915 Payments for next 28 years: 96,915 PV; 28 N; 10 I/YR; PMT display 10,414 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 48

Variations: Bullet loans Take out a 100,000 mortgage at 10%. Payments are computed on a 30-year basis. The loan must be paid off at the end of the fifth year. The first four payments: 100,000 PV; 30 N; 10 I/YR; PMT display 10,608. 0 1 2 3 4 5 10,608 10,608 10,608 10,608 x PV of all payments must be 100,000. 10,608 PMT; 4 N; 10%; PV display 33,626 100,000 33,626 = PV of x; 66,374 = PV of x 66,374 PV; 5 N; 10 I/YR; FV display 106,896 = x May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 49

Variations: Loans with negative amortization Take out a 100,000, 10%, 30 year mortgage. Make no payments for the first 2 years.* *Interest will continue to accrue on the unpaid balance. Amortization in first two years 1 2 Beg balance 100,000 110,000 Interest 10,000 11,000 Payment 0 0 Amortization 10,000 11,000 End balance 110,000 121,000 The last 28 payments will be: 121,000 PV; 30 N; 10 I/YR; PMT display 13,002. May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 50

Multiple Uneven Cash Flows Using the HP-10b Calculator Use the NPV function ( net present value ). The cash flows are entered using the CFj key [Clear] 0 CFj 1,000 CFj 2,000 CFj 3,000 CFj 10 I/YR [shift] NPV displays 4,815.92 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 51

Valuing an investment Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment? 0 CFj 40 CFj 75 CFj 15 I/YR [shift] NPV display 91.49 Why pay $100 for something that s only worth 91.49 [to you]? May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 52

Saving For Retirement You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years [at the end of the 40 th year]. How much would you be willing to invest today if you desire an interest rate of 12%? Time line: 0 1 39 40 41 44 0 0 0 25,000 25,000 25,000 0 CFj 0 CFj 39 [shift] Nj 25,000 CFj 5 [shift] Nj 12 I/YR [shift] NPV display 1,084.71 May 16, 2005 Copyright (c) 2005, Joel Hasbrouck, All rights reserved. 53