Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 0 Chapter 5 Introduction to Valuation: The Time Value of Money

Key Concepts and Skills Be able to compute the future value of an investment made today Be able to compute the present value of cash to be received at some future date Be able to compute the return on an investment

Chapter Outline Future Value and Compounding Present Value and Discounting More on Present and Future Values

TIME INTEREST TIME allows you the opportunity to postpone consumption and earn INTEREST. Why TIME? TIME Why is TIME such an important element in your decision?

Basic Definitions Present Value – The current value of future cash flows discounted at the appropriate discount rate. earlier money on a time line Future Value – The amount an investment is worth after one or more periods later money on a time line

Basic patterns of Cash flows 1.Single Amount: A lump sum amount either currently held or expected at some future date. 2.Annuity: A level periodic stream of cash flow 3.Mixed Streams: A stream of cash flows that is not an annuity, a stream of unequal periodic cash flows with no particular pattern. © 2012 Pearson Prentice Hall. All rights reserved. 5-5

Interest rate – “exchange rate” between earlier money and later money –Discount rate –Cost of capital –Opportunity cost of capital –Required return 6

Time lines are used to illustrate these relationships. A horizontal line on which time zero appears at the left most end and future periods are marked from left to right, can be used to depict investment cash flows. 7

Future Values Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year? –Interest = $ 1,000(.05) = $ 50 –Value in one year = principal + interest = $ 1, = $ 1,050 –Future Value (FV) = $ 1,000(1 +.05) = $ 1,050 Suppose you leave the money in for another year. How much will you have two years from now?  FV = $ 1,000(1.05)(1.05) = $ 1,000(1.05) 2 = $ 1,102.50

Future Values: General Formula FV = PV(1 + r) t –FV = future value –PV = present value –r = period interest rate, expressed as a decimal –T = number of periods Future value interest factor = (1 + r) t OR FVIF ( for Table)

Effects of Compounding Simple interest (interest is earned only on the original principal) Compound interest (interest is earned on principal and on interest received) Consider the previous example –FV with simple interest = $ 1, = $ 1,100 –FV with compound interest = $ 1, –The extra $ 2.50 comes from the interest of.05( $ 50) = $ 2.50 earned on the first interest payment

Figure 4.1

Future Values – Example 2 Suppose you invest the $1,000 from the previous example for 5 years. How much would you have?  FV = $1,000(1.05) 5 = $1, The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1,250, for a difference of $26.28.)

Future Values – Example 3 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, What is the effect of compounding? –Simple interest = $10 + $10(200)(.055) = $120 –Compounding added $447, to the value of the investment

Present Values How much do I have to invest today to have some amount in the future?  FV = PV(1 + r) t  Rearrange to solve for PV = FV / (1 + r) t  (1 + r) t or PVIF present value interest Factor (Table) When we talk about discounting, we mean finding the present value of some future amount. When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

PV – One-Period Example Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today? PV = $10,000 / (1.07) 1 = $9,345.79

Present Values – Example 2 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

Present Values – Example 3  Your parents set up a trust fund for you 10 years ago that is now worth $19, If the fund earned 7% per year, how much did your parents invest?  PV = $19, / (1.07) 10 = $10,000

Present value: Important relationship I For a given interest rate: –The longer the time period, the lower the present value. For a given r, as t increases, PV decreases. 4-18

Present value: Important relationship II For a given time period: –The higher the interest rate, the smaller the present value. For a given t, as r increases, PV decreases. Copyright  2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4-19

Single-Period PV Suppose you need $400 to buy textbooks next year. You can earn 7% on your money. How much do you have to put up today?

The Basic PV Equation - Refresher PV = FV / (1 + r) t There are four parts to this equation –PV, FV, r, and t –If we know any three, we can solve for the fourth

Discount Rate Often, we will want to know what the implied interest rate is in an investment Rearrange the basic PV equation and solve for r  FV = PV(1 + r) t  r = (FV / PV) 1/t – 1 If you are using formulas, you will want to make use of both the y x and the 1/x keys

Discount Rate – Example 1 You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest?  r = ($1,200 / $1,000) 1/5 – 1 = = 3.714%

Discount Rate – Example 2 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 = = 12.25%

Discount Rate – Example 3 Suppose you have a 1-year old son and you want to provide $75,000 in 17 years toward his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it?  r = ($75,000 / $5,000) 1/17 – 1 = = 17.27%

Finding the Number of Periods Start with basic equation and solve for t (remember your logs)  FV = PV(1 + r) t  t = ln(FV / PV) / ln(1 + r)

Number of Periods – Example 1 You want to purchase a new car and you are willing to pay $20,000. 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?  t = ln($20,000 / $15,000) / ln(1.1) = 3.02 years

Saving Up You would like to buy a new automobile. You have $50,000 or so, but the car costs $68,500. If you can earn 9%, how much do you have to invest today to buy the car in two years? Do you have enough? Assume the price will stay the same.

Evaluating Investments To give you an idea of how we will be using present and future values, considering the following simple investment. Your company proposes to buy an asset for $335. This investment is very safe. You would sell off the asset in three years for $400. You know you could invest the $335 elsewhere at 10% with very little risk. What do you think of the proposed investment?

Saving for College You estimate that you will need about $80,000 to send your child to school college in eight years. You have about $35,000 now. If you can earn 20% per year, will you make it? At what rate will you just reach your goal?

Only 18,262.5 Days to Retirement You would like to retire in 50 years as a millionaire. If you have $10,000 today, what rate of return do you need to earn to achieve your goal?

Valuing level cash flows Annuities and perpetuities 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 5-32

Annuities and perpetuities Basic formulas Perpetuity: PV = C/r Annuities: 5-33

Examples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings

Parts of an Annuity $100 $100 $100 (Ordinary Annuity) End End of Period 1 End End of Period 2 Today Equal Equal Cash Flows Each 1 Period Apart End End of Period 3

Parts of an Annuity $100 $100 $100 (Annuity Due) Beginning Beginning of Period 1 Beginning Beginning of Period 2 Today Equal Equal Cash Flows Each 1 Period Apart Beginning Beginning of Period 3

FVA 3 FVA 3 = $1,000(1.07) 2 + $1,000(1.07) 1 + $1,000(1.07) 0 $3,215 = $1,145 + $1,070 + $1,000 = $3,215 Example of an Ordinary Annuity -- FVA $1,000 $1,000 $1, $3,215 = FVA 3 7% $1,070 $1,145 Cash flows occur at the end of the period

FVA n FVA 3 $3,215 FVA n = R (FVIFA i%,n ) FVA 3 = $1,000 (FVIFA 7%,3 ) = $1,000 (3.215) = $3,215 Valuation Using Table III

PVA 3 PVA 3 = $1,000/(1.07) 1 + $1,000/(1.07) 2 + $1,000/(1.07) 3 $2, = $ $ $ = $2, Example of an Ordinary Annuity -- PVA $1,000 $1,000 $1, $2, = PVA 3 7% $ $ $ Cash flows occur at the end of the period

PVA n PVA 3 $2,624 PVA n = R (PVIFA i%,n ) PVA 3 = $1,000 (PVIFA 7%,3 ) = $1,000 (2.624) = $2,624 Valuation Using Table IV

Annuity Example 5.5 You can afford $632 per month. Going rate = 1%/month for 48 months. How much can you borrow? You borrow money TODAY so you need to compute the present value. 5-41

Annuity—Sweepstakes example Suppose you win the Publishers Clearinghouse $10 million. The money is paid in equal annual instalments of $ over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? –PV = [1 – 1/ ] /.05 = $

Quick quiz: Part 1 You know the payment amount for a loan and you want to know how much was borrowed. –Do you compute a present value or a future value? 5-43

Finding the number of payments $1000 is due on a credit card Payment = $20 month minimum Rate = 1.5% per month –How long would it take to pay off the $1000? –Formula solution: 1000 = 20(1 – 1/1.015 t ) / = 1 – 1 / t 1 / t =.25 1 /.25 = t t = ln(1/.25) / ln(1.015) = months = 7.75 years 5-44

Finding the number of payments— Another example Suppose you borrow $2,000 at 5% and you are going to make annual payments of $ How long before you pay off the loan? –2000 = (1 – 1/1.05 t ) /.05 – = 1 – 1/1.05 t –1/1.05 t = – = 1.05 t –t = ln( )/ln(1.05) = 3 years

Solving for r Suppose you borrow $1,000 and loan arrangement requires you to pay $282 per year for the next 4 years.If the payment made at the end of each year, what interest rate are you paying on the loan? Ans r=5% 46

Annuity due An annuity for which the cash flows occur at the beginning of the period. You are saving for a new house and you put $ per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years? 5-47

Annuity due formula 48

Perpetuities— Example 5.7 An annuity in which the cash flows continue forever. Perpetuity formula: PV = C / r If $100 receives each year forever and interest rate is 8% pv of perpetuity? Ans:$1,

Quick quiz You are considering preferred stock that pays a quarterly dividend of $1.50. If your desired return is 3% per quarter, how much would you be willing to pay? –$1.50/0.03 = $

PV OF UNEVEN CASH FLOWS Mixed Streams 3-51 Pattern of Unequal periodic cash flows that reflect no particular pattern.

FV OF UNEVEN CASH FLOWS 3-52

“Piece-At-A-Time” $600 $600 $400 $400 $100 $600 $600 $400 $400 $100 10% $545.45$495.87$300.53$ $ $ = PV 0 of the Mixed Flow

“Group-At-A-Time” (#1) $600 $600 $400 $400 $100 $600 $600 $400 $400 $100 10% $1, $ $ $1, = PV 0 of Mixed Flow [Using Tables] $600(PVIFA 10%,2 ) = $600(1.736) = $1, $400(PVIFA 10%,2 )(PVIF 10%,2 ) = $400(1.736)(0.826) = $ $100 (PVIF 10%,5 ) = $100 (0.621) = $62.10

Read problem thoroughly 2. Create a time line 3. Put cash flows and arrows on time line 4. Determine if it is a PV or FV problem 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem Steps to Solve Time Value of Money Problems

General Formula: PV 0 FV n = PV 0 (1 + [r/m]) mn n: Number of Years m: Compounding Periods per Yeari: Annual Interest Rate FV n,m : FV at the end of Year n PV 0 PV 0 : PV of the Cash Flow today Frequency of Compounding

$1,000 Julie Miller has $1,000 to invest for 2 Years at an annual interest rate of 12%. 1,000 1, Annual FV 2 = 1,000(1 + [0.12/1]) (1)(2) = 1, ,000 1, Semi FV 2 = 1,000(1 + [0.12/2]) (2)(2) = 1, Impact of Frequency

,000 1, Qrtly FV 2 = 1,000(1 + [0.12/4]) (4)(2) = 1, ,000 1, Monthly FV 2 = 1,000(1 + [0.12/12]) (12)(2) = 1, ,000 1, Daily FV 2 = 1,000(1 + [0.12/365]) (365)(2) = 1, Impact of Frequency

FV n (continuous compounding) = PV x (e k x n ) where “e” has a value of Continuing with the previous example, find the future value of the $100 deposit after 5 years if interest is compounded continuously. Continuous Compounding With continuous compounding the number of compounding periods per year approaches infinity. Through the use of calculus, the equation thus becomes: FV n = 100 x (2.7183).12 x 5 = $182.22

Computing payments with APRs Suppose you want to buy a new computer system and the store is willing 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? 5-60

Future values with monthly compounding Suppose you deposit $50 a month into an account that has an interest rate of 9%, based on monthly compounding. How much will you have in the account in 35 years? 5-61

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 interest rate of 5.5% based on daily compounding, how much would you need to deposit? 5-62

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. 5-63

Annual percentage rate (APR) This is the annual rate that is quoted by law. By definition APR = period rate times the number of periods per year. So, 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. 5-64

Computing APRs What is the APR if the monthly rate is.5%? –.5(12) = 6% What is the APR if the semi-annual rate is.5%? –.5(2) = 1% What is the monthly rate if the APR is 12%, with monthly compounding? –12 / 12 = 1% –Can you divide the above APR by 2 to get the semi- annual rate? NO!!! You need an APR based on semi- annual compounding to find the semi-annual rate. 5-65

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. 5-66

EAR formula APR = the quoted rate m = number of compounds per year 5-67

Decisions, decisions… II Which savings accounts should you choose: –5.25%, with daily compounding –5.30%, with semiannual compounding First account: EAR = ( /365) 365 – 1 = 5.39% Second account: EAR = ( /2) 2 – 1 = 5.37% 5-68

Decisions, decisions… II (cont.) 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 = FV = 100( ) 365 = $ –Second account: Semiannual rate =.0539 / 2 =.0265 FV = 100(1.0265) 2 = $ You will have more money in the first account. 5-69

Computing APRs from EARs If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: m = number of compounding periods per year 5-70

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? 5-71

Why is it important to consider effective rates of return? See how the effective return varies between investments with the same nominal rate, but different compounding intervals. EAR ANNUAL 10.00% EAR SEMIANNUALLY 10.25% EAR QUARTERLY 10.38% EAR MONTHLY 10.47% EAR DAILY (365) 10.52% 5-72

EAR with continuous compounding EAR=e^r-1 73

Calculate the payment per period. 2.Determine the interest in Period t. (Loan Balance at t – 1) x (i% / m) principal payment 3.Compute principal payment in Period t. (Payment - Interest from Step 2) principal payment 4.Determine ending balance in Period t. (Balance - principal payment from Step 3) 5.Start again at Step 2 and repeat. Steps to Amortizing a Loan

Amortised loan with fixed payment— Example Each payment covers the interest expense plus reduces principal. Consider a 5-year loan with annual payments. The interest rate is 9% and the principal amount is $5000. –What is the annual payment? 5000 = PMT[1 – 1 / ] /.09  PMT =

Amortised loan with fixed payment Example: Amortisation table 5-76

Calculate Debt Outstanding 2.Calculate Debt Outstanding – The quantity of outstanding debt may be used in financing the day-to-day activities of the firm. 1. Determine Interest Expense – Interest expenses may reduce taxable income of the firm. Usefulness of Amortization