NPV and The Time Value of Money

P.V. VISWANATH

FOR A FIRST COURSE IN FINANCE

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Learning Objectives

NPV and IRR How do we decide to invest in a project or not?

Using the Annuity Formula Valuing Mortgages and Similar payment plans

Valuing Simple Financial Securities

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NPV and the Internal Rate of Return

The NPV decision rule says: Accept a project if NPV>0.

There is another decision rule based on the Internal Rate of Return (IRR).

The IRR is the rate of return that makes the NPV = 0.To understand what the IRR is, we can use the

concept of the NPV profile.The NPV profile is the function that shows the NPV of

the project for different discount rates.Then, the IRR is simply the discount rate where the

NPV profile intersects the X-axis.That is, the discount rate for which the NPV is zero.

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Internal Rate of Return

Suppose we are looking at a new project and you have estimated the following cash flows: Year 0: -165,000 (required initial investment) Year 1: 63,120 Year 2: 70,800 Year 3: 91,080

Suppose the required rate of return is 12%. Then we can compute the NPV as the sum of the discounted present values of these cashflows.

NPV = -165000 + 63120/(1.12) + 70,800/(1.12)2 + 91080/(1.12)3 = 12,627.42

If we use different discount rates, we will get different NPVs, as shown in the next graph.

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NPV Profile For The Project

Clearly, the IRR decision rule corresponding to the NPV rule is: Accept a project if the IRR > Required Rate of Return.

-20,000

-10,000

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

Discount Rate

NP

V

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Terminology: Present and Future Value

Present Value – earlier money on a time lineFuture Value – later money on a time line

0 2 3 54 61

100

100 100

100 100100

If a project yields $100 a year for 6 years, we may want to know the value of those flows as of year 1; then the year 1 value would be a present value.

If we want to know the value of those flows as of year 6, that year 6 value would be a future value.

If we wanted to know the value of the year 4 payment of $100 as of year 2, then we are thinking of the year 4 money as future value, and the year 2 dollars as present value.

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Rate Terminology

Interest rate – “exchange rate” between earlier money and later money (normally the later money is certain).

Discount Rate – rate used to convert future value to present value. Compounding rate – rate used to convert present value to future value. Required rate of return – the rate of return that investors demand for

providing the firm with funds for investment. This is from the investors’ point of view. The higher the rate of return available, the more investors are willing to supply.

Cost of capital – the rate at which the firm obtains funds for investment; this is from the firm’s point of view. The lower the rate that firms have to pay, the more funds they will demand since more investment projects will meet the hurdle rate of return, i.e. the cost of firms’ funds.

The total amount of funds that will be lent will be equal to the amount at which the investors’ required rate of return will equal the amount that the firms is willing to pay. Hence in equilibrium, the cost of capital will be equal to the investors’ required rate of return.

Opportunity cost of capital – the rate that the firm has to pay investors in order to obtain an additional $ of funds, i.e. this is the marginal cost of capital. This is the rate that investors demand from this firm because if the firm doesn’t pay this this much, they can get that return from other demanders of capital.

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Relation between rates

If capital markets are in equilibrium, the rate that the firm has to pay to obtain additional funds will be equal to the rate that investors will demand for providing those funds. This will be “the” market rate.

Hence this is the single rate that should be used to convert future values to present values and vice-versa.

Hence this should be the discount rate used to convert future project (or security) cashflows into present values.

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Future Values

Suppose you invest $1000 for one year at 5% per year. What is the future value in one year? The compounding rate is given as 5%. Hence the value of current

dollars in terms of future dollars is 1.05 future dollars per current dollar.

Hence the future value is 1000(1.05) = $1050. Suppose you leave the money in for another year. How much will

you have two years from now? Now think of money next year as present value and the money in two

years as future value. Hence the price of one-year-from-now money in terms of two-years-from-now money is 1.05.

Hence 1050 of one-year-from-now dollars in terms of two years-from-now dollars is 1050(1.05) = 1000 (1.05)(1.05) = 1000(1.05)2 = 1102.50

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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

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Effects of Compounding

Simple interest Compound interest The notion of compound interest is relevant when money is invested for more

than one period. After one period, the original amount increases by the amount of the interest paid

for the use of the money over that period. After two periods, the borrower has the use of both the original amount invested

and the interest accrued for the first period. Hence interest is paid on both quantities.

This is why if the interest rate is r% per period, then a $1 today grows to $(1+r) tomorrow and $(1+r)2 in two periods.

(1+r)2 = 1+2r+r2 . The 2r is the “simple” interest for each of the two periods and the r2 = r x r is the interest for the second period on the $r of interest earned in the first period.

This computation is done automatically when we use the formula FV(C in t periods) = C(1+r)t

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Growth of $100 over time

From Brealey, Myers and Allen, “Principles of Corporate Finance”

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Growth of principal at different rates of interest

From Brealey, Myers and Allen, “Principles of Corporate Finance”

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Future Values – Example 2

Suppose you invest the $1000 from the previous example for 5 years. How much would you have? FV = 1000(1.05)5 = 1276.28

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 $1250, for a difference of $26.28.)

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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 = 447,189.84

What is the effect of compounding? Without compounding the future value would have

been the original $10 plus the accrued interest of 10(0.055)(200), or 10 + 110 = $120.

Compounding caused the future value to be higher by an amount of $447,069.84!

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Future Value as a General Growth Formula

Suppose your company expects to increase unit sales of books by 15% per year for the next 5 years. If you currently sell 3 million books in one year, how many books do you expect to sell in 5 years? FV = 3,000,000(1.15)5 = 6,034,072

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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

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.

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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 = 9345.79

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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

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Present Values – Example 3

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? PV = 19,671.51 / (1.07)10 = 10,000

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PV – Important Relationship I

For a given interest rate – the longer the time period, the lower the present value What is the present value of $500 to be received

in 5 years? 10 years? The discount rate is 10% 5 years: PV = 500 / (1.1)5 = 310.46 10 years: PV = 500 / (1.1)10 = 192.77

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PV – Important Relationship II

For a given time period – the higher the interest rate, the smaller the present value What is the present value of $500 received in 5

years if the interest rate is 10%? 15%? Rate = 10%: PV = 500 / (1.1)5 = 310.46 Rate = 15%; PV = 500 / (1.15)5 = 248.58

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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

FV = PV(1+r) t

r = (FV/PV)1/t – 1t = ln(FV/PV) ln(1+r)

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Discount 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? r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%

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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 = .122462 = 12.25%

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Discount Rate – Example 3

Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5000 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 = .172688 = 17.27%

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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)

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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

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Number of Periods – Example 2

Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% in closing costs. If the type of house you want costs 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?

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Example 2 Continued

How much do you need to have in the future? Down payment = .1(150,000) = 15,000 Closing costs = .05(150,000 – 15,000) = 6,750 Total needed = 15,000 + 6,750 = 21,750

Using the formula t = ln(21,750/15,000) / ln(1.075) = 5.14 years

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Present Value of an Annuity

The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present, and adding up the present values. Alternatively, there is a short cut that can be used in the calculation [A = Annuity; r = Discount Rate; n = Number of years]

nrr

AnrAPVAnnuityanofPV

)1(

11),,(

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Example: PV of an Annuity

The present value of an annuity of $1,000 at the end of each year for the next five years, assuming a discount rate of 10% is -

PV of $1000 each year for next 5 years = $1000

1 - 1

(1.10)5

.10

$3,791

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Annuity, given Present Value

The reverse of this problem, is when the present value is known and the annuity is to be estimated - A(PV,r,n).

Annuity given Present Value = A(PV, r,n) = PV r

1 - 1

(1 + r)n

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Computing Monthly Payment on a Mortgage

Suppose you borrow $200,000 to buy a house on a 30-year mortgage with monthly payments. The annual percentage rate on the loan is 8%.

The monthly payments on this loan, with the payments occurring at the end of each month, can be calculated using this equation: Monthly interest rate on loan = APR/12 = 0.08/12

= 0.0067Monthly Payment on Mortgage = $200,000 0.0067

1 - 1

(1.0067)360

$1473.11

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Future Value of an Annuity

The future value of an end-of-the-period annuity can also be calculated as follows-

FV of an Annuity = FV(A,r,n) = A (1 + r)n - 1

r

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An Example

Thus, the future value of $1,000 at the end of each year for the next five years, at the end of the fifth year is (assuming a 10% discount rate) -FV of $1,000 each year for next 5 years = $1000

(1.10)5 - 1

.10

= $6,105

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Annuity, given Future Value

If you are given the future value and you are looking for an annuity, you can use the following formula:

Annuity given Future Value = A(FV, r,n) = FV r

(1+ r)n - 1

Note, however, that the two formulas, Annuity, given Future Value and Present Value, given annuity can be derived from each other, quite easily. You may want to simply work with a single formula.

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Application : Saving for College Tuition

Assume that you want to send your newborn child to a private college (when he gets to be 18 years old). The tuition costs are $16000/year now and that these costs are expected to rise 5% a year for the next 18 years. Assume that you can invest, after taxes, at 8%. Expected tuition cost/year 18 years from now = 16000*(1.05)18 =

$38,506 PV of four years of tuition costs at $38,506/year = $38,506 *

PV(A ,8%,4 years) = $127,537 If you need to set aside a lump sum now, the amount you would

need to set aside would be - Amount one needs to set apart now = $127,357/(1.08)18 = $31,916

If set aside as an annuity each year, starting one year from now - If set apart as an annuity = $127,537 * A(FV,8%,18 years) = $3,405

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Valuing a Straight Bond

You are trying to value a straight bond with a fifteen year maturity and a 10.75% coupon rate. The current interest rate on bonds of this risk level is 8.5%.PV of cash flows on bond = 107.50* PV(A,8.5%,15 years) +

1000/1.08515 = $ 1186.85 If interest rates rise to 10%,

PV of cash flows on bond = 107.50* PV(A,10%,15 years)+ 1000/1.1015 = $1,057.05

Percentage change in price = -10.94% If interest rate fall to 7%,

PV of cash flows on bond = 107.50* PV(A,7%,15 years)+ 1000/1.0715 = $1,341.55

Percentage change in price = +13.03%

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Valuing a Consol Bond

A consol bond is a bond that has no maturity and pays a fixed coupon. Assume that you have a 6% coupon console bond. The value of this bond, if the interest rate is 9%, is as follows -Value of Consol Bond = $60 / .09 = $667

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V. Growing Perpetuities

A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity is -

where CF1 is the expected cash flow next year, g is the constant growth rate and r is the discount rate.

PV of Growing Perpetuity = CF1

(r - g)