Net Present Value

Single Period Example – Review

• You have the opportunity to invest in your cousin Ralph’s lobster ranch. He has informed you that for a small $125,000 investment now you can expect to receive $137,000 in one year.

• If you require a 9% return on such investments should you invest with Ralph?

• What is the value of this opportunity?• If you require a 10% return, what is the value to

you now of being able to invest with Ralph?

Time Value of Money• As we have seen, because there are more options

open to someone who has money today rather than money next year there is a time value of money.– A dollar today is worth more than a dollar tomorrow.

This lesson is one of the few absolutes in finance.

• This is manifest in positive expected rates of return on investments which require the sacrifice of money today for money later.

• It also implies that you can’t add together cash flows that occur at different points in time until you properly discount them.

The Two-period Case

• One payment several periods from now:– We talked about getting cash next year, what if

it doesn’t come till two years from now?– One illustration: if will value a time 1

cash flow as of time zero, will value a time 2 cash payment as of time 1. We know how to change a time 1 value to a time 0 value:

)1(1

rC

222

2 )1()1(

)1()(

r

C

r

rCCPV

)1(2

rC

The Two-period Case

• One payment several periods from now:– A second view: if you have $100 cash today, and a

bank will give you 7% interest per year and you leave the money in the bank for two years, how much will you have?

– Answer: $100(1.07)(1.07) = $114.49. So $114.49 is the future value of $100 of current cash (at 7%). Algebra tells us that the present value of the future $114.49 must be $100. Calculate this as $114.49/(1.07)2 = $100.

– Notation: PV(C2) = C2/(1+r)2

– Generally: PV(Ct) = Ct/(1+r)t and FVt(C0) = C0(1+r)t

Multi-period Examples

• If you invest $15 for 20 years at 9% with no withdrawls what will be the final balance (future value)?

$15(1.09)20 = $84.07

• If you will receive $25,000 in 6 years and the relevant interest rate is 11%, what is the present value of this future payment?

$25,000/(1.11)6 = $13,366.02

Simple vs. Compound Interest

• Suppose that I have had some finance training and I know better than to stuff my $100,000 under my mattress. Instead I put it in the bank for 12 years at an 8% interest rate. Not having stayed till the end of the course, however, at the end of each year I withdraw the interest I earn and stuff it under my mattress. How much will I have at the end of the 12 years?– I’ll still have my $100,000 of principal and at the end of each of the

12 years I will have put $100,000(.08) = $8,000 under the mattress, leaving $100,000 + 12*$8,000 = $196,000.

• If I made no withdrawals during the 12 years I’d have $100,000(1.08)12 = $251,817.01– What drives the difference?

Simple vs. Compound Interest

This also applies in discounting. When discounting cash flow in the future, we must consider the opportunity cost of not earning interest on interest when we invest in a project and not the financial market.

Present Value of a Series of Future Cash Flows• What happens if we have an investment that

provides positive cash flows at many future dates?• Its very easy, discount all the future cash flows to

the present, then just add them up.– We can and should do this because once we have

discounted them, their present values all represent cash values today. Since all the values are as of the same time they can be added.

– In other words all our work discounting has restated all the future cash flows as their equivalent amounts at a common point in time.

Present Value of a Series of Future Cash Flows• Those are the words, here are the symbols:

• For NPV the adjustment is obvious:

T

tt

t

t

TT

T

r

C

r

C

r

C

r

C

r

CPV

1

33

32

2

2

1

1

)1(

)1()1()1()1(

T

tt

t

t

TT

T

r

CC

r

C

r

C

r

C

r

CCNPV

10

33

32

2

2

1

10

)1(

)1()1()1()1(

Who got the Better Contract?Emmitt or Thurman?

93 94 95 960

1

2

3

4

5

6

7

8

($ millions)

Thurman

Emmitt

Thurman 4 2.7 2.7 4.1

Emmitt 7 2.2 2.4 2

Comparing the pay packages(assuming 8% interest and all payments

at beginning of each year)

Thurman

PV = 4 + 2.7/(1.08) + 2.7/(1.08)2+ 4.1/(1.08)3 = $12.1 mil

Emmitt

PV = 7 + 2.2/(1.08) + 2.4/(1.08)2 + 2/(1.08)3 = $12.7 million

Discounted Cash Flow Analysis• The “Emmitt vs Thurman” comparison is an example of

Discounted Cash Flow (DCF) analysis. DCF analysis has many uses in the business world, particularly in corporate finance. Most business decisions can in some way be expressed as $X now versus $Y later.– Homework: Re-compute the PV’s in the example using 2% as the

appropriate rate. What happened and why?

• To use DCF we need to know three things:– The amount of the expected cash flows– The timing of the cash flows– The proper discount (interest) rate

• Should reflect market conditions and may include a premium for risk.

• DCF allows comparison of values of alternative cash flow streams in terms of dollars today.

Time Value – Extension

• Occasionally you will encounter decisions for which you know all of the cash flows and/or values and need to solve for an interest rate.

• Example:– A risk-free zero coupon bond is currently selling for $680.58. The

bond will make a single payment of $1,000 at the end of year 5. What is the interest rate (yield) on this bond?

• $680.58 = $1,000/(1+r)5

• 0.68058 = 1/(1+r)5

• 1.469335 = (1+r)5, 1.08 = 1+r, and r = 8%

– Here, we were able to solve for r directly. More generally, for instruments that have more than one future payment, the only way to solve for r is by trial and error or iteration.

Alternate Compounding Periods

Interest is sometimes “compounded” over periods other than annually. In terms of bank account examples, this means the interest is credited to the account more frequently.

Caveat: All of the time value of money formulas use the implicit assumption that the compounding interval is the same as the payment interval. E.g.:

• Mortgage loans call for monthly payments.

• Bonds make coupon payments semiannually.

Alternate Compounding Periods (Cont.)

• Let m denote the number of compounding intervals per year, and r be the stated (or simple) annual rate.

• The relation between present and future values is stated as:

FVn = PV(1 + r/m)nm

E.g., if PV = 1000, r = .12 and m = 1, then FV2 is:

FV2 = 1000(1 + .12)2 = $1254.40,

while if m = 4 (quarterly compounding), then

FV2 = 1000(1 + .12/4)2*4

FV2 = 1000(1 + .03)8 = $1266.77

Example

• Find the PV of $500 to be received in 5 years, with:

• 12% stated annual rate, annual compounding,.

71.283$

12.1

5005

PV

12% stated annual rate, quarterly compounding,

84.276$

)4/12(.1

50020

PV

PV

500

1 12 22010

(. / )$279.

12% stated annual rate, semiannual compounding,

Stated And Effective Annual Rates Notice that the use of more frequent compounding acts as

if to (or effectively) increase(s) the interest rate. The Effective Annual Rate (EAR) is the annual interest

rate that would produce the same answer with annual compounding as is obtained with more frequent compounding. It can be obtained by:

EAR = (1 + r/m)m - 1

so if r = .12 and m = 4, then EAR = (1.03)4 - 1 = .1255. What we call the effective annual rate is just that rate that

if you received it for a year with only annual compounding you would wind up with the same amount of money.

Or, what annual rate would you effectively have when the compounding is more frequently than once per year.

Example

• A bank quotes a mortgage rate of 8% (the stated annual rate), but will compute monthly loan payments using standard time value formulas. This implies monthly compounding. What is the effective annual interest rate on the loan?

EAR

1

0 08

121 0 0830

12..

So the loan effectively costs you 8.30% per year.

Valuing Streams of Structured Future Cash Flows

• Now we are going to discuss the valuation of certain highly structured cash flow streams.

• The resulting valuation formulas are useful for simplifying the analysis of certain situations.

• Pay attention to the exact timing of the cash flows, the formulas don’t work unless you get this right.– Drawing diagrams of the cash flows can be useful.

• These formulas can make life easier and so are worth understanding.

Perpetuity• A stream of equal payments, starting in one period,

and made each period, forever. Forever??

• Remember, this gives the value of this stream of cash flows at time 0, one period before the first payment arrives.

…

0 1 2 3

C C C

r

CPV

Growing Perpetuity

• A growing perpetuity is a stream of periodic payments that grow at a constant rate and continue forever.

• The present value of a perpetuity that pays the amount C1 next period, grows at the rate g indefinitely when the discount rate is r is:

…

0 1 2 3

C1 C1(1+g) C1(1+g)2

gr

CPV

1

Examples

Perpetuity: $100 per period forever discounted at 10% per period

$100 $100 $100

0 1 2 3

PV = C/r = $100/0.10 = $1,000

Growing perpetuity: $100 received at time t = 1, growing at 2% per period forever and discounted at 10% per period

$100 $102 $104.04

0 1 2 3

PV = C/(r –g ) = $100/(0.10 – 0.02) = $1,250

…

…

Verification of the Perpetuity Example Answers

Level PerpetuityYear BofY Bal INT@10% Payment EofY Bal

1 1000 100 100 10002 1000 100 100 10003 1000 100 100 10004 1000 100 100 1000

Growing PerpetuityYear BOY Bal. INT@10% Payment EOY Bal.

1 1250 125 100 12752 1275 127.5 102 1300.53 1300.5 130.05 104.04 1326.514 1326.51 132.651 106.1208 1353.04

Place the present value in a bank account, and recreate the Place the present value in a bank account, and recreate the payments. Let’s stop at 4 years since “forever” would take a while.payments. Let’s stop at 4 years since “forever” would take a while.

Note that the account balance is growing. At what rate? Why must this happen?Note that the account balance is growing. At what rate? Why must this happen?

Annuities

• An annuity is a series of equal payments, starting next period, and made each period for a specified number of periods.

– If payments occur at the end of each period it is an ordinary annuity or an annuity in arrears.

– If the payments occur at the beginning of each period it is an annuity in advance or an annuity due.

0 1 2 3

C C C

Valuing Annuities

• We can do a lot of grunt work or we can notice that a T period annuity is just the difference between a standard perpetuity and one whose first payment comes at date T+1.

• The present value of a T period annuity paying a periodic cash flow of C when the discount rate is r is:

• If we have an annuity due instead, the net effect is that every payment occurs one period sooner, so the value of each payment (and the sum) is higher by a factor of (1+r).

• Or we can add C to the value of a T-1 period annuity.

TT rr

C

rrrCPV

)1(

11

)1(

11

Annuity Example

• Compute the present value of a 3 year ordinary annuity with payments of $100 at r = 10%.

or,

68.248$1.1

1100$

1.1

1100$

1.1

1100$

32PV

68.248)1.1(1.0

1

1.0

1100$

3

PV

Annuity Due Example

• What if the last example had the payments at the beginning of each period not the end?

• Or,

• Or,

55.273$1.1

1100$

1.1

1100$100$

2PV

55.273$)1.1(68.248$ PV

55.273$)1.1(1.0

1

1.0

1100$100$

2

PV

Example: A five year annuity paying $2000 per year, with r = 5%.

• Valuing the payments individually we get:

1 2,000.00 1,904.76 2 2,000.00 1,814.06 3 2,000.00 1,727.68 4 2,000.00 1,645.40 5 2,000.00 1,567.05

_________8,658.95

$8,658.95=)0.05(1.05

1-

0.05

12000=PV 5

Using the annuity formula we get:

Alternatively, suppose you were given $8,658.95 today instead of the annuity

year principal interest PMT Ending Bal1 8,658.95$ 432.95$ (2,000.00)$ 7,091.90$ 2 7,091.90$ 354.60$ (2,000.00)$ 5,446.50$ 3 5,446.50$ 272.32$ (2,000.00)$ 3,718.82$ 4 3,718.82$ 185.94$ (2,000.00)$ 1,904.76$ 5 1,904.76$ 95.24$ (2,000.00)$ 0.00$

• Notice that you can exactly replicate the annuity cash flows by investing the present value to earn 5%.

• This demonstrates that present value calculations provide a literal equality, in that the future cash flows can be converted into the present value and vice versa, if (and only if) the selected discount rate is representative of actual capital market conditions.

Growing Annuities

• A stream of payments each period for a fixed number of periods where the payment grows each year at a constant rate.

…

0 1 2 T-1 T

C1 C1(1+g) C1(1+g)T-2 C1(1+g)T-1

PV Cr g r g

g

r

T

1

1 1 1

1

T

r

g

gr

CPV

1

111

Example

• What is the present value of a 20 year annuity with the first payment equal to $500, where the payments grow by 2% each year, when the interest rate is 10%?

…

0 1 2 T-1 T=20

500 500(1.02) 500(1.02)18 500(1.02)19

$4,869.52

0.1010.021

0.020.101

0.020.101

500PV20

A Valuation ProblemWhat is the value of a 10-year annuity that pays $300 a year at the end of each year, if the first payment is deferred until 6 years from now, and if the discount rate is 10%?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

300 300 300 • • • • • • 300

The value of the annuity payments as of five years from now is:

Now discount this equivalent payment back 5 years to time zero:

$1,843.370.10)0.10(1

10.10

1300PV 105

$1144.580.10)(1

1843.37PV

50

Application: Retirement Planning• You have determined that you will require $2.5 million

when you retire 25 years from now. Assuming an interest rate of r = 7%, how much should you set aside each year from now till retirement?– Step 1: Determine the present equivalent of the targeted $2.5

million.

PV = $2,500,500/(1.07)25

PV = $2,500,000/5.42743 = $460,623

– Step 2: Determine the annuity payment that has an equivalent present value:

C

C

C

526,39$

65358.11623,460$

)07.1(07.

1

07.

1623,460$

25

Retirement Planning cont…

• Now suppose that you expect your income to grow at 4% and you want to let your retirement contributions grow with your earnings. How large will the first contribution be? How about the last?

.614,69$04.1158,27$

and,158,27$

960728.16623,460$

50882185.03.

623,460$

07.1

04.11

04.07.623,460$

2425

1

1

1

25

1

C

C

C

C

C

A College Planning Example

• You have determined that you will need $60,000 per year for four years to send your daughter to college. The first of the four payments will be made 18 years from now and the last will be made 21 years from now. You wish to fund this obligation by making equal annual deposits over the 21 years. You expect to earn 8% per year on the deposits.– Step 1: Determine the t = 17 value of the obligation.

– Step 2: Determine the equivalent t = 0 amount.

727,198$)312127.3(60000)08.01(08.0

1

08.0

160000 417

PV

710,53$70002.3

198727)08.1(

19872717

College Planning cont…

• Step 3: Determine the 21-year annuity that is equivalent to the stipulated present value.

C

C

C

362,5$

016803.10710,53$

)08.1(08.

1

08.

1710,53$

21

Present Value Homework Problem

• Your child will enter college 5 years from now. Tuition is expected to be $15,000 per year for (hopefully) 4 years (t=5,6,7,8).

• You plan to make equal yearly deposits into an account at the end of each of the next 4 years (t=1,2,3,4) to fund tuition. The interest rate is 7%.

• How much must you deposit each year?• What if tuition were growing over the 4 years?• Think about:

– How to decide whether/when to refinance your house?

Application: Leasing vs Buying a Car

• Saab 9-3 five-door/five-speed, CD• Lease Terms (Source WSJ 8/6/98)• Up front fees $1,748 including down payment

(t=0).• Refundable security deposit of $300• 38 monthly payments of $299 (t=1, 2, …, 38)• Residual value of $16,454• Annual interest rate: 8%

...

T=0 T=35T=2T=1 T=38

-1,748 -300-2,048

-299 -299 -299 -299 -299 -299-16,454 300-16,154

-$10,001=)0670.0067(1.0

1-

0.0067

1-299=PVA 38

Payments are an annuity:

Present value of residual value and security deposit:

$12,534(1.0067)

154,16PV

38

Lease vs Buy cont…

• Present value of the lease is:

-$2,048 + (-$10,001) + (-$12,534) =

-$24,583• What does this number mean?• If we could purchase the car for less than $24,583

we are better off buying.• Does it matter if we pay cash or borrow if/when

we purchase in making the decision?

Time Value of Money Summary• Discounted cash flow (or present value) analysis is the

foundation for valuing assets or comparing opportunities.• To use DCF we need to know three things:

– Forecasted cash flows– Timing of cash flows– Discount rate (reflecting current capital market conditions and risk)

• If there are no simple cash flow patterns, then each cash flow must be valued individually and the present values summed. Look for constant or growing annuities or perpetuities for short-cuts.

• Different streams of cash flows can be meaningfully compared only after they are each converted to their equivalent present (or common point in time) values.