FIN 40153: Intermediate Financial Management

INTRODUCTION TO VALUATION

(TIME VALUE OF MONEY)

(BASED ON RWJ CHAPTER 4)

Money has “time value”

Why? Because interest rates (and expected returns)

are positive.

Implies that: A dollar today is worth more than a dollar

tomorrow. Cash flows at different points in time cannot

be added together.

Basic Tools for Valuation: Future and Present Values (Discounting)

Terminology:

r is the interest rate or as is also referred to as the discount rate PV will be the value today or the present value of a future payment FVn will be the future value of money invested today n years from

now If you invest PV today then n years from now it will be worth

FVn = PV(1+r)n

This same basic formula can be used to determine the current amount that is equivalent to (i.e. can grow to) a stated future amount, since

PV = FVn/(1+r)n

Example

You need $20 million five years from now to fund a capital investment. If r = 6%, what amount can be set aside now to fund the investment?

PV = 20.0/(1.06)5 = 20.0/1.33823 = $14.945 million.

What if you can’t really earn r = 6%?

Example My Uncle Pete wants me to loan him $15,000 today so he can buy

some frozen fruit for his new smoothie stand. He says that he is willing to get the full $15,000 back to me in exactly three years, and is willing to give me $3,500 in three years on top of that for interest. What is the annual rate of interest that I am receiving on my loan to Pete under his proposal?

18,500 = 15,000(1+r)3 = 18,500 (18,500/15,000) = (1+r)3  r = (18,500/15,000)1/3 -1 = 0.0724 = 7.24%

On HP10BII: Amber C All�

-15,000 PV 18,500 FV 3 N 0 PMT Push the I/YR key to get 7.24% Note: Must be sure that P/YR is set = 1 !!!

Present Value Of A Series of Future Cash Flows

The present value of a stream of cash flows can be found using the following general valuation formula:

N

tt

t

t

NN

N

r

C

r

C

r

C

r

C

r

CPV

1

33

32

2

2

1

1

)1(=

)1(...

)1()1()1(

In words: (1) discount each cash flow back to the present using the appropriate discount rate, and (2) sum the present values. The latter step is OK, because each future cash flow has been restated as its equivalent amount at a common point in time.

Valuing Streams of Structured Future Cash Flows

Perpetuity A stream of equal payments, made at equal time intervals, forever.

The present value of a perpetuity that pays the amount C at the end of each period, when the discount rate is r, is:

…

0 1 2 3

C C C

r

CPV

Note that this gives the value as of time 0, one period before the first payment arrives.

Example

You wish to endow a Permanent Chair in finance at TCU. The chair needs $150,000/yr forever (in perpetuity).

The trustees of TCU can invest your money at 6%/yr through guaranteed investment contracts with the NeverBankrupt Insurance Co.

How much money do you need (what is the PV of the perpetuity) ?

PV = C / r = 150,000 / 0.06 = $2,500,000

Valuing Streams of Cash Flows

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, growing 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

Suppose that the Finance professor to be hired for the Chair wants 5% raises per year guaranteed:

Growing perpetuity: $150,000 received at time t = 1, growing at 5% per period forever and discounted at 6% per period

C C(1 + g) C(1 + g) 2

0 1 2 3

PV = C/(r –g ) = $150,000/(0.06 – 0.05) = $15,000,000

…

Sidebar: Valuation by Earnings or Cash Flow “Multiples”

Analysts and investors sometimes take a shortcut approach to valuation: multiply earnings or cash flow for the current year

by an appropriate “multiple”. The best known of these is the “price/earnings

ratio”. Valuation by multiple in the case of infinite-

lived, constant growth, cash flows:

grMwhere

CMPV

1

,* 1

Multiples Appropriate for Valuing Infinite-Lived Cash Flows with Constant

Growthgrowth rate (g) -2% 0% 2% 4% 6% 8% 10% 12% 14%interest rate

6% 12.5 16.7 25.0 50.0 8% 10.0 12.5 16.7 25.0 50.0

10% 8.3 10.0 12.5 16.7 25.0 50.012% 7.1 8.3 10.0 12.5 16.7 25.0 50.014% 6.3 7.1 8.3 10.0 12.5 16.7 25.0 50.016% 5.6 6.3 7.1 8.3 10.0 12.5 16.7 25.0 50.018% 5.0 5.6 6.3 7.1 8.3 10.0 12.5 16.7 25.020% 4.5 5.0 5.6 6.3 7.1 8.3 10.0 12.5 16.722% 4.2 4.5 5.0 5.6 6.3 7.1 8.3 10.0 12.524% 3.8 4.2 4.5 5.0 5.6 6.3 7.1 8.3 10.0

Larger multiples are justified by higher cash flow growth,or by lower interest (discount) rates

An annuity is a series of equal payments made at fixed intervals for a specified number of periods e.g., $100 at the end of each of the next three

years If payments occur at the end of each period it is an

Ordinary Annuity. If payments occur at the beginning of each period

it is an Annuity Due.

Annuities

0 1 2 3

100 100 100Ordinary Annuity

Valuing Annuities The present value of a T period annuity paying a periodic

cash flow of C when the discount rate is r is:

Trrr

CPV)1(

11

The textbook uses the shorthand notation ATr% to denote the factor

in brackets. If we have an annuity due instead, the net effect is that every payment occurs one period sooner, so the value of each payment is higher by the factor (1+r).To obtain the PV with beginning-of-period payments, simply multiply the answer obtained based on end-of-period payments by the factor (1+r).

Annuity 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

$8658.95=)0.05(1.05

1-

0.05

12000=PV 5

Using the annuity formula we get:

Do on HP: PMT=2000, N=5, I/YR=5, FV=0, push PV and get 8658.95 (note it displays as a negative number)

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 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 regular payments that grows at a constant rate, g.

• The present value of a T period growing annuity that pays C1 next period, with subsequent payments growing at rate g, when the discount rate is r is:

…

0 1 2 T-1 T

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

T

r

g

grgrCPV

1

1111

T

r

g

gr

cPV

1

111

or

Example (growing annuity)

A new venture is expected to generate $1 million in free cash flow during its first year. Subsequent cash flows will grow by 4% per year due to market growth and cost savings. Cash flows will continue for 20 years. The appropriate discount rate is 14%.

701,405,8$15943.1000,000,1014.1

04.11

04.14.

000,000,120

PV

Annuities with Deferred Starts

The standard formulas for level and growing annuities assume that the first payment occurs one period from now.

If the first payment is deferred until later than one period from now, the standard formulas can still be used, but:

The answer denotes the value as of one period before the first payment, not the value now.

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:

37.843,1$)10.01(10.0

1

10.0

1300

105

PV

58.1144$)10.01(

37.184350

PV

Application of Time Value Analysis: Planning for Retirement.

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 have already saved? set aside each year from now until retirement?Determine the present equivalent of the targeted 2.5 million.

PV = 2,500,000/(1.07)25

PV = 2,500,000/5.42743 = $460,623.Hmmm, you don’t have that much saved??Determine the annuity payment that has an equivalent present value:

C

C

C

39526

65358.11460623

)07.1(07.

1

07.

1460623

25

Retirement Planning Example, Continued.

You expect your earnings to grow by 4% per year. Your retirement goal is unchanged but, instead of making equal payments, you wish to make payments that grow along with your earnings. How large should the first payment be? How large should the 25th payment be?

.69614$04.127158

,27158$

960728.16460623

50882185.03.

460623

07.1

04.11

04.07.460623

2425

1

1

1

25

1

C

andC

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 r = 8% per year.

Time Line:

0 1 2 3 17 18 19 20 21 22

60 60 60 60

College planning, cont. • Step 1: Determine the time 17 value of the obligation:

.727,198$312127.360000)08.1(08.0

1

08.0

160000

417

PV

• Step 2: Determine the equivalent time zero amount:

710,53$70002.3

198727

08.1

19872717 PV

College planning example, continued.

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

C

C

C

362,5$

016803.10710,53

)08.1(08.

1

08.

1710,53

21

Verification of Answer Obtained in College-Planning Example

Year BOY Bal. Int. @8% Dep. (EOY) Pymt (EOY) EOY Bal.

1 0 0 5362 0 53622 5362 429 5362 0 111533 11153 892 5362 0 174074 17407 1393 5362 0 241625 24162 1933 5362 0 314576 31457 2517 5362 0 393357 39335 3147 5362 0 478448 47844 3828 5362 0 570349 57034 4563 5362 0 66958

10 66958 5357 5362 0 7767711 77677 6214 5362 0 8925312 89253 7140 5362 0 10175513 101755 8140 5362 0 11525814 115258 9221 5362 0 12984015 129840 10387 5362 0 14559016 145590 11647 5362 0 16259917 162599 13008 5362 0 18096918 180969 14477 5362 60000 14080819 140808 11265 5362 60000 9743520 97435 7795 5362 60000 5059221 50592 4047 5362 60000 1

Time Value Summary

Discounted cash flow, or present value, analysis is the foundation for valuing assets or comparing opportunities.

To use DCF you need to know three things Forecasted cash flows Timing of cash flows Discount rate (reflects 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. You can look for annuities or growing annuities to allow short cuts.

Different streams of cash flows can be meaningfully compared (or equated) after they are each converted to their equivalent present values.