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Corporate Finance:Time Value of Money

Professor Scott Hoover

Business Administration 221

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Basic Intuition example: Suppose that you are offered $15,000

in one year and that the cash flow is risk-free. The risk-free interest rate is 4%. What is the present value (PV) of the cash flow? V0 = $15,000 x PVIF4%,1 = $15,000 / 1.04 = $14,423.08 Why?

If someone offered you $14,423.08 today, you would be able to invest it at 4% interest to get $14,423.081.04 = $15,000. So, you are indifferent between the two.

both cash flows must have a value of $14,423.08.

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Important point: if we are indifferent between two things, they must have the same value at each and every point in time.

Implication: we can determine the value of a given cash flow at any point in time by multiplying or dividing by 1+R raised to some power.

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example: Suppose that a security will pay you $100 in five years. You can invest or borrow money at 8% annually. What is the value of the cash flow today?

V0 = $100/1.085 = $68.06 Why?

If we had $68.06 today…could invest to receive $68.061.085 = $100 in five years.

We are indifferent between the two.

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Does it matter what our cash flow needs are? No!

If we need money today…- borrow $68.06- would owe $100 in five years- use security proceeds to meet obligation net cash flow of $68.06 today, no net future obligation.

implication: as long as markets are sufficiently developed, the value of any security will be independent of the characteristics of any individual investor.

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What will the value of the cash flow be in nine years? V9 = V5 1.084 = $136.05

Why? Because we can invest the $100 we receive in five years at 8% for four years.

implication: Vt = VT(1+R)(t-T)

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The Present Value of Cash Flows example:

10% interest rate What is the PV of an annual series of three $1000

cash flows that begin one year from today? timeline:

V0 = 1000/1.1 + 1000/1.12 + 1000/1.13 = $2,486.85 (Notice that we can treat each cash flow separately). Why is the present value of the cash flows equal to $2,486.85?

Date: 0 1 2 3

Cash Flow: $1000 $1000 $1000

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Suppose that we invest the $2,486.85 at 10% and then withdraw $1000 each of the next three years. At the end of three years, we would have exactly $0 in the account.

The result is that we can achieve exactly the same cash flows, whether we have the $1000 annuity or the $2,486.85.

Date Sub-Balance Withdrawal Balance

0 $2,486.85 $0 $2,486.85

1 $2,735.54 $1000 $1,735.54

2 $1909.09 $1000 $909.09

3 $1000.00 $1000 $0.00

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example: If the appropriate interest rate is 8%, what is the present value of an annual series of twelve $100 cash flows that begin one year from today.timeline:

V0 = 100/1.08 + 100/1.082 +…+ 100/1.0812 = $753.61 Surely, there must an easier way to do these calculations…..

Date: 0 1 2 3 … 11 12

CF: $100 $100 $100 $100 $100 $100

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another example: Suppose that you expect to receive a $1 cash flow in one year, followed by 8% growth for the next 15 years (16 total cash flows). If R=10%, what is the present value of the cash flows? timeline:

V0 = $1/1.1 + $1.08/1.12 + … + $3.17/1.116 = $12.72

Is there an easier way?

Date: 0 1 2 3 … 15 16

CF: $1 $1.08 $1.16 $2.94 $3.17

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Yes! We skip the derivation here, but suppose the following: We expect to receive C in one year, followed by cash

flows that grow at the rate g each year thereafter. There are n total cash flows. The present value of the cash flows is …

PV = C [1 – {(1+g)/(1+R)}n] / (R-g)

EXTREMELY IMPORTANT POINT: Notice that the formula gives you the value of the growing annuity one period before the first cash flow.

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The factor that is multiplied by C is called the “Present Value Interest Factor for Growing Annuities” (PVIFGA)

When n, we get the famous Gordon model:

PV = C / (R-g).

When g=0, we get the PVIFA:

PV = CPVIFAR,n where PVIFAR,n = (1 – 1/(1+R)n) / R

When n and g=0, we get the perpetuity formula:

PV = C/R

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Redoing our previous examples… example: $100 for next 12 years, 8% interest:

PV = $100 (1 - 1/(1.08)12) / .08 = $753.61 This equation is much, much easier to use for long

annuities.

example: $1 next year, 8% growth, 16 cash flows, 10% interest PV = $1 (1 - 1/(1.08/1.10)16) / 0.02 = $12.72

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The Future Value of Cash Flows Another example: Suppose you would like to deposit

$1,000 per year for the next 20 years (starting in one year). If you will receive 8% per year over the twenty years, how much money will you have when you make the last payment? timeline:

Date: 0 1 2 3 … 19 20

CF: $1000 $1000 $1000 $1000 $1000 $1000

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FV = $10001.0819 + $10001.0818 + $10001.0817 + … + $10001.081 + $1000 = $45,761.96

Alternatively, we could calculate the present value and then multiply by 1.0820: PV = $1000PVIFA8%,20 = $9818.15

V20 = $9818.151.0820 = $45,761.96 Implication: We can accomplish any valuation with a two

step process. 1. Calculate the value of the cash flow stream at any

point in time. 2. Convert to the desire time by discounting or

compounding the interest (i.e., use Vt = VT(1+R)(t-T)).

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We can then define the “Future Value Interest Factor for Growing Annuities” (FVIFGA) as

FVIFGAR,g,n = PVIFGAR,g,n (1+R)n

= [(1+R)n – (1+g)n]/(R-g)

EXTREMELY IMPORTANT POINT: Notice that the formula gives you the value of the growing annuity at the time of the last cash flow.

The FVIFGA reduces to the Future Value Interest Factor for Annuities (FVIFA) when g=0.

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Recap Present value of a single cash flow:

V0 = Ct/(1+R)t

Present value of a growing annuity: V0 = C1 PVIFGAR,g,n

Present value of an infinite life growing annuity: V0 = C1 PVIFGAR,g, = C1/(R-g)

Present value of an annuity: V0 = C1 PVIFGAR,0,n = C1 PVIFAR,n

Present value of a perpetuity: V0 = C1 / R

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Future value of a single cash flow: Vt = C0 (1+R)t

Future value of a growing annuity: Vn = C1 FVIFGAR,g,n

Future value of an annuity: Vn = C1 FVIFGAR,0,n = C1 FVIFAR,n

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Examples #1: Suppose you will be receiving $1000

payments for eight years beginning in two years. If R=10%, what is V0? V1 = $1000 PVIFA10%,8 = $5,334.93

Why? The formula gives the value one period before the first cash flow.

V0 = V1/1.1 = $4,849.93

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#2: Suppose you plan to begin saving $1500 per year starting two years from now. You will make 5 total deposits. How much money will you have one year after your last deposit if the interest rate is 6%? timeline:

V6 = $1,500 FVIFA6%,5 = $8,455.64 Why? The formula gives the value at the time of the last cash flow.

V7 = V6 1.06 = $8,962.98

Date: 0 1 2 3 4 5 6 7

CF: $1500 $1500 $1500 $1500 $1500

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#3: Suppose R=7%. How much money must you save each of the next 30 years so that you have $2,000,000 when you retire? timeline:

V30 = C FVIFA7%,30 $2,000,000

C = $2,000,000 / FVIFA7%,30 = $21,173

Date: 0 1 2 3 … 28 29 30

CF: C C C C C C C

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#4: You would like to retire in 20 years with an annual retirement income of $120,000. You expect to live for another 30 years after that. How much must you invest each year (beginning in one year) if your investment account pays 12% interest annually? timeline:

V20 = $120,000 PVIFA12%,30 $966,622

C = $966,622 / FVIFA12%,20 = $13,416

Date: 0 1 2 … 20 21 … 50

CF: C C C C $120K $120K $120K

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#5: Redo the previous problem while controlling for inflation. We want $120,000 in today’s dollars every year during

retirement. Suppose that inflation is expected to be 2.5% per year. How much should we deposit each year under the assumption

that we increase the amount saved at the rate of inflation? timeline:

V20 = $120,000 1.02521 PVIFGA12%,2.5%,30 = $1,973,040

C = $1,973,040 / FVIFGA12%,2.5%,20 = $23,407 Note that these numbers are optimistic. Let’s consider more

realistic numbers. (See spreadsheet).

Date: 0 1 ... 20 21 ... 50

CF: C … C1.02519 $120,0001.02521 … $120,0001.02550