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Investments:Time Value of Money

Professor Scott Hoover

Business Administration 365

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Basic Intuition Recall that...

... V0(Ct) = Ct/(1+R)t e.g., C3=$1120, R=8%

V0 = $1120/1.083 = $889

... FVt(C0) = C0(1+R)t e.g., C0=$100, R=10%

V2 = $100×1.12 = $121

Why? Indifference!

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example: R: 8% twelve $100 CFs beginning one year from today. V0 = ? timeline:

V0 = 100/1.08 + 100/1.082 +…+ 100/1.0812 = $753.61

Date: 0 1 2 3 … 11 12

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

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another example: R: 10% $1 CF in one year followed by 8% growth for the next 15 years

(16 total CFs). V0 = ? 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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General example (for math junkies): We expect to receive n cash flows beginning with C in one year. That CF will grow at the rate g each year thereafter. What is the PV?

PV = C/(1+R) + C(1+g)/(1+R)2 + C(1+g)2/(1+R)3 + … + C(1+g)n-1/(1+R)n

Multiplying the above equation by (1+R)/(1+g) gives

PV(1+R)/(1+g) = C (1/(1+g) + 1/(1+R) + (1+g)/(1+R)2 + … + (1+g)n-2/(1+R)n-1)

Subtracting the first equation from the second and rearranging gives

EXTREMELY IMPORTANT POINT: The formula gives you the value of a constant-growth annuity one period before the first cash flow.

gR

R

g

CPV

n

1

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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 of perpetual growth: PV = C / (R-g).

When g=0, we get the PVIFA (recall accounting tables): PV = CPVIFA, where

When n and g=0, we get the perpetuity formula: PV = C/R

R

RPVIFA

n

1

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

V0 = $100 (1 - 1/(1.08)12) / .08 = $753.61

This equation is much easier to use for long annuities.

example: $1 next year, 8% growth, 16 cash flows, 10% interest V0 = $1 (1 - (1.08/1.10)16) / 0.02 = $12.72

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Another example: You plan to deposit $1,000 per year for the next 20 years

(starting in one year). 8% annual interest How much money will you have just after your last deposit? 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: V0 = $1000PVIFA8%,20 = $9,818.15

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 desired time by discounting or

compounding the interest over the appropriate number of periods.

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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, or

EXTREMELY IMPORTANT POINT: 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.

gR

gRFVIFGA

nn

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Examples #1: Suppose you will be receiving eight annual

$1000 payments 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 five 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

What don’t we like about this scenario?

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.

would withdraw $120,000 1.02521 = $201,549.80 during the first year of retirement.

Suppose that our annual savings increase at the rate of inflation. How much should we deposit each year under these assumptions? timeline:

V20 = $201,549.80 PVIFGA12%,2.5%,30 = $1,973,040

C = $1,973,040 / FVIFGA12%,2.5%,20 = $23,407

Let’s consider additional scenarios. (See spreadsheet).

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

CF: C … C1.02519 $120,0001.02521

= $201,549.80… $120,0001.02550

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Lessons Learned Annuity-based analyses are faulty and biased in favor of

saving too little Tremendous value in investing over long periods of time.

Start investing early! Planning to live forever is both doable and wise.

You need only small amount more money each year If and when you do die, you will leave a substantial

inheritance If you plan and execute properly

no need to shift to safer investments as you get older. need to save less each period! (counterintuitive)

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#6: Extending #5, suppose that we owe $10,000 today and that we anticipate paying for four years of our child’s college education beginning in 20 years. Average college cost today = $25,000. How does this change our problem?

We can calculate the value of those additional withdrawals at date 20 and incorporate it into the original problem. Why? Indifference!

V20, $10K debt = $10,000×1.1220 = $96,463 V20,college = ?

Expected first payment = $25,000×1.02520 = $40,965 V19, college = $40,965×PVIFGA12%,2.5%,4 = $128,721 V20, college = V19, college × 1.12 = $144,167 V20,all withdrawals = $96,463+$144,167 = $240,630

So, we need to plan to have $240,630 more than the $1,973,040 we computed in #5, or $2,213,670

This gives us C = $ 2,213,670 / FVIFGA12%,2.5%,20 = $26,262

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Note that we could easily incorporate this into our spreadsheet by entering the value today of those withdrawals. In doing so, we would need to enter it as a negative number. V0, additional withdrawals =

$10,000 + $25,000×1.02520×PVIFGA12%,2.5%,4/1.1219 = $24,945

Note that this is just $240,630/1.1220