Section #1 Time Value Solutions(1)

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Section #1 Solutions

Time Value Exercises

January 23rd, 2013

Copyright 2013 by Rich Curtis


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1. Michelle and The Time Value of Money

a. After agents fees and taxes, Michelle will have $5.28 M:

$10 M - Agent's Fees - Tax

= $10 M - .04($10 M) - .45(Taxable Income)

= $10 M - .04($10 M) - .45($10 M - .04($10 M))

= [$10 M - .04($10 M)] [1-.45]

= [$10 M] [1-.04] [1-.45]

= $5.28 M


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If Michelle can invest to earn 5.5% annually after-tax for 25 years, the$5.28 million will become $20.13 million:

FV = PV 1+r

N

= $5.28 M

1.055

25

= ($5.28 M) (3.81339)

= $20.1347 M

Is this consistent with the Rule of 72?

72

5.5= 13.09

Note:


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b. If we consider the cash flows at times 0, 1, 2, 3, and 4,her total Time 25 future value could be calculated as follows:

$5.28 M

(1.055)

25

+ $5.28 M

(1.055)

24

+ $5.28 M

(1.055)23

+ $5.28 M

(1.055)

22

+ $5.28 M

(1.055)

21

= $90.710 M


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Time

0 1 2 3 4 25

$5.28 M$5.28 M $5.28 M $5.28 M $5.28 M


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One could also use the formula for the future value of an annuity toget the Time 4 value of the annuity:

FVt=4 = $5.28 M1.055

5 - 1

.055

= $29.4682 M

Remember that the formula for the future value of an annuity givesyou the value as of the date of the final cash flow, Time 4.

Then use the formula for the future value of a single cash flow to findthe Time 25 future value of the annuity:

FVt=25 = $29.4682 M 1.055

21

= $90.710 M


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Time

0 1 2 3 4 25

$5.28 M$5.28 M $5.28 M $5.28 M $5.28 M

(1.055)21

29.5 M 90.7 M


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One could also a) PV the Time 0 cash flow and b) add the PV of theTime 1-4 annuity, if one was very careful:

PVt=0 = $5.28 M + $5.28 M

1 - 1

1.055( )4

.055

= $23.7872 M

Then take this Time 0 present value out to Time 25 by using theformula for the future value of a single cash flow:

FVt=25 = PVt=0(1+r)25

= $23.7872 M (1.055)25

= $90.710 M


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Time

0 1 2 3 4 25

$5.28 M$5.28 M $5.28 M $5.28 M $5.28 M

(1.055)25


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Lastly, you could use the formula for the PV of an annuity to bringall the cash flows to Time -1 (since the first cash flow is at Time 0):

Then future value the $22,5471 M ahead 26 years, to get theTime 25 value of the annuity:

PVt=-1 = $5.28 M

1 -1

1.055( )5

.055

= $22.5471 M

FVt=25 = PVt=-1 (1+r)26

= $22.5471 M (1.055)26

= $90.710 M


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Time

-1 0 1 2 3 25

$5.28 M $5.28 M$5.28 M $5.28 M $5.28 M

4

(1.055)26


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We must solve for the discount rate r in the following expression:

PV = FV

1+r( )N

$125 = $36,000,000

(1+r)98

2. Sunflowers

(1+r)98 = $36,000,000$125

1 + r = $36,000,000$125

198

r = .13686 or 13.686%


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The equation above could also be solved using logarithms or bytrial and error.

Note: You may be asked a question like this at Finance interviews.


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3. Amerks Dividend Discount Stock Valuation(See RWJ, 2008, Chapter 8)

P0 =D1

(1+ke)+

D1(1+g)

(1+ke)2+

D1(1+g)2

(1+ke)3+ ...

Time: 1 2 3

Cash Flow: D1 D1(1+g) D1(1+g)2

=D

1ke -g

if ke > g


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a.

P0 =

D1

ke g

= $2.14 .05

= $22.22


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b.

P0 =

D1

ke g

= $2.12 .05

= $28.57


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Stock Pricesversus

Cost of Equity

(D1=$2, g=.05)

$0.00

$10.00

$20.00

$30.00

$40.00

$50.00

$60.00

0.00 0.05 0.10 0.15 0.20 0.25

Costs of Equity

Stock Prices


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c.

P0=

D

1ke g

ke g =D

1P0

g = ke -D

1

P0

= .14 - $2

$15

g = .00667 or .667%

ke -D1

P0

= g


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Note:

g in the formula is the growth rate fromTime 1 to Infinity.

The growth rate from Time 0 to Time 1,

g0,1 , is not explicitly in the formula butdefines the magnitude of D1. That is,the D1 we use equals D0(1+g0,1).

In fact, in (c) the dividend grows at 5%between Time 0 and Time 1 (since weassumed that the market uses a D1of $2/share) and the dividend grows at.667% after Time 1.


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d. In parts (a) and (b) the growth rate g was determined as follows:

If internal investments return 30% annually rather than 20% annually,then the growth rate increases to 7.5%:

If the dividends are estimated to grow at 7.5% not only fromTime 1 to Infinity, but also from Time 0 to Time 1, we have:

g = b = (.20) (.25) = .05

g = b = (.30) (.25) = .075

D1 = D

0(1+g

0,1) = $1.9048 (1.075) = $2.048

D1 also equals (1-b) E1.


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The stock price is given by:

P0 =

D1

ke g

= $2.048.14 .075

= $31.51


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Note:

1. In (d) the dividends are assumed to grow at 7.5%between Time 0 and Time 1 AND at 7.5% from

Time 1 to Infinity.

If dividends instead grew at 5% from Time 0 toTime 1 and at 7.5% from Time 1 to Infinity, then:

P0 =

D1

ke g

=

$2

.14 .075

= $30.77

Note that changing g0,1 , the growth rate from Time 0to Time 1 only changed D1, not the g in the formula

(which holds from Time 1 to Infinity)!!

D1 = D

0(1+g

0,1) = $1.9048 (1.05) = $2

and the estimated Time 0 stock price is:


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2. The numbers show how sensitive the stock price is to

changes in g. If a firm misses their earnings or salesnumbers by a measly penny, analyst may revise theirestimates of g downward with a dramatic effect onthe theoretical stock price.


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Stock Pricesversus

Growth Rates

(D1=$2, ke=.14)

$0.00

$50.00

$100.00

$150.00

$200.00

$250.00

$300.00

$350.00

$400.00

$450.00

0.00 0.05 0.10 0.15

Growth Rates

Stock Prices


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4. Multiple and Continuous Compounding

a.FV = $100(1+.06)

= $106

The effective annual interest rate is 6.00%.

One could also use the formula for EAR inthe Time Value notes:

EAR = 1 + APRp

p

- 1

= 1 + .061

1

- 1

= .06


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b.


Once again, one could also use the formulafor EAR in the Time Value notes:

FV = $100 1+.06

12

12

= $106.168

EAR = 1+APR

p

p- 1

= 1+.0612

12

- 1

= .06168


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c.


Once again, one could also use thecontinuous compounding formula for EAR inthe Time Value notes:

FV = $100eAPR(t)

= $100e(.06)(1)

= $106.184

EAR = eAPR - 1

= e(.06) - 1

= .06184


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5. Factoring Accounts Receivable

For every $100 of accounts receivable sold to the factor, the contractwould result in a cash flow to your firm of +$97.50 at time 0 an $100sixty days later. (The negative cash flow is due to the fact that youre

selling the accounts receivable and thus are giving up cash you wouldhave received 60 days later.)

Time: 0 60 Days

Cash Flows: +$97.50 -$100

Whats the interest rate over that 60-day period?


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Time: 0 1

Cash Flows: +$100 -$110

If you borrowed $100 at Time 0 and repaid $110 at Time 1,

you would be paying $10 in interest:

Eyeballing it, its perhaps obvious that the interest rateover that period is 10%. To get .10 you divide thedollar interest cost, $10, by the amount borrowed.

Dont divide by the amount repaid.

$10$100

= .10


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$2.50$97.50

= .025641

Since in the factoring problem, you are effectively paying $2.50 ininterest for every $97.50 in financing, the effective cost of borrowingover the 60-day period is:

Remember to divide the interest paid by the amount borrowed, notthe amount repaid!!


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Remember that if the monthly rate was 1% (i.e. an APR of 12%), wewould calculate the effective annual rate as:

1 + Effective Annual Rate = (1.01)12

= 1.1268

Effective Annual Rate = .1268 or 12.68%

The calculation on the next slide is the same, except that we use a60-day interest rate (instead of a monthly interest rate) and wecompound 6.0833 times per year (instead of 12 times per year).

Now, lets annualize that 60-day rate


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Given a 60-day rate of 2.5641%, the effective annual cost ofborrowing is given by:

1 + Effective Annual Rate = (1.025641)

365

60

= (1.025641)6.0833

= 1.1665

Effective Annual Rate = .1665 or 16.65%


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Theres also another way to solve for the effective annualcost of the loan:

$97.50 =$100

1+ r( )60

365

1+ r( )60

365 =$100

$97.50

1+ r( )60

365 = 1.02564

1 + r = (1.02564)

365

60

1 + r = (1.02564)6.08333

r = .1665 or 16.65%


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The End

Section #1 Time Value Solutions(1)

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