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Solutions to Present Value Problems

**Present Value: Solutions
Problem 1
**a. Current Savings Needed = $ 500,000/1.110 = 192,772$
b. Annuity Needed = $ 500,000 (APV,10%,10 years) = 31,373$

**Problem 2
**Present Value of $ 1,500 growing at 5% a year for next 15 years = 18,093$
Future Value = $ 18093 (1.08^15) = 57,394$

**Problem 3
**Annual Percentage Rate = 8%
Monthly Rate = 8%/12 = 0.67%
Monthly Payment needed for 30 years = $ 200,000(APV,0.67%,360) = 1,473$

**Problem 4
***a. Discounted Price Deal
*Monthly Cost of borrowing $ 18,000 at 9% APR = 373.65$
[A monthly rate of 0.75% is used]
*b. Special Financing Deal *17.98245614
Monthly Cost of borrowing $ 20,000 at 3% APR = 359.37$
The second deal is the better one.

**Problem 5
**a. Year-end Annuity Needed to have $ 100 million available in 10 years= 6.58$
[FV = $ 100, r = 9%, n = 10 years]
b. Year-beginning Annuity Needed to have $ 100 million in 10 years = 6.04$

**Problem 6
**Value of 15-year corporate bond; 9% coupon rate; 8 % market interest rate
Assuming coupons are paid semi-annually,

Value of Bond = 45*(1-1.04^(-30))/.04+1000/1.04^30 = 1,086.46$ If market interest rates increase to 10%,

Value of Bond = 45*(1-1.05^(-30))/.05+1000/1.05^30 = 923.14$ The bonds will trade at par only if the market interest rate = coupon rate.

**Problem 7
**Value of Stock = 1.50 (1.06)/ (.13 - .06) = 22.71$

**Problem 8
**Value of Dividends during high growth period = $ 1.00 (1.15)(1-1.15^5/1.125^5)/(.125-.15)

5.34$ Expected Dividends in year 6 = $ 1.00 (1.15)^5*1.06*2 = 4.26$ Expected Terminal Price = $ 4.26/(.125-.06) = 65.54$ Value of Stock = $ 5.34 + $ 65.54/1.125^5 = 41.70$

**Problem 9
**Expected Rate of Return = (1000/300)^(1/10) - 1 = 12.79%

**Problem 10
**Effective Annualized Interst Rate = (1+.09/52)^52 - 1 = 9.41%

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Solutions to Present Value Problems

**Problem 11
**Annuity given current savings of $ 250,000 and n=25 = 17,738.11$

**Problem 12
**PV of first annuity - $ 20,000 a year for next 10 years = 128,353.15$
PV of second annuity discounted back 10 years = 81,326.64$
Sum of the present values of the annuities = 209,679.79$
*If annuities are paid at the start of each period,
*PV of first annuity - $ 20,000 at beginning of each year= 148,353.15$
PV of second annuity discounted back 10 years = 88,646.04$
Sum of the present values of the annuities = 236,999.19$

**Problem 13
**PV of deficit reduction can be computed as follows –

Year Deficit Reduction PV 1 25.00$ 23.15$ 2 30.00$ 25.72$ 3 35.00$ 27.78$ 4 40.00$ 29.40$ 5 45.00$ 30.63$ 6 55.00$ 34.66$ 7 60.00$ 35.01$ 8 65.00$ 35.12$ 9 70.00$ 35.02$

10 75.00$ 34.74$ Sum 500.00$ 311.22$

The true deficit reduction is $ 311.22 million.

**Problem 14
**a. Annuity needed at 6% = 1.89669896 (in billions)
b. Annuity needed at 8% = 1.72573722 (in billions)

Savings = 0.17096174 (in billions) This cannot be viewed as real savings, since there will be greater risk associated with the higher-return investments.

**Problem 15
**a. Year Nominal PV

0 $5.50 $5.50 1 $4.00 $3.74 2 $4.00 $3.49 3 $4.00 $3.27 4 $4.00 $3.05 5 $7.00 $4.99

$28.50 $24.04 b. Let the sign up bonus be reduced by X. Then the cash flow in year 5 will have to be raised by X + 1.5 million, to get the nominal value of the contract to be equal to $30 million. Since the present value cannot change,

X - (X+1.5)/1.075 = 0 X (1.075 - 1) = 1.5

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Solutions to Present Value Problems

X = 1.5/ (1.075 -1) = $3.73 million The sign up bonus has to be reduced by $3.73 million and the final year's cash flow has to be increased by $5.23 million, to arrive at a contract with a nominal value of $30 million and a present value of $24.04 million.

**Problem 16
**Chatham South Orange

Mortgage $300,000 $200,000 Monthly Payment $2,201 $1,468 Annual Payments $26,416 $17,610 Property Tax $6,000 $12,000 Total Payment $32,416 $29,610 b. Mortgage payments will end after 30 years. Property taxes are not only a perpetuity; they are a growing perpetuity. Therefore, they are likely to be more onerous. c. If property taxes are expected to grow at 3% annually forever,

PV of property taxes = Property tax * (1 +g) / (r -g) For Chatham, PV of property tax = $6000 *1.03/(.08-.03) = $123,600 For South Orange, PV of property tax = $12,000 *1.03/(.08-.03) = $247,200

To make the comparison, add these to the house prices, Cost of the Chatham house = $400,000 + $123,600 = $523,600 Cost of the South Orange house = $300,000 + $247,200 = $547,200

The Chatham house is cheaper.

**Problem 17
**a. Monthly Payments at 10% on current loan = 1,755.14$
b. Monthly Payments at 9% on refinanced mortgage = 1,609.25$
Monthly Savings from refinancing = 145.90$
c. Present Value of Savings at 8% for 60 months = 7,195.56$
Refinancing Cost = 3% of $ 200,000 = $6,000
d. Annual Savings needed to cover $ 6000 in refinancing cost= 121.66$
Monthly Payment with Savings = $ 1755.14 - $ 121.66 = 1,633.48$
Interest Rate at which Monthly Payment is $ 1633.48 = 9.17%

**Problem 18
**a. Present Value of Cash Outflows after age 65 = $ 300,000 + PV of $ 35,000 each year for 35 years =

707,909.89$ b. FV of Current Savings of $ 50,000 = 503,132.84$ Shortfall at the end of the 30th year = 204,777.16$ Annuity needed each year for next 30 years for FV of $ 204777 = 1,807.66$ c. Without the current savings, Annuity needed each year for 25 years for FV of $ 707910 = 9,683.34$

**Problem 19
**a. Estimated Funds at end of 10 years:

FV of $ 5 million at end of 10th year = 10.79$ (in millions) FV of inflows of $ 2 million each year for next 5 years = 17.24$ - FV of outflows of $ 3 million each year for years 6-10 = 17.60$ = Funds at end of the 10th year = 10.43$

b. Perpetuity that can be paid out of these funds = $ 10.43 (.08) = 0.83$

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Solutions to Present Value Problems

**Problem 20
**a. Amount needed in the bank to withdraw $ 80,000 each year for 25 years = 1,127,516$
b. Future Value of Existing Savings in the Bank = 407,224$
Shortfall in Savings = $ 1127516 - $ 407224 = 720,292$
Annual Savings needed to get FV of $ 720,292 = 57,267$
c. If interest rates drop to 4% after the 10th year,
Annuity based upon interest rate of 4% and PV of $ 1,127,516 = 72,174.48$

**Problem 21
**Year Coupon Face Value PV

1 50.00$ 46.30$ 2 50.00$ 42.87$ 3 50.00$ 39.69$ 4 50.00$ 36.75$ 5 50.00$ 34.03$ 6 60.00$ 37.81$ 7 70.00$ 40.84$ 8 80.00$ 43.22$ 9 90.00$ 45.02$

10 100.00$ 1,000.00$ 509.51$ Sum = 876.05$

**Problem 22
**a. Value of Store = $ 100,000 (1.05)/(.10-.05) = 2,100,000$
b. Growth rate needed to justify a value of $ 2.5 million,

100000(1+g)/(.10-g) = 2500000 Solving for g,

g = 5.77%

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