PV 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%
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
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
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%

Last Update: 1/20/98

Name: Aswath Damodaran