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