Question

Present Value. Use a calculator for each of these problems.

(a) You are moving to Bozeman and you are going to stay here forever. You would like to find an apartment. You can either buy it or rent it. The monthly rent is $500 and the monthly interest rate is 0.1%. Alternatively, you can purchase the apartment, paying $600,000. How are you going to finance your new accommodation? Argue using the PV formula.

(b) You take a loan to buy a car that costs $4000. What is your monthly payment if you want to pay back the loan after 3 years (36 payments) and the monthly interest rate is 0.5%. (Hint: think annuity)

(c) You are hired by Merrill Lynch to help assess the value of a T-bond (a bond issued by the Treasury Department) with the face value F=$1000, coupon c = $100 (paid anually until T − 1 and face value F paid at time T) and time to maturity equal to T = 10 years. The interest rate is equal to r = 10%. Find the PV of such a bond. Is it a good or bad deal to buy such a bond for $900? Explain.

(d) You want to receive $40,000 per year when retired (you will be retired from 61-80). How much do you have to save between 21-60 years if the interest rate is 5%?

(e) You save $20,000 per year when working and plan to work from 21-60. How much will you consume per year when you retire? Assume you will be retired from 61-80. The interest rate is 5%.

Answer 1

Part a) Rent vs buy decision

This is classic example of Perpetuity. Since Mr. Bozeman is intending to settle permanently. There will be perpetual cash outflows of monthly rent for him.

Present value of a perpetuity = Perpetuity/Discount rate

Here, if he rents PV of Perpetuity = (500)/0.001 = (5,00,000)

In case of buying the property, Bozeman needs to take a loan of $6,00,000 at 0.1% monthly interest rate. Here the repayment length is not given so lets assume 5 years.

Now we need to calculate Present value of interest payments for 5 years

Annual Interest rate = 0.1% *12 = 1.2%

Annual interest payments = $6,00,000 *1.2% = $7200

On summation of PV of outflows using PVF of 1.2% we have PV of CAsh outflow = 5997+4997+4162 + 3470 +2887 = ($21513)

Here we cannot compare PV of perpetuity and PV of interest outflows directly because of time difference. However, we must go with buying the house through mortgage financing as rental will be perpetual and Interest outflows will come to an end by 5 years or Loan period. Additionally, there will be a terminal sale value of the house in case of purchase which wont be there in case of renting the house.

Part b) Calculation of Monthly payments using Annuity

Loan amount $4000

Monthly interest rate = 0.5% or 6% Annual interest rate = r

No of time periods = 36 months or 3 years

Annual installment = Loan amount/PVFA_i,r

Therefore, taking PVFA_3,0.06 from the annuity table = 2.673

Annual installment = 4000/2.673 = $1496

Monthly installment would be = 1496/12 = $125

Part c) Present value of the bond

Bond value = Present value of interest + Present value of Maturity value

Face value of the bond = $1000

Coupon rate = REquired rate of return for investor = 10%

Time = 10 years, Interest payments till Year 9 $100 every year.

Below is the table showing both the values for 10 years

year	Inflow	PVF(r=10%)	PV
1	100	0.909	90.9
2	100	0.826	82.6
3	100	0.751	75.1
4	100	0.683	68.3
5	100	0.62	62
6	100	0.564	56.4
7	100	0.513	51.3
8	100	0.466	46.6
9	100	0.424	42.4
10	1000	0.385	385
Present value of the bond			$960.6

As we observe that the Present value of inflows from the bonds is higher than the buy price of $900, it is a good deal for investor to buy it now.

Part e) Yearly consumption on retirement

Yearly saving = 20,000 for 19 years

Rate of interest = 5% or 0.05

Future value of Savings = Savings *( 1+i)ⁿ

FV = 20,000 * (1+0.05)³⁹

Taking CVF for 39 years = 6.705

FV = $134100

Number of years retired = 80-61 = 19 years

Therefore savings which can be used per year = 134100/19 = $7058