Question

Question 1 (a)]

Find the monthly payment on a $300,000 mortgage quoted at 14 percent and amortized over 25 years.

(b) Someone promises to pay you $1,000,000 every year (at the end of the year) for the next 15 years. The first payment will be one year from today. The relevant effective annual interest rate is 7.0%. What does the time line look like for this problem? What would you pay, today, for this promise?

(c)   Show that the answer in Example 5 is the same as the difference between the present value of the following two perpetuities:

• Perpetuity 1 pays $1,000,000 at the end every year with the first payment made in 1year.
• Perpetuity 2 pays $1,000,000 at the end every year with the first payment made in 16years.

The relevant annual effective interest rate is 7.0%.

(d) You are quoted an interest rate of 12% per annum compounding semiannually. What is the equivalent effective annual interest rate? What is the equivalent per annum interest rate compounding monthly?

(e)   You expect to receive $10,000 as a bonus after 5 years on the job. You have calculated the present value of this bonus and the answer is $8000. What discount rate did you use in your calculation?

(f) Suppose you borrow $10,000 at the annual interest rate of 9%, and you are required to pay it back in 60 equal monthly installments, the first one is due at the end of the first month. How much is the monthly installment?

(g) You want to buy a piece of land for $12,000 cash. The owner would allow you to pay for it in six annual installments of $2300 each, the first one right now. Which method is cheaper for you if the time value of money is 12%?

Answer 1

Answer 1(a)

First of all we have to find the Annuity factor of 14% on monthly compounding basis for 25 years

monthly rate is 14%/12 = 1.167 and total months are 25 x 12 = 300 months

Annuity factor of 1.167% for 300 month is =1/(1+r)¹+1/(1+r)²+1/(1+r)³......... up to 1/(1+r)³⁰⁰

=83.052

monthly payment = $300,000/83.052 = $ 3,612.29

Answer 1(b)

For this we have to calculate the present value of the total cash in flow at interest rate of 7% for 15 years

formula for this = cash in flow x P.V. annuity factor for 15 year @7%

Year	P V Factor	Amount	PV of Amount
1	0.935	10,00,000.00	9,35,000.00
2	0.873	10,00,000.00	8,73,000.00
3	0.816	10,00,000.00	8,16,000.00
4	0.763	10,00,000.00	7,63,000.00
5	0.713	10,00,000.00	7,13,000.00
6	0.666	10,00,000.00	6,66,000.00
7	0.623	10,00,000.00	6,23,000.00
8	0.582	10,00,000.00	5,82,000.00
9	0.544	10,00,000.00	5,44,000.00
10	0.508	10,00,000.00	5,08,000.00
11	0.475	10,00,000.00	4,75,000.00
12	0.444	10,00,000.00	4,44,000.00
13	0.415	10,00,000.00	4,15,000.00
14	0.388	10,00,000.00	3,88,000.00
15	0.362	10,00,000.00	3,62,000.00
			91,07,000.00

Today we will pay $ 91,07,000.00

time line

Beginning

1 year

2 year

3 year

4 year

5 year

6 year

7 year

8 year

9 year

10 year

11 year

12 year

13 year

14 year

15 year

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

$ 10 lac

Answer 1(d)

Compunding Interest Rate = (1+(R/100))ⁿ -1

here, R = Rate of interest (in our care R=6% ( i.e. 12% / 2)

n= time, (in our case n=2 as the interest rate is compounding on monthly basis)

then by applying the above formula = (1+(6/100))² -1 = 12.36%

For monthly compounding the n = 12

and the interest rate = 12% /12 = 1%

= 1+(1/100))¹² -1 = 12.68%

Answer 1(e)

Present Value of money = PV= FV/(1+r)ⁿ

then, 8000 = 10000/ (1+r)⁵

(1+r)⁵ = 10000/8000

1+r = (1.25)^1/5 = 1.0456

now r = 4.56%

Answer 1(f)

First of all we have to calculate the annuity factor of the 0.75% (i.e. 9%/12 as the repayment on monthly basis) for 60 month

= 1/ (1+r)¹ + 1/(1+r)² .......... up to 1/(1+r)60  = 48.173

Now the amount of installment is = Loan amount / Annuity factor

=$10000/48.173 =$ 207.585 (approx.)

Answer 1(g)

In this case at 1st option total outflow of cash = $12000.00

At 2nd option our outflow of cash = 10591.50

Year	PV factor	Installment	Adjusted Cash Outflow
1	1.0000	2,300.00	2,300.00
2	0.8930	2,300.00	2,053.90
3	0.7970	2,300.00	1,833.10
4	0.7120	2,300.00	1,637.60
5	0.6360	2,300.00	1,462.80
6	0.5670	2,300.00	1,304.10
			10,591.50

Hence in the given situations the 2nd option is profitable as it has less cash out flow as compare to 1st case.

Note : 1st installment given at starting of the year than here PV Factor = 1

and the 2nd installment given at starting of the 2nd year which is equal to end of 1year

Answer 1 (c)

as the given question have any 5 no example and it seems that it is 2 nd part of the same, which is not provided in the given question. so we have incomplete information to solve the said problem.