Question

a) What is the EAR corresponding to a nominal rate of 8% compounded semiannually? Compounded quarterly? Compounded daily?

b) Your client is 40 years old; and she wants to begin saving for retirement, with the first payment to come one year from now. She can save $5,000 per year; and you advise her to invest it in the stock market, which you expect to provide an average return of 9% in the future. If she follows your advice, how much money will she have at 65?

c) You have $60,000 to put as a down payment on a new house that costs $480,000, and you have been quoted the following terms: 2.99% Annual Percentage Rate (APR), for 30 years. If you decide to purchase this home, what will your monthly payment be? Additionally, over the life of the loan what would your total interest expense be?

d) An investment will pay $ 1, 500 at the end of each year for 20 years, and on the date of the last payment will also make a separate payment of $40,000. If your required rate of return on this investment is 4%, how much would you be willing to pay for the investment today?

e) A client has $300,000 in an account that earns 8% per year, compounded monthly. The client's 35th birthday was yesterday and she will retire when the account value is $1 million. At what age can she retire if she puts no more money in the account? At what age can she retire if she puts $250 per month into the account every month, beginning one month from today?

Answer 1

a) 8% compounded semiannually is equivalent to 4% effective rate every 6 months

hence, EAR = (1+0.04)^2-1 =0.0816 or 8.16%

Quarterly compounded rate (r) is given by

(1+r/4)^4 = 1.0816

=> r= 0.0792156 or 7.92%

Daily compounded rate (r) is given by

(1+r/365)^365 = 1.0816

r= 0.078449856 or 7.84%

b) Amount in future = The future value of annuity if one saves $5000 at the end of each year for 25 years and earns 9% p.a.

= 5000*1.09^24+ 5000*1.09^23+....+5000

=5000/0.09*(1.09^25-1)

=$423504.48  

c) Total loan = $480000 - $60000 = $420000

Monthly interest rate = 2.99%/12 = 0.002492

No. of payments = 30*12 = 360

Monthly payment (A) is given by

A/0.002492*(1-1/1.002492^360) = 420000

=> A = $1768.43

Total Amount paid in 360 installments = $1768.43*360 =$636650.13

So,Total Interest Expense = $636650.13 - $420000 = $216650.13

d) Amount to be paid for investment today = present value of Cashflows from investments

= 1500/0.04*(1-1/1.04^20)+40000/1.04^20

= $38640.97

e) Interest rate per month =8%/12 =0.00667

Let the no. of months after which she retires is n

If she doesn't put any amount

Future value of $300000 > $1 million

=> 300000*1.00667^n> 1000000.

=> 1.00667^n>3.33

Taking natural log of both sides

=> n> ln(3.333)/ln(1.006667)

n > 181.197

So, n= 182

So, She can retire after 182 months if she does not put any money in the account i.e when she is 50 years 2 months old

If she puts $250 every month

Future value of $300000 + deposits after n months > $1 million

=> 300000*1.00667^n + 250/0.00667*(1.00667^n -1) > 1000000.

=> 337500*1.00667^n - 37500>1000000

=> 1.00667^n >3.0740

Taking natural log of both sides

=> n> ln(3.074)/ln(1.006667)

n > 169.011

So, n= 170

So, She can retire after 170 months if she puts $250 every month in the account i.e. when she is 49 years 2 months old.