Question

Question 5.

(a) Assume that your savings account offers you an APR of 6% with quarterly compounding. What is the present value of $1, 000 to be received 3 quarters from today?

(b) Your bank charges you 0.5% per month on your car loan. What APR must the bank report on your loan? What is the effective annual interest rate (EAR) on your loan? Explain why the EAR is higher than the APR.

(c) You have been given the once-in-a-lifetime opportunity to invest $2, 000 in a certificate of deposit paying an APR of 24% with monthly compounding. Exactly how many months will it take for your $2, 000 to grow to $10, 000?

Answer 1

a) Amount to be received in three quarters = Future value = $1000. To find the present value of $1000

APR = 6% with quarterly compounding

Quarterly rate = APR / 4 = 6% / 4 = r = 1.5% per quarter

No of quarters = n = 3

We know that

Present value of $1000 = Future value / (1+ quarterly rate)ⁿ = 1000 / (1+1.5%)³ = 1000 / (1.015)³ = 1000 / 1.045678375 = 956.3169 = $956.32 (rounded to two decimal places)

b) Periodic rate = Monthly rate = 0.5% per month, No of compounding periods = n = 12

We know that APR = Periodic rate x no of periods in a year = 0.50 x 12 = 6.00%

Hence APR = 6%

EAR = (1+Periodic rate)ⁿ - 1 = (1+0.50%)¹² - 1 = (1.0050)¹² - 1 = 1.061677 - 1 = 0.061677% = 6.1677% = 6.17% (rounded to two decimal places)

Hence EAR = 6.17%

APR is based on simple interest and does not take into consideration compounding. Interest rate for a year can be calculated by just multiplying periodic rate by number of periods in a year In this question APR is calculated by the product of monthly rate and no of months in a year. On the other hand EAR is based on compound interest. In EAR interest for first month is calculated using monthly rate. At the end of first month, interest for first month is added to initial principal, to arrive at principal for calculating the interest for second month. In other words Principal for calculating a interest for a month is arrived by the sum of principal for calculating interest for previous month and interest for previous month. Thus interest for every successive month is higher that its its previous month. This is results higher total rate for the year as compared to APR. That is why EAR is higher than APR.

c) APR = 24% compounded monthly, Monthly rate = APR / 12 = 24% / 12 = 2% per month

Initial investment = Present value = $2000, Amount to accumulate = Future value = $10000

We can find number of months to accumulate $10000 by using NPER function in excel

Formula to be used in excel: =NPER(rate,pmt,-pv,fv)

In the formula pmt = 0 as there are no equal annual deposits to accumulate $10000

Using NPER function in excel, we get number of months to accumulate $10000 = 81.27 months

Hence Number of month to accumulate $10000 = 81.27 months