Question

1. What effect does compounding interest more frequently than annually have on future value and b) the effective annual rate (EAR) and Why?
2. Jason borrows RM15,000 from a bank at 10% annually compounded interest to be repaid in six equal instalments. Calculate the interest paid in the second year.
3. Suppose you want to buy a house 5 years from now. The average price of your dream house is RM500,000 and its price is growing at 5% per year. To accumulate enough money to buy your dream house with cash at the end of fifth year, how much will you deposit into an account of the end of every year paying annual interest at 12%?
4. Razak is 30 years old and he has decided that it is the time to plan for his retirement at age of 65. He expects a 15 year retirement period and concluded that he would need RM25,000 per year during his retirement years in order to live comfortably. How much should Razak saves at the end of each year in an account paying 6% interest per annum to meet his goal at age 65?
5. Last December, Azhar received an annual bonus of RM1,500. These annyal bonuses are expected to grow by 5% annually, for the next 5 years. How much will Azhar have at the end of the fifth year if he invests his annual bonuses (including the most recent bonus) in a project paying 8% per year?

Answer 1

a]

EAR = (1 + (r/n))ⁿ - 1

where r = annual nominal rate of interest

n = number of compounding periods per year

future value = present value * (1 + EAR)^{number of years}

Therefore, as the compounding frequency (n) increases, the EAR increases and as a result, the future value increases.

Higher the compounding frequency, higher the EAR and future value

b]

The interest amount in the 2nd year payment is calculated using IPMT function in Excel :

rate = 10% (annual rate)

per = 2 (we are calculating the interest amount in the 2nd year payment)

nper = 6 (6 year loan with 1 payment each year)

pv = 15000 (loan amount)

IPMT is calculated to be RM 1,305.59

c]

future value = present value * (1 + growth rate)^{number of years}

Price of house after 5 years = 500,000 * (1 + 5%)⁵ = RM 638,140.78

Future value of annuity = P * [(1 + r)ⁿ - 1] / r,

where P = periodic payment. We need to calculate this.

r = periodic rate of interest. This is 12%

n = number of periods. This is 5

638,140.78 = P * [(1 + 12%)⁵ - 1] / 12%

P =  638,140.78 * 12% / [(1 + 12%)⁵ - 1]

P = RM 100,449.57

Amount to deposit at end of each year = RM 100,449.57

d]

First, we calculate the amount required at retirement to enable the yearly withdrawals during retirement. The amount required at retirement is calculated using PV function in Excel :

rate = 6% (rate of return earned)

nper = 15 (number of years in retirement)

pmt = -25000 (yearly withdrawal. This is entered with a negative sign because it is a withdrawal)

PV is calculated to be RM 242,806.22

Next, we calculate the yearly saving required to accumulate the required amount at retirement. The yearly saving required is calculated using PMT function in Excel :

rate = 6% (rate of return earned)

nper = 35 (number of years until retirement)

pv = 0 (amount currently saved is zero)

fv = 242806.22 (required amount at retirement)

PMT is calculated to be RM 2,178.91

Annual savings required = RM 2,178.91