Question

1. An amount of $15,000 is deposited into an account which pays 5.2% interest.
   1. How many periods will it take the account to grow to at least $20,000 if interest is compounded semiannually?

2. If interest is compounded semiannually and simple interest is paid for part of a period, how long before the account is worth exactly $20,000? Use a 360 day year for the simple interest.

3. If interest is compounded continuously, how long before the account is worth exactly $50,000? Use a 360-day year.

Answer 1

Principal Amount = $15,000

Interest Rate = 5.2% p.a.

(A)

Semi - Annual Interest Rate = 5.2% / 2 = 2.6%

Amount = Principal * (1 + Interest Rate)^{Number of Periods}

20000 = 15000 * (1 + 2.6%)^{Number of Periods}

Number of Periods = log (20000 / 15000) / log(1.026)

Number of Periods = 11.208 Periods i.e. 11.208 / 2 = 5.604 Years Approx.

(B)

As per our last answer it would take slightly more than 11 semi annual periods. So, for 11 periods interest would be compounded but after that only simple interest will be paid. So,

Amount after 11 periods = Principal * (1 + Interest Rate)¹¹

Amount after 11 periods = 15000 * (1 + 2.6%)¹¹

Amount after 11 periods = $19893.55

Interest to be earned to get 20000 = 20000 - 19893.55 = $106.45

Simple Interest =

Time =

Time =

Time = 49.13 Days i.e. 0.136 year

Total Time taken = 5.5 Years + 0.136 = 5.636 Years

(C)

When interest rate is continuously compounded

Amount = Principal * e^{(Interest Rate * Time Period)}

50000 = 15000 * e^{(5.2% * Time Period)}

Time Period = Log(50000 / 15000) / 5.2% = 23.153 Years.