Ben plans to save $300 per month for the next 30 years to have a nest egg for re
ID: 3228395 • Letter: B
Question
Ben plans to save $300 per month for the next 30 years to have a nest egg for retirement. His sister Stevie has decided to wait and start saving $600 per month for half the time figuring she will do at least as well as her brother with this strategy. Which of the following statements are false ? ( Select all options that apply. You may want to make up an APR and test the options using Excel's FV function.)
A. The ending balance on both accounts will be the same.
B. Stevie will have earned more interest than Ben.
C. At retirement Stevie will have more money in her savings plan than Ben, since she is investing more money each month.
D. Ben and Stevie will have invested the same amount in the end.
E. Ben will have earned more interest than Stevie.
Explanation / Answer
Solution
At the very outset, two more additional information are necessary –
1. Interest rates for both
2. Whether interest is ‘simple’ or ‘compound’
On the first, we will assume the rates are the same for both. Since the questions hover around comparative statements, actual rate need not be known.
For the second, no short-cut – we need to work out under both cases separately and then answer the questions.
Back-up Theory
Interest Calculations
Simple Interest
If a sum, P, is invested for T years at an interest rate of r% per annum, the interest,
S = PTr/100 = PTi, ……………………………………………………………………(1)
where i = interest rate expressed in decimal form [i.e., r = 10 => i = 0.1]
Amount at the end of T years, A = P + S = P(1 + Ti) ………………………………..(2)
Regular Periodic Investment (Recurring Deposit)
If a sum, P, is invested every year (at the start of the year) for T years at an interest rate of r% per annum, the interest,
S = P{Ti + (T - 1)i + (T - 2)i + (T - 3)i + …… + 1i } = PiT(T + 1)/2 …………………(3)
Amount at the end of T years, A = P + S = P[1 + {iT(T + 1)/2}] ……………………(4)
Compound Interest
If a sum, P, is invested for T years at an interest rate of r% per annum,
Amount at the end of T years, A = P(1 + i)T ………………………………..(5)
Interest, C = A – P = P{(1 + i)T – 1}………………………………..(6)
Regular Periodic Investment (Recurring Deposit)
If a sum, P, is invested every year (at the start of the year) for T years at an interest rate of r% per annum, the interest,
A = P{(1 + i)T + (1 + i)T-1 + (1 + i)T-2 + (1 + i)T-3 + ……. + (1 + i)1}
= P(1 + i){(1 + i)T - 1)}/{(1 + i) – 1} = P[(1 + i){(1 + i)T - 1)}/i] …………………………………… …………………(7)
Interest, C = A – P = P{[(1 + i){(1 + i)T - 1)}/i] – 1}………………………………..(8)
Now, to work out solution,
Total amount invested by Ben = 300 x 12 x 30 = 108000
Total amount invested by Stevie = 600 x 12 x 15 = 108000 …………………………..(9)
Total Simple Interest earned by Ben
= PiT(T + 1)/2 where P = 300, T = 12 x 30 = 360 and i = i/12
= 1759500i …………………………………………………………………………….(10)
Total Simple Interest earned by Stevie
= PiT(T + 1)/2 where P = 600, T = 12 x 15 = 180 and i = i/12
= 814500i …………………………………………………………………………….(11)
By (10) and (11),
Total Simple Interest earned by Ben > Total Simple Interest earned by Stevie ………(12)
Since compound interest grows faster than simple interest, (12) =>
Total Compound Interest earned by Ben > Total Compound Interest earned by Stevie ….(13)
(9) to (13) => A of Ben > A of Stevie …………………………………………………..(14)
A. The ending balance on both accounts will be the same. [FALSE by (14)]
B. Stevie will have earned more interest than Ben.[FALSE by (12),(13)]
C. At retirement Stevie will have more money in her savings plan than Ben, since she is investing more money each month. [FALSE by (14)]
D. Ben and Stevie will have invested the same amount in the end.[True by (9)]
E. Ben will have earned more interest than Stevie. [True by (9)]
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