Question

Use the following information about to help answer problems 23-24. TIME CFs 0 2 233 3 4 $ 2,200.00 222 $ 3,279.58 . Next Year (T1), you will deposit $2,200.00 in a savings account Over the short term, you will increase the amount you deposit each year by a constant growth rate so that at the end of year 5 (T-5) you will make a deposit of $3,279.58 . You will make no deposits beyond the $3,279.58 at the end of year 5 (T 5) You earn a rate of return of 12.0% on your savings every year . 23.The deposit you will make at the end of year 4 (T-4) is closest to: a. $2,968 b. $3,091 c. $3,346 d. $3,462 e. $3,848 24.The future value of your savings account at the end of year 6 (T-6) is closest to a. $13,565 b. $17,790 c. $18,121 d. $19,120 e. $20,459

Answer 1

23.

• Given that,
  • Amount deposited at the end of Year 1 (T = 1) = $2200
  • Amount to be deposited at the end of each year has to be increased at a constant growth rate = r
  • Amount deposited at the end of Year 5 (T = 5) = $3279.58
  • Therefore, Amount deposited at the end of
    • Year 2 (T=2) = 2200 (1+r)
    • Year 3 (T=3) = 2200 (1+r)²
    • Year 4 (T=4) = 2200 (1+r)³
    • Year 5 (T=5) = 3279.58 = 2200 (1+r)⁴

=> 3279.58/2200 = (1+r)⁴

  => 1.49071818 = (1+r)⁴

=> r = 0.10496 = 10.496%

• 
  
• Therefore, Amount deposited at the end of Year 4 (T=4) = 2200 (1+r)³ = 2200 (1+0.10496)³ = 2967.98 = 2968 (approx.)
• Correct answer is a. $2,968

What we have done above is nothing but use of Present value and Future value formula.

FV = PV (1+r)ⁿ  , where FV is the end value, that is 3279.58 ; PV is 2200 ; r = rate of interest = constant growth rate ; n = number of periods = 5-1 = 4

So, 3279.58 = 2200 (1+r)⁴ gave us the constant growth rate, r = 10.496%. Then we applied the same formula for computing the Amount deposited at the end of Year 4, where n = 4-1 = 3

24. Computation of Rate of Return on Savings

YEAR (T= )	Amount at beginning of the year  [A]	Savings Interest earned @12% per annum and deposited at the end of the year [B]	Amount deposited at the end of the Year   [C]	Amount in the Savings Bank Account [A+B+C]
0	0	0	0	0
1	0	0	2,200	2,200
2	2,200	2200*0.12 = 264	2,431	4,895
3	4,895	4895*0.12 = 587.64	2,686.16	8,168.8
4	8,168.8	8168.8 *0.12 = 980.26	2,968	12,117
5	12,117	12117*0.12 = 1454.04	3,279.58	16,850.68
6	16,850.68	16850.68 *0.12 = 2022.08	0	18,872 (approx)

We realise that none of the options matched our answer. It would be wise to select the option which is closest to our answer.

So, we will select Option d. $19,120

If you are not new to the concept of Discounting factor, then we can use the Future Value Factor (FVF) to make the calculations faster

YEAR (T= )	Amount deposited at the end of the year	No. of Years for which Amount will remain deposited in the Bank Account	FVF @12% per year	Future Value
0	0	6	(1+0.12)6 = 0	0
1	2,200	5	(1+0.12)5= 1.7623	3,877
2	2,431	4	(1+0.12)4= 1.5735	3,825
3	2,686.16	3	(1+0.12)3= 1.4049	3,773
4	2,968	2	(1+0.12)2= 1.2544	3,723
5	3279.58	1	(1+0.12)1= 1.12	3,673
	TOTAL			18,872

Thus, we get the same answer. This certainly means that the options given are inaccurate.