Question

1. Suppose you are saving for your down payment on a new car. Over the next four years you will save increasing amounts and expect to earn 12% on your savings. Today you are ready to deposit $1,000, at the end of the year you will deposit $1,100, then $1,210 and finally in the third year 1,331.
   1. Calculate the ending balance at the start of year 4 by compounding each cashflow and taking the sum in year four.
   2. Calculate the ending balance at the start of year 4 by using the growing annuity formula. (Hint you will need to find the growth rate and think carefully about the timing of the cashflows.)

Answer 1

Part (a)

Please see the table below. Please be guided by the second column titled “Linkage” to understand the mathematics. The last row highlighted in yellow is your answer. Figures in parenthesis, if any, mean negative values. All financials are in $.

Start of year	Time until start of year 4	Amount ($)	Future Value ($)
t	n = 4 - t	PV	FV = PV x (1 + 12%)^n
0	4	1,000.00	1,573.52
1	3	1,100.00	1,545.42
2	2	1,210.00	1,517.82
3	1	1,331.00	1,490.72
Total			6,127.48

Part (b)

The FV of a growing annuity due is given by:

A(1+r) [(1 + r)" – (1+g)" r-9

A = first annuity = 1,000; g = growth rate in annuity = 1,100 / 1,000 - 1 = 10%, r = 12%, n = 4

Hence, FV

1,000(1 +0.12) 2(1+0.12) -(1+0.10) 10.12 -0.10

= $ 6,127.48