Question

Jen wants to accumulate 400,000 at the end of 16 years. She deposits CX+10,000) into an investment account at the end of each of the first 8 years. She then deposits X into the same account at the end of each of the final 8 years. The interest rate on the account is a 6% nominal rate compounded annually. Find X 7.

Answer 1

Since you have posted multiple unrelated questions, i will answer the first one.

Question No. 7

Let's look at first investment made at the end of year 1. So this investment has N = 16 - 1 = 15 years to grow until maturity (i.e end of year 16). Interest rate, I = 6%.

So the FV factor associated with this investment = FV factor = (1 + I)^N = (1 + 6%)¹⁵ = 2.3966

So, the future value of investment made at the end of year 1 = 2.3966 x amount invested = 2.3966 x (X + 10,000). We will now extend the same logic for investments made subsequently. We will thus create the following FV factor table:

End of year, P	Number of years until maturity	FV factor
P	N = 16 - P	FV = (1+6%)N
1	15	2.3966
2	14	2.2609
3	13	2.1329
4	12	2.0122
5	11	1.8983
6	10	1.7908
7	9	1.6895
8	8	1.5938
Subtotal for P = 1 to 8		15.7751
End of year, P	Number of years until maturity	FV factor
P	N = 16 - P	FV = (1+6%)N
9	7	1.5036
10	6	1.4185
11	5	1.3382
12	4	1.2625
13	3	1.1910
14	2	1.1236
15	1	1.0600
16	0	1.0000
Sub total for P = 9 to 16		9.8975

We have split the table in two parts because amount invested per year is different for the two blocks of time:

• For P =1 to P =8; amount invested per annum = X + 10,000; so the future value of investments = Amount invested per annum x Sum of FV factors corresponding to the block of first 8 years = (X + 10,000) x 15.7751; Please see the table above for calculation of 15.7751
• For P =9 to P =16; amount invested per annum = X; so the future value of investments = Amount invested per annum x Sum of FV factors corresponding to the block of next 8 years, starting from year 9 to year 16 = X x 9.8975; Please see the table above for calculation of 9.8975

So, the future value of these investments should add up to the target value of 400,000.

Hence, the equation in hand is:

(X + 10,000) x 15.7751 + X x 9.8975 = 400,000

Hence, 25.6725X + 157,751 = 400,000

Hence, X = 9,436.13