# Calculating accumulated value of an annuity with changing payments?

In a series of 40 payments the first 10 payments are 10 each, the second 10 payments are 20 each, the third 10 payments are 30 each, and the final 10 payments are 40 each. The payments are equally spaced and the interest rate is 5% per payment period. Find the accumulated value at the time of the final payment.

This is what I did, but I got the wrong answer and I'm not sure what I'm doing wrong:

``````10[((1.05)^10 - 1)/0.05] x (1.05)^30 + 20[((1.05)^10 - 1)/0.05] x (1.05)^20 +
30[((1.05)^10 - 1)/0.05] x (1.05)^10 + 40[((1.05)^10 - 1)/0.05] x (1.05) =
``````

2353.98

• Is this a homework question? – Eric Feb 11 '14 at 16:02
• No I'm working out problems for the FM/2 Exam – user13026 Feb 11 '14 at 16:05
• Welcome to Money.SE. The equation text above is awful. Are you doing this on a spreadsheet? A single line equation is going to be cumbersome and tough to adjust. – JTP - Apologise to Monica Feb 11 '14 at 16:24
• You did one tiny thing wrong (That's all it takes!) You evaluated the four annuities, and carried the results forward 30, and 20, and 10, and 1 period. Does something stand out as not fitting the pattern? – DJohnM Feb 11 '14 at 22:14

The first payment of 10 accumulates 40 cycles of interest, i.e. `10*1.05^40` and the last payment of 40 accumulates 1 cycle of interest, `40*1.05`. To get the total these are added together along with all the payments in-between :-

``````10*1.05^40 + 10*1.05^39 + 10*1.05^38 + 10*1.05^37 + 10*1.05^36 +
10*1.05^35 + 10*1.05^34 + 10*1.05^33 + 10*1.05^32 + 10*1.05^31 +
20*1.05^30 + 20*1.05^29 + 20*1.05^28 + 20*1.05^27 + 20*1.05^26 +
20*1.05^25 + 20*1.05^24 + 20*1.05^23 + 20*1.05^22 + 20*1.05^21 +
30*1.05^20 + 30*1.05^19 + 30*1.05^18 + 30*1.05^17 + 30*1.05^16 +
30*1.05^15 + 30*1.05^14 + 30*1.05^13 + 30*1.05^12 + 30*1.05^11 +
40*1.05^10 + 40*1.05^9 + 40*1.05^8 + 40*1.05^7 + 40*1.05^6 +
40*1.05^5 + 40*1.05^4 + 40*1.05^3 + 40*1.05^2 + 40*1.05 =
``````

2445.27

In shorter form :- Alternatively expressed :- So solving for future value, v :- Using this result the future value can be calculated more succinctly :-

``````10*1.05*(1.05^10 - 1)/0.05 + 10*1.05*(1.05^20 - 1)/0.05 +
10*1.05*(1.05^30 - 1)/0.05 + 10*1.05*(1.05^40 - 1)/0.05 =
``````

2445.27

Note

The above finds the total after 40 periods, giving the last payment one period in which to accrue interest. If, as you state, you actually want to know the total at the time the last payment is made, then you would have to deduct one cycle of interest from each term, i.e first term `10*1.05^39`; last term, simply 40. I imagine you probably want the full 40 periods though.

``````10*1.05^39 + 10*1.05^38 + 10*1.05^37 + 10*1.05^36 + 10*1.05^35 +
10*1.05^34 + 10*1.05^33 + 10*1.05^32 + 10*1.05^31 + 10*1.05^30 +
20*1.05^29 + 20*1.05^28 + 20*1.05^27 + 20*1.05^26 + 20*1.05^25 +
20*1.05^24 + 20*1.05^23 + 20*1.05^22 + 20*1.05^21 + 20*1.05^20 +
30*1.05^19 + 30*1.05^18 + 30*1.05^17 + 30*1.05^16 + 30*1.05^15 +
30*1.05^14 + 30*1.05^13 + 30*1.05^12 + 30*1.05^11 + 30*1.05^10 +
40*1.05^9 + 40*1.05^8 + 40*1.05^7 + 40*1.05^6 + 40*1.05^5 +
40*1.05^4 + 40*1.05^3 + 40*1.05^2 + 40*1.05 + 40 =
``````

2328.82

(User58220 has posted the formula for this version.)

Another, perhaps simpler way to look at and calculate the result.

Find the future value, at the time of the last payment, of 4 ordinary annuities, all with identical payments of \$10 each, and an interest rate of 5% per payment period. The four annuities have lengths of 40, 30, 20 and 10 payments, and all end at the same moment.

So you need to evaluate

``````FV = 10 * (1.05^N - 1) / 0.05
``````

for N = 10, 20, 30, and 40, and add the results...

You should get `\$2328.8246`

• Yes, that works out. Seems odd that it works out to 39 periods. What's the point of the 40th payment when you get it straight back? (I would guess it's for the convenience of the mathematical shortcut.) – Chris Degnen Feb 11 '14 at 23:59
• Well, the OP asked specifically for the value at the time of the last payment, and that's what the future value of an ordinary annuity is defined as... – DJohnM Feb 12 '14 at 5:12
• Yes, you both seem to be using a standard formula. – Chris Degnen Feb 12 '14 at 6:50
• To include a 40th interest period this formula can be used: `FV = 10 * 1.05 * (1.05^N - 1) / 0.05`. – Chris Degnen Feb 12 '14 at 9:18
• The extra `1.05` term is mentioned here: "To calculate the future value of an annuity due, multiply the result by (1+i)." Also diagrams here. – Chris Degnen Feb 12 '14 at 10:39