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Find s5 if δt = 0.02t for 0 ≤ t ≤ 5.

https://i.stack.imgur.com/7Yw72.jpg

My answer sheet gives me 5.7726 and I don't know how it got that answer. Can someone please explain how my method is incorrect?

I was asked to find the accumulated value of an annuity lasting 5 payment periods given the force of interest over the interval 0 and 5.

As my work shows, I first found the value of the accumulated interest a(t) to which I then converted to the effective rate and finally, I had all the necessary variables to determine the accumulated value of an annuity with n=5.

I don't see how I am wrong here.

swordlordswamplord
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You can't use the effective interest rate to find $s_{\overline{5}|}$ because the interest rate is not constant and the amount invested is not constant. The force of interest is increasing over time, and the amout invested is also increasing over time. Instead of using "shortcut" annuity formulas you need to calculate $s_{\overline{5}|}$ as the accumulated value of the cash flows. You will have cash flows of $1$ and times $1,2,3,4,5$ and so their accumulated values at time $5$ are: $$s_{\overline{5}|} = 1\cdot \frac{a(5)}{a(1)} + 1 \cdot \frac{a(5)}{a(2)} +1 \cdot \frac{a(5)}{a(3)} +1 \cdot \frac{a(5)}{a(4)} +1 \cdot \frac{a(5)}{a(5)}.$$ Using this formula plus the expression you found for $a(t)$ gives the correct answer of $5.7726$.

kccu
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