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Find the future value of a five year annuity ($s_{(n)}$) if $\delta _t=0.02t$ for $0 \le t \le 5$.

What I know is

$\delta_t= \frac{A'(t)}{A(t)}$

$A(5)=\frac{0.02}{0.02. X5}=0.2$

I am not even sure of this. And then I do not know know to proceed with the annuity part.

Tosh
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1 Answers1

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From $\delta(t)=\frac{a'(t)}{a(t)}=\frac{\mathrm d\log a(t)}{\mathrm d t}$ we have the accumulation function $$ a(t)=\mathrm e^{\int_0^5\delta(s)\mathrm d s}=\mathrm e^{0.02\int_0^5 s\mathrm d s}=\mathrm e^{0.01\,t^2} $$ and then the future value $$ s_{\overline{n}|}=\sum_{t=1}^n a(n-t)=\sum_{t=0}^{n-1} a(t)=1+\mathrm e^{0.01}+\mathrm e^{0.04}+\mathrm e^{0.09}+\mathrm e^{0.016}=5.3185 $$

alexjo
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