Payments made out in even daily intervals, increasing .69%, for 365 days with the first payment one day, next two days, etc

Question

I have a sequence of $365$ "contracts" guaranteeing me a payout, starting out at $.0069$, with the first one paying out for one day, the next one for two days. There are, thus, $365$ payments paid out on day one, then $364$ on day two, and so on through $365$ days. These contracts are each $.69\%$ larger than the previous one. So, the one paying out for $365$ days pays out an amount $1.0069^{365}$ times the amount paying out for just one day.

How would I calculate this sequence of cash flows without having to plug in $365$ payments into my calculator? And how would I do this in MS Excel?

Answer 1

Let $P$ the value the first payment (the same value for each contract) and $g=0.69\%$ and $n=365$. We have the values

• 1st day: $w_1=nP$,
• 2nd day: $w_2=(n-1)P(1+g)$
• 3rd day: $w_3=(n-2)P(1+g)^2$
• ....
• $k$-th day: $w_n=(n-k+1)P(1+g)^{k-1}$

Summing up we have the value $$ \textsf{V}=\sum_{k=1}^{n}\,w_k=\sum_{k=1}^{n}\,P\times(n-k+1)\times(1+g)^{k-1} $$ and observing that $\sum_{k=1}^{n}\,a^{k}=a\frac{a^n-1}{a-1}$ and $\sum_{k=1}^{n}\,k\,a^{k}=a\frac{na^{n+1}-(n+1)a^n+1}{(a-1)^2}$ we have $$ \textsf{V}=P\frac{(1+g)^{n}+g\left[(1+g)^n-n-1\right]-1}{g^2} $$