# Annuity immediate

An annuity immediate has quarterly payments of $1000$ for $25$ years at a rate of $6 \%$ converted quarterly. Find its present value.

The effective interest rate $i$ is $0.06/4$. So let $v = \frac{1}{1+i}$. Then wouldn't the present value be $1000 \left(\frac{1-v^{25}}{i} \right)$?

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The present value of an income stream with $n$ equally spaced payments, with an interest of $r$ at each payment, each payment of $C$, is obtained by computing the present value of each payment and adding it up. Here, you have $r=0.06/4 = 0.015$, as you note, so you get $$1000 + \frac{1000}{1+r} + \frac{1000}{(1+r)^2} + \cdots + \frac{1000}{(1+r)^{100}},$$ since there are 100 payments (one payment per quarter, 25 years, beginning now).

This is just $$1000\left(1+\frac{1}{1+r} + \left(\frac{1}{1+r}\right)^2 + \cdots + \left(\frac{1}{1+r}\right)^{100}\right)$$ which is $$1000\left( \frac{1 - \frac{1}{(1+r)^{101}}}{1-\frac{1}{1+r}}\right).$$

If you only get 1000 payments (no payment at the end of the last quarter 25 years from now) then the sum only goes up to exponent 99; adjust accordingly.

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If the first payment is received at the end of the first quarter, then the present value is $$1000\left(\frac{1-v^{100}}{i}\right).$$

If the first payment is received at the beginning of the first quarter (this seems to be the case, since you stated "immediate"), then the present value is $$1000\left(\frac{1+i-v^{99}}{i}\right),$$ assuming a total of 100 payments are made.

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For starters, there are 100 periods, not 25. If you just reduce each payment to present value, what is the value of the nth payment? Can you then sum this from 1 to 100?

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