Business Mathematics and Statistics · Ch 7 — Financial Mathematics (Annuities; Stocks, Shares, Debentures and Brokerage)

Present Value of an Annuity

The present value of an annuity is the single lump sum, invested today at the same rate of interest, that would be exactly enough to fund every future instalment of the annuity — the value 'discounted back' to the present rather than accumulated forward to the end.

Note

Present Value of an Ordinary Annuity

$V = P\left[\dfrac{1-(1+i)^{-n}}{i}\right]$

where $P$ , $i$ , $n$ carry the same meaning as before.

This is the formula behind every loan repaid in equal instalments (an EMI): the loan amount sanctioned today is exactly the present value of the stream of future instalments the borrower promises to pay, at the loan's own interest rate. Rearranging the formula to solve for $P$ given a known present value $V$ (the loan amount) is exactly how an EMI is calculated in practice:

P = \dfrac{V\cdot i}{1-(1+i)^{-n}}

For an annuity due, exactly as with the future value, every instalment is discounted one period less (since it arrives one period earlier), so the present value picks up one extra factor of $(1+i)$ :

$V_{\text{due}} = P\left[\dfrac{1-(1+i)^{-n}}{i}\right](1+i)$ …

Definition 5Present Value of an Annuity

The single lump sum invested today, at the annuity's interest rate, that is exactly sufficient to fund every future instalment: $V = P\left[\dfrac{1-(1+i)^{-n}}{i}\right]$ for an ordinary annuity. This is …

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