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I'm reading a book that gives three formulas. In all of these formulas:

  • $x$ is the amount of money to be received in the future
  • $n$ is the number of years
  • $r$ is the interest rate.

The first is the present value of a lump sum: $PV = x / (1 + r) ^n$. Pretty self explanatory. Suppose I wanted $100$ dollars in $20$ years and could invest money at a $5\%$ rate of return, what amount of money would I have to invest today to have $100$ dollars in $20$ years? $100 / (1.05) ^{20} = 37.69$. Alternatively, if I have $100$ dollars today and it depreciates at the rate of $5\%$ a year for $20$ years, it would be worth $37.69$ dollars in $20$ years.

The second formula is the formula for the present value of a perpetuity: $PV = x / r$ ($x$ is the amount of money to be received at the end of each year forever. So getting $100$ dollars at the end of each year forever would be the same as getting $2000$ dollars today, assuming I can invest that at the rate of $5\%$ per year.

The third formula appears to combine the two: the formula for the present value of an annuity: $PV = (x / r) - ((x / (1 + r) ^ n) / r)$ ($x$ is the amount of money received at the end of each year for $r$ years).

Notice that the left hand side is exactly the formula for the present value of a perpetuity. The right hand side is the present value of a lump sum divided by $r$.

I'm trying to understand the third formula. How would someone derive it? Why do we subtract the present value of a lump sum from the present value of a perpetuity? I understand intuitively that an annuity must have a lower present value than a perpetuity. If $r = 20$, there needs to be some way to indicate that payments stop after $20$ years. I'm just not sure how this is incorporated into this formula.

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Take your second statement: $\dfrac{x}{r}$ is the present value of a perpetuity paying $x$ starting from now (first payment in year $1$)

Now consider the present value of a perpetuity paying $x$ with first payment in year $n+1$. This is the value in year $n$ of $\dfrac{x}{r}$, discounted back to now so dividing by $(1+r)^n$ to give $\dfrac{x}{(1+r)^n r}$

So the present value of an annuity paying $x$ for the first $n$ years is the difference between these, namely $\dfrac{x}{r}-\dfrac{x}{(1+r)^n r}$

An alternative approach would be to consider the sum of the present values of the $n$ payments $$S= \dfrac{x}{(1+r)} + \dfrac{x}{(1+r)^2} + \cdots + \dfrac{x}{(1+r)^{n}}$$ and you might calculate this using the following to allow cancelation $$(1+r)S - S = x - \dfrac{x}{(1+r)^n}$$ which since $(1+r)S-S=rS$ becomes $$S=\dfrac{x}{r}-\dfrac{x}{(1+r)^n r}$$

It may be worth noting that the second term tends to zero if $n$ increases, so the value or cost of a long-term annuity approaches that of a perpetuity

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