this is my first actuarial question so correct any mistakes I make in formatting!
We have a perpetuity with annual payments. The first payment is $ \$500$ and then payments increase by $ \$25$ each year until they become level at $ \$900$. We want to find the value of this perpetuity at the time of the first payment where the annual effective interest rate is $ 5 \%$.
- This has been my reasoning so far...
First, I thought of this as two separate annuities. I could just consider the payments $ \$500 \to \$900$ as an arithmetically increasing annuity-immediate and then consider the rest of the perpetuity as a regular level perpetuity. Summing the present values of these together and then multiplying by $(1.05)$ would give me the value at the time of the first payment.
However, I remember there was a trick to doing this sort of problem (I mean there is a shorthand for calculating the PV of an increasing and then level perpetuity).
The formula for a perpetuity that increases by $1$ and then levels at time $n$ is (I think) $PV = \frac{\text{annuity-due}}{i}$.
Does anyone know what I'm talking about? Can anyone explain how to use this short-hand for this problem?
Thank you!
Three separate parts. The first is a level $\$475$ perpetuity whose value is:$$\frac{\$475}{d}$$ Then you can value the increasing annuity, for $16$ years, beginning now with a payment of $\$25$. The formula for this is $$$25 \cdot \frac{\overset{..}{a}_{\overline {16} \rceil}-16v^{16}}{d}$$ Then the last bit is the perpetuity of $\$25 \cdot 16$, beginning $16$ years from now. This is valued (presently) at $$\$25 \cdot \frac{16}{d} \cdot v^{16}$$ Add the second and the third part together and you get your formula. Easy no?