Present Value of Perpetuity Really really struggling with the following question:
Suppose you receive $\$44000$ per year, but get a $\$1000$ raise every $k = 9$ years. That is, you are paid $\$44000$ in years $1, 2, ..., k$, and paid $\$44000 + \$1000$ in years $k+1, k+2, ..., 2k$, and paid $\$44000+\$2000$ in years $2k+1, 2k+2, ..., 3k$, and so on into perpetuity.  Given an interest rate $2.7\%$, what is the present value of these cash flows?  (to nearest $\$0.01$) (Hint: work out a formula rather than just calculating numerically. Can you convert this question to the PV of an annuity of length 9 years?)
The answer should be $\$1,766,315.72$.
I tried calculating annuity for $n=9$ first and get the annuity of $\$44,000+\$1000$ and $\$44,000+\$2000$ separately. Is that what I do? If yes, how?
Any help would be appreciated! Thanks.
 A: This is an increasing perpetuity that has the following cash flow, in thousands:  $$PV = 44 (v + v^2 + \cdots + v^9) + 45 v^{9} (v + v^2 + \cdots + v^9) + 46 v^{18} (v + v^2 + \cdots + v^9) + \cdots$$ where $v = 1/(1+i)$ is the annual present value discount factor.  Note we are assuming that payments are made in arrears, so this is a perpetuity-immediate and the present value is calculated with respect to one year before the first payment of $44$.  Clearly, we may perform a factorization:
$$PV = (v + v^2 + \cdots + v^9)(44 + 45 v^{9} + 46 v^{18} + \cdots ),$$ and the first factor is simply $$\require{enclose} a_{\enclose{actuarial}{9}i} = \frac{1 - v^9}{i} \approx 7.89619.$$  The second factor is $$44(1 + v^9 + v^{18} + \cdots) + (v^9 + 2v^{18} + 3v^{27} + \cdots) = 44 \ddot a_{\enclose{actuarial}{\infty} j} + (Ia)_{\enclose{actuarial}{\infty} j},$$ where $j = (1+i)^{9} - 1$ is the effective nine-year rate of interest.  Since a perpetuity-due has a present value of $$\ddot a_{\enclose{actuarial}{\infty} j} = 1 + \frac{1}{j},$$ and an increasing perpetuity-immediate has a present value of $$(Ia)_{\enclose{actuarial}{\infty} j} = \frac{1}{j} + \frac{1}{j^2},$$ the second factor in our equation of value is $$44(4.6905) + 17.3103 \approx 223.692.$$  Therefore the present value is $$PV \approx (7.89619) ( 223.692) = 1766.31472367$$ in units of thousands of dollars.

As an addendum, the formulas for the present values of the perpetuities can be found as follows.  As usual, $i$ is some periodic interest rate and $v = 1/(1+i)$.  We assume $i > 0$, otherwise the present value is infinite.  Then $0 < v < 1$ and
First, $$\ddot a_{\enclose{actuarial}{\infty} i} = 1 + v + v^2 + \cdots = \frac{1}{1-v} = \frac{1}{1 - \frac{1}{1+i}} = \frac{1+i}{i} = 1 + \frac{1}{i},$$ from the formula for the sum of an infinite geometric series.
Hence, $$a_{\enclose{actuarial}{\infty} i} = v + v^2 + v^3 + \cdots = \frac{1}{i},$$ which is the present value of a perpetuity-immediate.
Finally, let $$x = (Ia)_{\enclose{actuarial}{\infty} i} = v + 2v^2 + 3v^3 + \cdots.$$  Then $$vx = v^2 + 2v^3 + 3v^4 + \cdots,$$ and $$
vx + a_{\enclose{actuarial}{\infty} i} = (v^2 + 2v^3 + 3v^4 + \cdots) + (v + v^2 + v^3 + \cdots) = v + 2v^2 + 3v^3 + \cdots = x.$$  Solving for $x$ yields $$x = (Ia)_{\enclose{actuarial}{\infty} i} = a_{\enclose{actuarial}{\infty}i} \cdot \frac{1}{1-v} = (a_{\enclose{actuarial}{\infty}i}) (\ddot a_{\enclose{actuarial}{\infty}i}) = \frac{1}{i} + \frac{1}{i^2},$$ as claimed.
