Conceptually, the way amortization of loan payments works is that the lender should come out even at the end of the loan when you account for future value of money based on the loan's interest rate. If the lender loaned $A$ dollars at an interest rate per period $r$ over $n$ periods, the future value (at the end of the loan) of the money lent is $A(1+r)^n$. If the payment is $p$, then the future value of the first payment is $p(1+r)^{n-1}$ (paid at the end of the first period), for the second is $p(1+r)^{n-2}$, for the $k$th payment is $p(1+r)^{n-k}$, up to the last payment which is paid at the end of the loan so its future value is $p(1+r)^0=p$. If you take the sum of these future values of all the payments, it should be equal to the future value of the original loan. This equation can be solved for $p$ in terms of the general $A$, $r$, and $n$ to get a generalized amortization formula.
Now, on a period-to-period basis, the balance is defined by the initial balance $b_0=A$ and the recurrence relation $b_{k}=(1+r)b_{k-1}-p$. That is, the change from one period to the next is to add the interest due and subtract the payment. Knowing the parameters of the loan and the payment, you can use this to find the balance after a specific number of periods, as well as the breakdown of interest and principal in each payment.
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When the perpetuity begins to pay out when the annuitant turns $60$ years of age, the schedule of payments is monthly, not annually. I quote:
The benefit pays an annual total of 48000 with level payments at the start of every month.
The $48000$ is therefore divided into $12$ equal payments of $4000$.
Consequently, the equation of value at the time that she turns $60$ is:
$$P(1+i)^{40} + (1.06)P(1+i)^{39} + \cdots + (1.06)^{39}P(1+i) = 4000 + 4000v^{1/12} + 4000v^{2/12} + \cdots,$$
where $v = 1/(1+i)$ is the annual present value discount factor. The first thing to note is that I set the valuation time at age $60$, rather than at age $20$, because this makes the calculation a little cleaner. On the left-hand side is the accumulated value of a geometrically increasing annuity-due with annual payments. On the right-hand side is the present value of a perpetuity-due with monthly payments.
The geometric annuity-due is expressible as a level payment annuity-due with modified interest rate; e.g. factoring out $(1.06)^{40} P$, we get $$P(1+i)^{40} + (1.06)P(1+i)^{39} + \cdots + (1.06)^{39}P(1+i) = (1.06)^{40} P \left( \left(\frac{1+i}{1.06}\right)^{40} + \left(\frac{1+i}{1.06}\right)^{39} + \cdots + \frac{1+i}{1.06}\right).$$ So if $i = 0.08$, the factor in parentheses is a level-annuity-due with effective annual interest rate $$j = \frac{1.08}{1.06} - 1 \approx 0.01887,$$ and the level payment is adjusted to be $(1.06)^{40}P$. Thus this side of the equation of value is $$(1.06)^{40} P \ddot s_{\enclose{actuarial}{40}j} = (1.06)^{40} P (1+j) \frac{(1+j)^{40} - 1}{j}.$$
Meanwhile, the right-hand side is a perpetuity-due with effective monthly rate $$k = (1+i)^{1/12} - 1 \approx 0.00643403,$$ which you can verify yields an effective annual rate of $i$, since $(1+k)^{12} = 1+i$. So the equivalent monthly present value discount factor is $$v^{1/12} = \frac{1}{1+k} \approx 0.993607.$$ Then the right-hand side has present value
$$4000 \ddot a_{\enclose{actuarial}{\infty} k} = 4000 \left(1 + \frac{1}{k}\right).$$
Hence the required level payment is $$P = \frac{4000 \left(1 + \frac{1}{k}\right)j}{(1.06)^{40} (1+j)((1+j)^{40} - 1)} \approx \frac{4000 (156.424)(0.01887)}{(10.2857)(1.01887)(1.11211)} \approx 1012.9496.$$
Best Answer
Let $v = 1/(1+i)$ be the annual present value discount factor. Then the present value is expressed as the cash flow $$PV = 1000 + 1000(0.97)v^4 + 1000(0.97)^2 v^8 + 1000(0.97)^3 v^{12} + \cdots.$$ Note that "end of the fourth year" means that four years have elapsed from the time of the first payment, for the reason that if we say "end of the first year," the payment occurs at time $t = 1$.
In actuarial notation, we would have $$PV = 1000 \ddot a_{\overline{\infty}\rceil j},$$ where $j = ((0.97)v^4)^{-1} - 1$ is the effective periodic rate corresponding to the effective periodic present value discount factor of a level payment of $1000$, adjusted for the decrease in payments, the payment frequency, and the annual rate of interest. Since $\ddot a_{\overline{\infty}\rceil j} = 1+ \frac{1}{j}$, we immediately obtain $$PV = 1000(1 + (((0.97)(1.08)^{-4})^{-1} - 1)^{-1}) = 3484.07.$$