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Given the following 4 characteristics determine the loan amount.

  1. First payment of $2000 due one year from now.

  2. Next 7 payments are 3% larger than the preceding payments.

  3. Next 7 payments are 3% lower than the preceding payments.

  4. Loan has annual effective rate i = 7%

So the main problem I have is with criteria number 3. I am not sure how to evaluate a geometric progression that is decreasing.

For instance, a geometric progression that is increasing as in the second criteria

$$ \sum_{t=1}^{n} a(1+g)^{t-1}(1+i)^{-t} $$

$$ = \sum_{t=1}^{n}a(1+i)^{-1} \left( \frac{1+g}{1+i} \right)^{t-1} $$

$$ = a(1+i)^{-1}\left( \frac{1- \left( \frac{1+g}{1+i} \right)^n}{1-\left( \frac{1+g}{1+i} \right)} \right) $$

$$ = a\left( \frac{1- \left( \frac{1+g}{1+i} \right)^n}{i-g} \right) $$

So the way I understand it verbally, is that the annuity has a discount rate of 1/1.07 compounded per payment period. This means that for each future payment period the payment value decreases however, since payments are increasing by 3% the discount now becomes 1.03/1.07

Now if payments are supposedly decreasing by 3% then would it simply be 1/1.10 for the discount?

swordlordswamplord
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1 Answers1

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If we assume payments are made on an annual basis at $i = 0.07$, and you want the present value of the cash flow, we have $$\begin{align} PV &= 2000\left( v + 1.03 v^2 + \cdots + (1.03)^6 v^7 + (1.03)^7 v^8 \right.\\ & \quad \left. + \;(1.03)^7(0.97) v^9 + (1.03)^8 (0.97)^2 v^{10} + \cdots + (1.03)^7 (0.97)^7 v^{15} \right).\end{align}$$ Note that by "3% lower," I assume that this means we subtract $3\%$ of the previous amount; i.e., multiply by $97\%$. Also, $v = (1+i)^{-1} = (1.07)^{-1}$ is the effective annual present value discount factor.

To write this actuarial notation, we can go about it in multiple ways. One way is

$$\require{enclose} PV = 2000 \left( \frac{1}{1.03} a_{\enclose{actuarial}{8} j_1} + (1.03)^7 v^8 a_{\enclose{actuarial}{7} j_2} \right),$$ where $$j_1 = \frac{0.07 - 0.03}{1 + 0.03} = \frac{4}{103} \approx 0.038835, \\ j_2 = \frac{0.07 + 0.03}{1 - 0.03} = \frac{10}{97} \approx 0.103093$$ are modified annual interest rates for each annuity-immediate. Payments that occur in geometric progression are essentially equivalent to level payment annuities with a modified interest rate; e.g., if the common ratio is $r > 0$ then the cash flow $$v + r v^2 + r^2 v^3 + \cdots + r^{n-1} v^n = \frac{1}{r}(rv + (rv)^2 + \cdots + (rv)^n).$$ So the present value discount factor is $v' = rv$ rather than simply $v$, and the effective interest per period is $$\frac{1}{1 + j} = \frac{r}{1 + i},$$ or $$j = \frac{1 + i}{r} - 1.$$ In your case, $r = 1.03$ for the first eight payments, and $r = 0.97$ for the second set of seven. This gives us $$PV \approx 2000 \left( \frac{6.76525}{1.03} + (1.03)^7 \frac{4.81924}{(1.07)^8} \right) \approx 20035.6.$$

heropup
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