# Monthly payment sinking fund

Wanda takes out a 28-year mortgage for $178000 at an interest rate of j_1 = 8.1% (i = 0.081). After the 10th monthly mortgage payment, she decides to make some changes. To repay the loan, she will make 94 more mortgage payments (additional to the 10 she has already made). At the same time as the 11th payment (that is, one month from the 10th payment), she plans to set up a sinking-fund that will pay off the remainder of the mortgage on the same day as she makes the last mortgage payment. If the sinking-fund earns interest at j_{12} = 6% (i = 0.06/12). What will be the amount of each monthly sinking-fund deposit? Note: She will make her first deposit into the sinking-fund on the same day that she makes her 11th mortgage payment. So I need to use the equation Rs = A / ((1+0.06/12)^(94)-1)/((0.06/12)) Rs = A / 119.624308 I'm having problems solving for A. So I converted the yearly interest rate into monthly which is i = 6.51165312E^-3. Found that the monthly payments are$1306.66 but not sure how to get A after 10 months or 11 months, the wording is a bit confusing. Anyone please help?

the value of the sinking fund after 94 payments, equals the balance of the loan after 104 payments have been made.

when you take out the loan

$178,000 = P\sum_\limits{i=1}^{336} (1+r)^{-i}\\ 178,000(1+r)^{336}((1+r)-1) = P [(1+r)^{336}-1]\\ P = 178,000 \frac {r(1+r)^{336}}{(1+r)^{336}-1}$

$P = 1341.43$

and 104 months in you owe.

$P\sum_\limits{i=1}^{232} (1+r)^{-i}\\ 178,000 \frac {r(1+r)^{336}}{(1+r)^{336}-1}\frac {(1+r)^{232}-1}{r(1+r)^{232}}\\ 178,000 \frac {(1+r)^{104}((1+r)^{232}-1)}{(1+r)^{336}-1}$

$157,000$

And the value of your sinking fund.
$A\sum_\limits{i=2}^{94} (1+y)^{i}\\ A\frac {(1+y)^{94} - (1+y)}{y}$

Where A are your monthly payments.

Equals the balance due on your mortgage.

$A = 178,000 \frac {(1+r)^{104}((1+r)^{232}-1)}{(1+r)^{336}-1}\frac {y}{(1+y)^{94} - (1+y)}$

$1,323.51$

• What did you come up as the final answer for the sinking fund deposit? – Alex Vincent Mar 4 '17 at 3:03
• I also thought the set up for the sinking fund was after the 10th payment, so the 10 payments extra is not inclusive, I may be incorrect though. I find this question very confusing – Alex Vincent Mar 4 '17 at 3:05
• I made a mistake on the set up of the sinking fund. You only make 94 payments. Nonetheless, it is a repeated exercise in the NPV of a series of constant cash flows. – Doug M Mar 4 '17 at 3:07
• Apparantly 1312.44 isn't the right answer for the quesion – Alex Vincent Mar 4 '17 at 3:11
• I have a timing mismatch. She doesn't pay into the sinking fund on the last payment date. – Doug M Mar 4 '17 at 3:27

Let be $L=178,000$, $n=28\times 12=336$, $i=\frac{i^{(12)}}{12}=\frac{8.1\%}{12}=0.68\%$. The monthly installment is $$P=\frac{L}{a_{\overline{n}|i}}=1,341.433$$ where $a_{\overline{n}|i}=\frac{1-v^n}{i}$ and $v=\frac{1}{1+i}$.

After the 10th payment the remaining loan is $$L'=P\,a_{\overline{n-10}|i}=176,557.47$$ Wanda decides to repay this loan with sinking fund method over $m=94$ periods by a sinking fund that charges rate of interest $i$ to the loan and credits rate of interest $j=\frac{j^{(12)}}{12}=\frac{6\%}{12}=0.5\%$ to the sinking fund by periodic payment of amount $$P'=L'\left(i+\frac{1}{s_{\overline{m}|j}}\right)= 2,667.70$$ where $s_{\overline{m}|j}=\frac{(1+j)^m-1}{j}$ $$L'i= 1,191.76$$ is the interest payment and $$\frac{L'}{s_{\overline{m}|j}}= 1,475.93$$ is the sinking fund deposit.