Monthly payment sinking fund

Question

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?

Answer 1

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.

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

$1,323.51$

Answer 2

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.