# Financial mathematics increasing annuities

I am having trouble solving this problem

Julie bought a house with a 100,000 mortgage for 30 years being repaid with payments at the end of each month at an interest rate of 8% compounded monthly. If Julie pays an extra 100 each month, what is the outstanding balance at the end of 10 years immediately after the 120th payment?

My attempt:

I first want to find the deposit per month.

I let $D$ be the deposit per month and since it increases by 100 each payment, I used an increasing annuity,

$D*100(Ia_{30|0.08}) = 100,000$

However, the $D$ I got was 8.12, which is clearly not right.

Can someone help?

• Many financial advice columns have described the strategy of paying $X$ amount extra with each month's mortgage payment. What they always mean by "and extra $100$" is that instead of paying $734$ each month (for example), you pay $834$. They do not mean that you pay $734$, then $834$, then $934$, etc. Mar 28, 2016 at 12:46
• I don't know how you are calculating $Ia_{30|0.08}$, but it seems that usually it represents the future value of $30$ payments of $1,2,3,\ldots,30$ at $8\%$ interest per payment period. So $8.12\times100(Ia_{30|0.08})$ would represent a payment of $812$ for the first year, $1624$ for the second year, etc., a total of $377580$, which seems in line for the cost of interest on this mortgage. But of course that payment schedule is completely unlike any interpretation of the question you're trying to answer. Mar 28, 2016 at 13:06
• It isn't an increasing annuity. The additional payment is 100 each month. Mar 28, 2016 at 14:46

At a nominal interest rate of $i^{(12)}=8\%$ compounded monthly, the effective interest rate per month is $i_m=\frac{i^{(12)}}{12}=0.67\%$. Let $L=100,000$. Without extra payment we have for $n=12\times 30=360$ months $$L=P\,a_{\overline{n}|i_m}\quad\Longrightarrow P=\frac{L}{a_{\overline{n}|i_m}}=\frac{100,000}{a_{\overline{360}|0.67\%}}=\frac{100,000}{136.28}=733.76$$ With an extra monthly payment $Q=100$ we have a periodic payment $P+Q=833.76$. Then we can find the outstanding loan balance $B_t$ $$B_{t}=L(1+i_m)^{t}-(P+Q)\,s_{\overline{t}|i_m}$$ and at $t=120$ we have $$B_{120}=\overbrace{100,000\times\underbrace{(1+0.67\%)^{120}}_{2.22}}^{221,964.02}\;-\,\overbrace{833.76\times\underbrace{\,s_{\overline{120}|0.67\%}}_{182.95}}^{152,533.92}=69,430.10$$ Without the extra payment the OLB would be $$B_{t}=L(1+i_m)^{t}-P\,s_{\overline{t}|i_m}$$ and at $t=120$ we would have $$B_{120}=\overbrace{100,000\times\underbrace{(1+0.67\%)^{120}}_{2.22}}^{221,964.02}\;-\,\overbrace{733.76\times\underbrace{\,s_{\overline{120}|0.67\%}}_{182.95}}^{134,239.32}=87,724.70$$ Using the prospective method we have $$B_t=P\,a_{\overline{n-t}|i_m}\quad\Longrightarrow\; B_{120}=733.76\,a_{\overline{240}|0.67\%}=733.76\times 119.55=87,724.70$$ that is obviously equal to the value founded with the retrospective method.

With an extra payment we will shorten the length $n$ of the loan repayment to a new $N$ $$L=(P+Q)\,a_{\overline{N}|i_m}=(P+Q)\frac{1-(1+i_m)^{-N}}{i_m}$$ and then solving for $N$ $$N=-\frac{\log\left(1-i_m\frac{L}{P+Q}\right)}{\log(1+i_m)}=241.9084704\approxeq 242$$ that is the lenght of the loan $n=360$ has been shortened to $N=240$ if we make an extra payment.

Using the retrospective method we have $$B_t=(P+Q)\,a_{\overline{N-t}|i_m}\quad\Longrightarrow B_{120}=833.76\,a_{\overline{122}|0.67\%}=833.76\times 83.27=69,430.10$$

• I was wondering, how do you find the outstanding balance using the prospective method? Mar 31, 2016 at 12:09
• I tried doing $833.76*(a_{240|0.067}+a_{120|0.067})$ but I am unable to get the answer Mar 31, 2016 at 12:10
• I've added some details Mar 31, 2016 at 13:09

HINT:

I have no formulas, but good old spreadsheet works wonders: I calculated the monthly principal to be paid: $277.7777 Then the sum of the all the interest compounded monthly over the thirty years:$120,333.3333

TOTAL principal+interest=$220,333.3333 So amount to pay per month is$612.037 during 360 months.

If you decide to instead pay $712.037 during 120 months, you will have outstanding balance at month 121 of$120,603.7037

(The decimals are there to show data is not rounded to cents).

Depending on the bank or country practices... I forgot to estimate the outstanding balance if those $100 go directly to principal: you will finish to pay in 317 months, thus reducing both the time and interest. Outstanding Principal just after month 120 (no interest)=$62,145.11041