# Tricky mortgage rate question

Can anybody suggest me how I might solve the following equation?

$$(1+x)^{300} -125x(1+x)^{300} = 1$$

where x is the unknown that I want to solve for.

The actual question in full is

Q. Assume that your are going to retire in 25 years time. You want a mortgage of £100,000 now to extend and renovate your house but want to have it paid in full before you retire. The maximum repayment per month that your budge will allow id £800. What is the rate of interest you need from your bank to have the loan repaid in 300 monthly repayments (i.e 25 years).

So If the present value of each installment forms geometric sequence:

100,000 = 800/(1+i) + 800/(1+i)^2....

Where i is the monthly interest rate.

Then we can use the formula for Amortization - mortgages and loans:

A = Pi(1+i)^t/((1+i)^t -1)

Where :

A = repayment amount, P is the principle i is the term rate, and t is the term.

Then we get :

800 = 100,000i(1+i)^300/((1+i)^300-1)

Simplifying gives us:

$$(1+i)^{300} -125i(1+i)^{300} = 1$$

By the way the answer is 8.75% (yearly rate) which converts to a monthly rate of .70146% i.e a value of i of .0070146 which does indeed satisfy the equation above.

• What is the unknown? Dec 14 '20 at 16:14
• Is $i$ a variable here? If so you should consider choosing a different letter, as it looks like the $\sqrt{-1}$ Dec 14 '20 at 16:15
• Since you chose "complex-numbers" for a tag, I'll suppose $i$ is the complex number such that $i^2=-1$. The first thing to do is to find $(1+i)^{300}$. The easiest way to do this is to find the polar form and use DeMoivre's theorem. Then come back to rectangular coordinates and finish off with a multiplication and a subtraction. Since I don't think this is an equation, rather an equality, then your answer should be $1$. Dec 14 '20 at 16:20
• What you have tried? Dec 14 '20 at 16:20
• Sorry i is a variable (interest rate) not a complex number.
– Bren
Dec 14 '20 at 16:28

You seem to be making a mistake in setting up the equation to be solved. If the per-period discount rate is $$v$$, then we have $$100000=800\sum_{k=0}^{299}v^k=800\frac{v^{300}-1}{v-1}$$ or $$125(1-v)=1-v^{300},\tag1$$ assuming $$v\neq1.$$ If $$x$$ is the per-period interest rate, then $$v = \frac 1{1+x}$$,and we can rewrite $$(1)$$ as $$125x(1+x)^{299}=(1+x)^{300}-1$$ so it seems you made a slight error in setting up your equation, or it might be a difference in when the first payment is due. In my experience the first payment on a mortage is due when the loan is received. In any event, it will be easier to deal with in the form $$(1)$$. We rewrite $$(1)$$ as $$v=\frac{124+v^{300}}{125}\tag2$$ There's no way to solve $$(2)$$ explicitly, but we can solve it by iteration. Write $$f(v)=\frac{124+v^{300}}125$$ and pick some starting value for $$v$$, say $$v=0$$. Then $$f(v)=f(0)=.992$$. Take this as the new guess for $$v$$ and compute $$f(v)$$ again. $$f(.992)=0.9927187727986595$$. Continue in this manner until we find a $$v$$ such that $$f(v)=v$$ to sufficiently many decimals. We quickly arrive at $$v=0.9929610747842791$$ which gives $$x=0.007088822910052306$$ This is the monthly interest rate, so the nominal annual rate on the mortgage will be $$12$$ times that, or $$0.08506587492062767$$ I imagine that the expected answer is $$8.5\%$$.

• Thanks, I'll take a look later, have something to do now.
– Bren
Dec 14 '20 at 17:55
• Ok so the answer in the text is 8.75%, I expect that APR rate definitions vary from country to Country. Here (1_r)^12 = (1+i) r% per month, i% per year . I Can see where you are going with the sequence, but an APR of 8.5 % does not add up where as one of 8.75 does. Taking your monthly rate of 0.007088822910052306, the value of the Mortgage (P) would be: 800/(0.007088822910052306) * [(1.007088822910052306^300 -1)/1.007088822910052306^300 = 99,313.67 Where as using a monthly rate of .0070146 (equivalent APR of 8.75%) we get: 800/(0.0070146) * [(1.0070146^300 -1)/1.0070146^300 = 100,040.46
– Bren
Dec 14 '20 at 22:30
• I wasn't giving the APR, but the nominal annual rate compounded monthly. The APR I get is $8.846\%$, by the calculation you describe. Dec 14 '20 at 22:31
• Sorry for the compressed comment stack exchange would not allow more characters. In essence I think the formula I used (taken from the state tables book) A = Pi(1+i)^t/((1+i)^t -1) is correct, and the derived polynomial is also correct. And as stated 0070146 is a solution to the polynomial. I guess I just had to use numerical methods to solve. Just surprised at that as this is a high school math text book question, where numerical solutions to polynomials is not in the text. But thanks for all the responses.
– Bren
Dec 14 '20 at 22:35
• I agree the question is not specific in terms of nominal or APR. Never the less the solution given is 8.75%, but again not specific APR or nominal.
– Bren
Dec 14 '20 at 22:37