Reducing mortgage length

by mark | 5 Aug 2023, 6:18 p.m.

Given everything that is going on at the minute, I've been spending more time talking about mortgages than I normally would. Part of this is just the terrors of middle age. But one thing stuck out:

You know, if you pay an extra payment right at the start of the mortgage, you knock an entire year off the term

Seemed mad to me. But lets work out. Maybe it isn't. 

The Theory of Interest

Now you should have been taught this in school. But if you weren't I will go over the very basics. It is not that hard. 

Interest rates are normally quoted as an Annual Percentage Rate (APR) for borrowing or an Annual Effective Rate (AER) for savings. There are very precisely defined rules about how these are calculated and displayed - e.g., if you see an APR of 3.2% it could mean you need to use 3.1999% in your calculations. I will inaccurately use the terms interchangeably. 

So. Interest. If someone says you have an interest rate of \(i\) then what does this actually mean? Well it means if you put some money away for a year it will grow by that much (or the amount you owe will grow by that much if you are a borrower). \(i = 5\%\) means you can put £100 away and end up with £105 in a year. That's it.

Interest is just the idea that a bird in the hand is worth two in the bush, with money instead of avians. Do you prefer £100 now or £105 in a year? If the interest rate is 5% then they are the same thing. If the interest rate is 6% then £100 now is better; if the interest rate is 4% then £105 in a year is better. Interest rates on mortgages and so on are actually set in this way. Major institutional lenders are constantly borrowing and lending vast sums, in complicated ways with derivative financial instruments, with the prices of deals between each other filtering down to the interest rate offered to mere mortals like us. 

So we know how to use i to go from one year to the next. But in the real world you would borrow £1,000 today and pay it back over several years. To understand this is where compound interest comes into play. It's not too hard I promise. £100 today, at a rate of 5%, is the same as £105 in a year; what about two years? We just have 

\(£100 \text{ today}\equiv£100\times(1 + 5\%)^2 = £110.25\)

And for three years it is

\(£100 \text{ today} \equiv £100 \times(1+5\%)^3 = £115.76\)

and so on. Thus, you can make £300 by having £105 returned in one year, £110.25 returned in two and £115.76 in three. But people prefer level payments. So we come to the real question: if I want to pay back £300 in three annual instalments at 5%, how much is each instalment? You would expect something like £110. But exactly? 

Annuity Factors

This is where some algebra comes in. One thing we always want is the value today of £1 in one (or two, or three, or...) years time. To save ink we introduce a special letter for this:

\(v^n = \frac 1 { (1+i)^n}\)

This also works for non annual periods. So the value of £1,234 at 5% in two and a half years is 

\(\text{value} = £1234 v^{2.5} = £1234 / \left(1.05^{2.5}\right) = £1092.30\)

That is, you could put £1,092.30 in a savings account that pays 5% for two and a half years and you would have £1,234 at that point. 

To figure out what our annual payment is, we want to solve the following equation:

\(£300 = P \left(v^1 + v^2 + v^3\right)\)

This is easy enough. Evaluate the thing in the brackets at 5% and divide £300 by it:

\(£300 = P( 0.95238 + 0.90703 + 0.86384) \\ P = £300 / 2.7232 \\ P = £110.16\)

There we go. Three payments of £110.16 a year apart is worth £300 today. 

Doing that tedious thing with the vs is impractical if you have very many (say six hundred monthly repayments on a fifty year mortgage). Fortunately you can use a thing called a geometric series to do the hard work for you. First thing - notice that we can write \(\frac v {1-v}\) as \(\frac1i\). We can do a summing up as:

\(\require{enclose} \text{value} =a_\enclose{actuarial} n = v + v^2 + v^3 + \cdots + v^n \)

So,

\(\require{enclose} a_\enclose{actuarial} n - va_\enclose{actuarial} n = v\left (1 - v^n\right) \\ a_\enclose{actuarial} n(1-v) = v\left (1 - v^n\right) \\ a_\enclose{actuarial} n = \frac{1-v^n}{i}\)

Great. So if we are paying £10,000 at the end of every year, for 25 years, at 5%, this is worth 

\(\require{enclose} 10000 a_\enclose{actuarial}{25} = 10000\frac{1 -v^{25}}{i} = 140,039\)

This is quite roughly how a mortgage works. You borrow £140k and you pay back £833.33 a month (it's complicated - see next section) for 25 years. The expression \(\require{enclose} a_\enclose{actuarial}n\) is called the annuity for n years. The funny angle bracket thing means it is for n years. Without the bracket it means something very different. 

Payments in advance and monthly

There are other things we can do. Often payments are made at the start, not end, of a period. There is a sum for that. Observe: 

\(\require{enclose} \ddot a_\enclose{actuarial} n = 1 + a_\enclose{actuarial}{n-1} =1 + \frac{1-v^{n-1}}{i} \\ = \frac{(i+1)-v^{n-1}}{i} \\ = \frac{1-v^n}{iv}\\ =\frac{1-v^n}{d}\)

where we have introduced d=iv the discount rate. If you get £1 in a year's time at a rate i, you put (1-d) into an account. 

Two more headaches. You usually see interest quoted annually but you pay monthly (or your investment pays interest every three months, or semi-annually). This needs you to use convertible interest rates. If you earn 1% every month and say this is 12% a year, you are really using an annual rate convertible monthly. This is notated as

\(\left(1 + \frac{i^{(12)}}{12}\right)^{12} = (1+i); \\ i^{(12)} = 12\left((1+i)^{1/12} - 1\right)\)

This seems arcane, but if you are paying your mortgage monthly and not annually then you can use 

\(\require{enclose} a_\enclose{actuarial}n^{(12)} = \frac{1-v^n}{i^{(12)}}\)

in your calculations and it will work. Similarly, for monthly payments at the start of the month, you can use a discount rate convertible monthly: 

\(\left(1-\frac{d^{(12)}}{12}\right)^{12} = (1-d); \\ 1-\frac{d^{(12)}}{12} = (1-d)^{1/12}; \\ \frac{d^{(12)}}{12} = 1 - (1-d)^{1/12}; \\ d^{(12)} = 12\left(1 - (1-d)^{1/12}\right)\)

and then use 

\(\require{enclose} \ddot a_\enclose{actuarial}n^{(12)} = \frac{1-v^n}{d^{(12)}}\)

in your sums and have it work. 

Final thing: if you are making payments so frequently they might as well be continuous, you can use the force of interest

\(\require{enclose} \delta = \ln(1+i);\\ \bar a_\enclose{actuarial}{n} = \frac{1-v^n}{\delta}\)

Shortening mortgage lengths

Phew. Now we can actually do some stuff. 

I want to know the following. I take out a mortgage, payable monthly in advance, for 25 years. My very first payment will be double because I found some cash down the back of the sofa. Just the first one. At what rate of interest will this reduce the term to 24 years? 

First thing is "what is my monthly payment". This is quite easy. We have borrowed P as the principal, so my monthly repayment is 

\(\require{enclose} P = 12R\ddot a^{(12)}_\enclose{actuarial}{25}; \\ R = P\, /\, 12\ddot a^{(12)}_\enclose{actuarial}{25}\)

We then pay this at the start, and knock a year off the term:

\(\require{enclose} P - R = 12R\ddot a^{(12)}_\enclose{actuarial}{24}\)

Unfortunately you just have to do this by trial and error. But I can tell you, for a 25 year mortgage, an interest rate of 10.7% means you can knock a year off the term by making a double payment at the start.

Principal: £250,000

Interest rate: 10.7%

Annuity factor \(\require{enclose} \ddot a^{(12)}_\enclose{actuarial}{25} = 9.10348\)

Monthly payment: £2,288.50

So pay this immediately at the start, and it comes off the principal, so our mortgage is really £247,711.50.

Annuity factor with one year off: \(\require{enclose} \ddot a^{(12)}_\enclose{actuarial}{24} = 9.02016\)

Value of 24 years of repayments is \(\require{enclose} 12R\ddot a^{(12)}_\enclose{actuarial}{24} = 9.02016 \times 12 \ £2,288.50 = £247,711,63 \)

So actually we over pay by 13p. 

This is only a rough calculation - mortgage firms do things like allow for different numbers of days in each month, leap years etc and I've just gone a month is exactly 1/12 of a year with no bank holidays to worry about. But yes - if your 25 year mortgage APR is 10.7% you can knock a year off it with a double payment right at the start.

If you have a 30 year mortgage you need 8.81% for this to work, and for a 35 year term you need 7.5%, which I think some people might actually be on.