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## Endowment Mortgages

Anyone taking out an endowment mortgage to buy a house borrows money from a bank promising to repay the loan after a 25 or so years, and raises the money to repay the loan by investing a certain amount of money each year, the amount being calculated so that after 25 years, the total investment will pay off the loan.
Suppose an amount £200,000 is borrowed. The bank will charge interest on this loan at
$r_1$
%, so that after 25 years the amount of money that needs to be repaid is
$200000 (1+r_1/100)^{25}$

If £
$A$
is to be invested at the start of each year, then in the first year £
$A$
will be invested at an interest rate of
$r_2$
.
After 25 years this will have grown to
$A(1+r_2/100)^{25}$

In the second year £
$A$
will be invested at an interest rate of
$r_2$
.
After 24 years this will have grown to £
$A(1+r_2/100)^{24}$

in the third year £
$A$
will be invested at an interest rate of
$r_2$
.
After 23 years this will have grown to £
$A(1+r_2/100)^{23}$

Carrying on like this, in the 25th year £
$A$
will be invested at an interest rate of
$r_2$
.
After 1 year this will have grown to £
$A(1+r_2/100)$

At the end of the 25th year the last instalment of £
$A$
will be paid. Adding these up - in reverse order - gives
$A+A(1+r_2/100)+A(1+r_2/100)^2+...+A(1+r_2)^{25}$

This is a geometric series with first term
$A$
, common ratio
$(1+r_2/100)$
and 26 terms.
The sum of the series is
$S_{26}=\frac{A(1-(1+r_2/100)^{26}}{1-(1+r_2/100)}= \frac{100A(1-r_2)^{26}}{r_2}$

This must be at least equal to the value of the loan after 25 years.
$\frac{100A(1-r_2)^{26}}{r_2} \geqslant 200000(1+r_1/100)^{25} \rightarrow A \geqslant \frac{2000r_2(1+r_1)^{25}}{(1-r_2/100)^{26}}$