## The Dependence of Property Value on Interest Rates

Suppose the interest rate is

\[r\]

and the rent is fixed at £10,000 paid once a year in advance. At the start of the second year £10,000 is paid. Because the property owner must wait a year and could invest the money at an interest rate of \[r\]

% the money is only worth £\[10,000/1.0r\]

now.At the start of the third year £10,000 is paid. Because the property owner must wait two years and could invest the money at an interest rate of

\[r\]

% the money is only worth £\[\10,000/(1.0r)^2\]

now.Continuing in this way, the total value of all future income at this interest rate is

\[I=10000+10000/1.0r+10000/(1.0r)^2 +....\]

This is a geometric series with first term

\[a=10000\]

and common ratio \[r= 1/1.0r\]

.The sum of a geometric series with first term

\[a\]

and common ratio \[r\]

is \[S= \frac{a}{1-r}\]

Hence

\[I=\frac{10000}{1-1/1.0r}\]

.If

\[r=7 \%\]

then \[I= 10000/(1-1/1.07) =152857\]

If

\[r=2 \%\]

then \[I= 10000/(1-1/1.02) =510000\]

If

\[r=0.05 \%\]

then \[I= 10000/(1-1/1.005) =2910000\]

This calculation ignores rent rises and is why property prices rocket when interest rates fall.