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.