A perpetuity immediate is an annuity immediate that continues indefinitely and its present value is denoted by $a_{∞|}$.
The present value of a perpetuity immediate that pays one unit at the end of every year is given by: \[a_{\infty|}=\frac{1}{i}\]
What is the present value of an annuity immediate that pays $£200$ indefinitely at rate of $3\%$?
\[a_{\infty\vert}=200\left(\frac{1}{0.03}\right)\approx{£6,666.67}.\]
A perpetuity due is an annuity due that continues indefinitely and its present value is denoted by $\ddot a_{\infty|}$.
The present value of a perpetuity due that pays one unit at the beginning of every year is given by: \[\ddot a_{\infty|}=\frac{1}{d}\]
What is the present value of an annuity due that pays $£200$ indefinitely at a rate of $3\%$?
\begin{align} &d=iv=\frac{0.03}{1.03}=0.02912621359\\ &\ddot a_{\infty\vert}=200\left(\frac{1}{0.02912621359}\right)\approx£6,866.67 \end{align}