10  Deferred Annuity

A deferred annuity is an annuity that does not commence until some amount of time has elapsed.

10.1 Deferred Annuity-Immediate

Consider a deferred annuity-immediate that makes payments of \(1\) at the end of each unit of time, for \(n\) units of time, commencing after \(k\) units of time have elapsed. At time \(k\), this annuity will be a regular annuity-immediate, and we can discount from time \(k\) to time \(0\) to calculate the value at time \(0\):

\[PV_k = a_{\overline{n|}} \implies PV_0 = v^k\times a_{\overline{n|}} \equiv \ _{k}|a_{\overline{n|}} \]

We denote \(_{k}|a_{\overline{n|}}\) to be the present value of an annuity-immediate that is deferred for k units of time

10.1.1 Calculator Guide

Example: The annual effective interest rate is \(6\%\). A deferred annuity-immediate makes payments of \(\$1,000\) per year for 15 years, after a deferral period of 5 years. Calculate the present value of the deferred annuity-immediate.

Solution: We find this value using the BA II Calculator. First we obtain the present value of a regular 15-year annuity immediate that pays \(\$1,000\) per year:

15 [N] 6 [I/Y] 1000 [PMT] [CPT] [PV] -> PV = -9,712.25

Now enter this value as a future value, put zero in for the present value and payments, and then discount the -9,712.25 back 5 years:

[FV] 0 [PV] 0 [PMT] 5 [N] [CPT] [PV] -> PV = 7,257.56

10.1.2 Deferred Perpetuity-Immediate

The present value of a deferred perpetuity-immediate can be found by taking the limit as \(n\) goes to infinity:

\[_k|a_{\overline{\infty|}} = \lim_{n\to\infty}\ _k|a_{\overline{n|}} = \frac{v^k}{i}\] ## Deferred Annuity-Due

Consider a deferred annuity-immediate that makes payments of \(1\) at the end of each unit of time, for \(n\) units of time, commencing after \(k\) units of time have elapsed. The first payment occurs at time \(k\), and at this time this will be a regular annuity-due, and we can discount from time \(k\) to time \(0\) to calculate the value at time \(0\):

\[PV_k = \ddot a_{\overline{n|}} \implies PV_0 = v^k \times \ddot a_{\overline{n|}} \equiv \ _k|\ddot a_{\overline{n|}}\] ### Deferred Perpetuity-Due

The present value of a deferred perpetuity-due can be found by taking the limit as \(n\) goes to infinity: \[_k|\ddot a_{\overline{\infty|}} = \lim_{n\to\infty}\ _k|\ddot a_{\overline{n|}} = \frac{v^k}{d}\]

Example: The nominal annual interest rate compounded monthly is \(9\%\). A deferred perpetuity-due make payments of \(\$100\) per month forever, after a deferral period of 3 years. Calculate the present value of the deferred perpetuity-due.

Solution: The monthly effective rate of interest is \(i^{(12)}/12 = 0.09/12 = 0.0075\). Measured in months, the deferral period of 3 years is 36 months: \(k=36\). The present value is: \[PV_0 = 100 \times\ _k|\ddot a_{\overline{\infty|}0.75\%} = 100\times\frac{1.0075^{-36}}{(0.0075 / 1.0075)} = 10,265.07\]