Deferred annuities

Where the annuity formula puts its answer

Lesson 7.1 showed the ordinary-annuity PV landing at $t=0$ — one period before the first payment. That is a property of the formula, not of the calendar: if the first payment is at $t=4$, =PV puts its answer at $t=3$. A deferred annuity — one whose first payment is more than a period away — is therefore valued in two steps: value the annuity at its own start, then discount that lump sum back to today as in Lesson 6.2.

$$P{V}_0=\underset{\text{lands at }t=(\text{first payment})-1}{\underbrace{PMT\times PVAF(r,n)}}\times \frac{1}{(1+r{)}^{\text{first payment}-1}}$$

This is the shape the final's "questions you may never have seen" tier is built from. Draw the timeline, mark the first payment, step back one, and the exponent is that step's position.

Worked example — a pension that starts in year 4

Maritime Ferries promises a retiring captain $2,000 a year for 5 years, the first payment at the end of year 4 (so payments at $t = 4, 5, 6, 7, 8$). At 6 %, what must the company set aside today?

Step 1. Value the five payments as an ordinary annuity: =-PV(0.06,5,2000) = 8,424.73. This value sits at $t=3$, one period before the first payment.

Step 2. Discount the 8,424.73 from $t=3$ to $t=0$: $8,424.73/1.06^3=8,424.73/1.191016=7,073.56$.

In one cell: =-PV(0.06,5,2000)/(1+0.06)^3. The deferral exponent is 3, not 4: the annuity formula has already discounted the first payment one period.

Worked example — eight payments from year 5

A settlement pays $1,500 a year for 8 years, first payment at the end of year 5, at 4 %. Annuity value at $t = 4$:‘=-PV(0.04,8,1500)‘=10,\mathrm{099.12.}Discountfouryears:$10{,}099.12 / 1.04^4 = \mathbf{8{,}632.77}$. =-PV(0.04,8,1500)/(1.04)^4.

Move the start and watch the answer move

Use the wrong exponent on the pension — 4 instead of 3 — and the answer drops to 6,673.17: one year of discounting too many. Forget the deferral entirely and report 8,424.73: the value as if the first payment were a year away, which overstates the PV of a stream that has not started. Push the settlement's first payment one year later, to year 6: =-PV(0.04,8,1500)/(1.04)^5 = 8,300.74 — a year's delay costs 332.03 of present value, the 4 % on 8,300.74.

The rule, cold

First payment at year $k$ → annuity PV lands at $k-1$ → discount $k-1$ periods. With monthly periods the same rule applies in months. If the first payment is at $t=1$, $k-1=0$ and there is no deferral: the ordinary annuity is the special case.