Present Value of a Delayed Annuity
A delayed, or deferred, annuity is a series of equal payments that does not start until some time in the future. You value it in two steps: find the annuity's value one period before the first payment, then discount that amount back to today.
What you will learn
- What a delayed (deferred) annuity is
- The two-step method: value the annuity, then discount it
- How to count the delay period without an off-by-one error
- How it compares with ordinary annuities and annuities due
The formula
- C
- equal payment per period
- r
- discount rate per period
- n
- number of payments
- d
- periods of delay; first payment at the end of period d + 1
Worked example
You will receive $1,000 a year for 5 years, with the first payment at the end of year 4. The discount rate is 6%. What is it worth today?
- Count the delay: the first payment is at year 4, so the annuity is valued at year 3, giving d = 3.
- Value the annuity at year 3: $1,000 × [1 − (1.06)⁻⁵] ÷ 0.06 ≈ $4,212.36.
- Discount 3 years back to today: (1.06)³ = 1.191016.
- PV₀ ≈ $4,212.36 ÷ 1.191016 ≈ $3,536.78.
Answer: About $3,536.78 today (rounded to the cent).
Common questions
How do you calculate the present value of a deferred annuity?
First value the payments as an ordinary annuity at one period before the first payment. Then discount that single amount back to today using PV = amount ÷ (1 + r)ᵈ, where d is the number of periods until that point.
What is the difference between a deferred annuity and a delayed annuity?
In finance problems they mean the same thing: a stream of equal payments that starts after a gap. Note that deferred annuity is also the name of a retirement insurance product, which is a different topic.
Why do you discount back only to one period before the first payment?
The ordinary annuity formula gives a value one period before the first payment. So if the first payment is at the end of year 4, the formula gives a year-3 value, and you discount 3 years, not 4.
Can you check a deferred annuity answer another way?
Yes. Discount each payment separately and add them up, or value an annuity for the full period and subtract the annuity for the delay years. For the example, both give about $3,536.78.