Finance · level 2
Annuity factors
An annuity is a perpetuity minus a perpetuity that starts later. The factor is bounded by 1/r, which is the fastest way to check your arithmetic.
Worked example: $1.3M a year for 5 years at 6%. Present value?
payment$1.3M
annuity factor4.21
PV$5.48M
Step by step
- 1
Annuity factor
(1 − 1.6^−5) / 0.06 = 4.21
- 2
Multiply
$1.3M × 4.21 = $5.48M
- 3
Sanity check
The perpetuity would be $21.67M; 5 years must be less than that.
$1.3M a year for 5 years at 6%. Present value? = 5,476,073
The theory behind it
Intuition
An annuity is a perpetuity minus a perpetuity that starts later, which is why the factor is (1 − (1+r)^−n)/r.
Common pitfalls
- ×Confusing ordinary annuity (end of period) with annuity-due (start of period).
- ×Forgetting to match the rate frequency to the payment frequency.
In the interview
Lease-versus-buy, debt service and subscription-value rounds.