Present value of an annuity (Barwert einer Rente)

Level 3 · Advanced

German termBarwert einer Rente

Don't confuse withCapitalized value of a flow

Present value of an annuity is the value today of a finite sequence of equal payments made at regular intervals. Each payment is discounted for its own waiting time, then the discounted amounts are added. The result depends on the payment amount, number and timing of payments, and a discount rate expressed for the same interval.

Why it matters

Pensions, leases, loan payments, and structured settlements are often quoted as monthly or annual flows. Their face-value sum ignores when each euro arrives. An annuity present value puts the stream and a lump sum on the same date, without claiming that the chosen discount rate is uniquely correct.

Worked example

Assume five year-end payments of €1,000 and an illustrative annual discount rate of 4%. For an ordinary annuity, the factor is:

The present value is therefore €1,000 × 4.451822 = €4,451.82. The undiscounted €5,000 total is larger because later payments are worth less at the comparison date.

Payment timing changes the answer. If each €1,000 arrives at the start rather than the end of each year, every payment is discounted for one fewer year. This annuity-due value is €4,451.82 × 1.04 = €4,629.89.

A comparison checklist

Input	Question to make explicit
Payment	Is it fixed, indexed, or uncertain?
Count	How many payments are contractually expected?
Timing	Beginning or end of each period?
Discount rate	Nominal or real, and does it reflect payment risk?
Taxes and fees	Are the compared cash flows before or after them?

Don't confuse with capitalized value of a flow

An annuity has a finite payment count. A capitalized-value shortcut may model a continuing or perpetual flow by dividing it by a rate. Treating a five-year stream as perpetual can overstate its value dramatically.

Check yourself

Five €1,000 year-end payments have an annuity factor of 4.451822. What is their present value in euros?

Why is an annuity due worth more than an otherwise identical ordinary annuity?

Each payment arrives one period earlierIt always has more paymentsIts discount rate is automatically zero

Which inputs are required for a finite annuity present value? Select all that apply.

Payment amountNumber and timing of paymentsDiscount rate for the same intervalA claim that the stream continues forever

What distinguishes present value of an annuity from capitalized value of a continuing flow?

An annuity has a finite payment countAn annuity never uses discountingCapitalized value is always smaller

Why it matters​

Worked example​

A comparison checklist​

Don't confuse with capitalized value of a flow​

Check yourself​

Sources​

Why it matters

Worked example

A comparison checklist

Don't confuse with capitalized value of a flow

Check yourself

Sources