Corporate Accounting and Financial Management · Time Value of Money

Present Value of a Single Sum and Annuity

Updated 11 October 2026 · Fact-checked

Present value is today's worth of money you will receive later, found by discounting at a given rate. For a single sum, PV = FV ÷ (1 + r)^n. For an annuity, multiply the equal payment by the PVIFA. For uneven flows, discount each year separately and add.

Understand Present Value of a Single Sum and Annuity

Money today is worth more than the same money later. You can invest it and earn interest. So a future amount must be reduced to compare it fairly with a rupee in hand now. This reduction is called discounting, and the result is the present value (PV).

Discounting is the reverse of compounding. If ₹1,000 grows to ₹1,210 in two years at 10%, then the present value of ₹1,210 receivable in two years is ₹1,000. The rate used is the discount rate, which is the return you could earn elsewhere.

An annuity is a series of equal cash flows at equal intervals, such as ₹10,000 every year for 5 years. You could discount each payment one by one. The PVIFA (present value interest factor of an annuity) does this in one step. It is simply the sum of the single-sum factors for each year.

A normal (ordinary) annuity pays at the end of each period. An annuity due pays at the start of each period, so every payment is one year closer and worth more. Uneven cash flows are different amounts in different years. You discount each one with its own PVIF and add the results.

In the exam you will usually be given a table of PVIF and PVIFA values. Your job is to pick the right factor for the correct rate and number of years, and to use it in the right place.

Key rules to remember

Present value of a single sum: PV = FV ÷ (1 + r)^n = FV × PVIF(r, n); PVIF(r, n) = 1 ÷ (1 + r)^n. r is the rate per period and n is the number of periods.
Present value of an ordinary annuity: PV = A × [1 − (1 + r)^−n] ÷ r = A × PVIFA(r, n); A is the equal payment at the end of each period.
PVIFA from PVIF: PVIFA(r, n) = Σ PVIF(r, t) for t = 1 to n; Adding the PVIF values of years 1 to n gives the PVIFA. Useful when only one table is given.
Present value of an annuity due: PV = A × PVIFA(r, n) × (1 + r); Equivalent method: A + A × PVIFA(r, n − 1). Payments are at the start of each period.
Present value of uneven cash flows: PV = Σ CFt × PVIF(r, t); Discount each year's cash flow with its own factor, then add.
Rate per period when interest is compounded more than once a year: r = annual rate ÷ m and n = years × m; m is the number of compounding periods in a year.

How to solve Present Value of a Single Sum and Annuity questions

Use this method for any present value question. It keeps you from picking the wrong factor or the wrong timing.

1Draw a quick timeline. Mark each cash flow and when it occurs (start or end of the year).
2Identify the type: single sum, ordinary annuity, annuity due or uneven flows. A mix is possible, so split it into parts.
3Find the rate per period and the number of periods. Adjust both if compounding is half-yearly or quarterly.
4Pick the factor: PVIF for a single sum or uneven flows, PVIFA for equal flows. Read it from the table at the right rate and year.
5Multiply the cash flow by the factor. For an annuity due, multiply by (1 + r) or use the n − 1 method.
6For uneven flows, compute each year's PV separately and add them.
7If the question asks for a decision, compare PV with the price or cost today and state your conclusion.

Quickest way: Table-lookup shortcut with a timeline check

When to use it: Use it when the question gives PVIF or PVIFA tables and time is short.

Underline the rate, the years and the word 'beginning' or 'end'.
If payments are equal, use a single PVIFA multiplication. Do not discount year by year.
If the payment is at the beginning, multiply the final PV by (1 + r).
If a deferred annuity starts after some years, take PVIFA for the last year of payments minus PVIFA for the years before the start.
For uneven flows, write a three-column table of year, cash flow and PVIF, then product. Total the last column.
Do a sense check: PV must be less than the total of undiscounted cash flows.

Common mistakes in Present Value of a Single Sum and Annuity

Using PVIFA for flows that are not equal.
Students see a series of cash flows and reach for the annuity factor out of habit.
Fix: Check that every amount is identical. If even one year differs, use PVIF for each year, or split the flow into an annuity plus a separate amount.
Treating an annuity due as an ordinary annuity.
The words 'beginning of each year' are missed in a hurry.
Fix: Underline the timing. For payments at the start, multiply the ordinary annuity PV by (1 + r).
Using the annual rate and years when interest is half-yearly or quarterly.
Students forget that both the rate and the number of periods change.
Fix: Divide the annual rate by m and multiply the years by m before looking up the table.
Reading the wrong row or column in the table.
Rates and years are close together and the table is dense.
Fix: Put a finger on the rate column and the year row, and write the factor beside the working before multiplying.
Discounting a deferred annuity by the wrong number of years.
Students apply PVIFA and forget that the annuity itself starts later.
Fix: Use PVIFA(n) minus PVIFA(years of delay), or take PVIFA for the annuity length and then multiply by the PVIF of the delay year.
Adding undiscounted amounts to discounted ones.
A cash flow at time zero needs no discounting, and students mix it up with the others.
Fix: Keep time-zero amounts as they are, with a factor of 1. Discount only the flows that occur later.

Worked examples

Example 1

A company will receive ₹50,000 at the end of each year for 4 years. The discount rate is 10%. Given PVIFA(10%, 4) = 3.170, find the present value if (a) the receipts are at the end of each year and (b) they are at the beginning of each year.

Show the solution

(a) The payments are equal and at year-end, so it is an ordinary annuity.
PV = 50,000 × 3.170 = ₹1,58,500.
(b) For an annuity due, multiply by (1 + r).
PV = 1,58,500 × 1.10 = ₹1,74,350.
Check: each payment comes a year earlier, so the PV must be higher. It is.

Answer: (a) ₹1,58,500; (b) ₹1,74,350.

Example 2

A project gives cash inflows of ₹40,000 at the end of year 1, ₹60,000 at the end of year 2 and ₹80,000 at the end of year 3. The discount rate is 10%. Given PVIF(10%, 1) = 0.909, PVIF(10%, 2) = 0.826 and PVIF(10%, 3) = 0.751, find the total present value. If the project costs ₹1,50,000 today, should it be accepted on this basis?

Show the solution

The flows are unequal, so discount each year with its own PVIF.
Year 1: 40,000 × 0.909 = ₹36,360.
Year 2: 60,000 × 0.826 = ₹49,560.
Year 3: 80,000 × 0.751 = ₹60,080.
Total PV = 36,360 + 49,560 + 60,080 = ₹1,46,000.
Compare with the cost: 1,46,000 − 1,50,000 = −₹4,000.
The present value of inflows is less than the cost today.

Answer: Present value of inflows is ₹1,46,000. It is ₹4,000 below the cost of ₹1,50,000, so the project should not be accepted on this basis.

Exam tips

Write the formula, the factor and the product on separate lines. Marks are given for method even if a table reading slips.
Always state the timing assumption in your answer, such as 'cash flows at year-end', when the question is silent.
Show the working for uneven flows in a table. It is faster to write and easier for the examiner to follow.
End with a one-line conclusion in words, such as accept or reject, or the amount payable today. Many answers lose marks by stopping at a number.
Learn the link between PVIF and PVIFA so you can build one from the other if only one table is printed.

Practice questions from Time Value of Money

Present Value of a Single Sum and Annuity in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Present Value of a Single Sum and Annuity: frequently asked questions

What is the difference between PVIF and PVIFA?

PVIF is the factor for one lump sum received at a single future date. PVIFA is the factor for equal payments over several periods and equals the sum of the PVIFs for those periods. Use PVIF for single or uneven flows and PVIFA for equal flows.

How do I find the present value of an annuity due?

Find the present value as if it were an ordinary annuity, then multiply by (1 + r). You can also take the first payment as it is and add the PVIFA for n − 1 periods applied to the payment.

Can I use PVIFA if the cash flows change every year?

No. PVIFA works only when every payment is the same and the gaps are equal. For changing amounts, discount each year with its own PVIF and add.

What discount rate should I use in the exam?

Use the rate given in the question. It may be called the required return, cost of capital, opportunity cost or interest rate. If compounding is not annual, convert the rate and the number of periods before using the table.