Corporate Accounting and Financial Management · Time Value of Money

Future Value of a Single Sum and Annuity Explained

Updated 11 October 2026 · Fact-checked

Future value is what money invested today grows to after n periods at interest rate i. For a lump sum, FV = P × (1 + i)^n. For equal payments at each period end, FV = A × [((1 + i)^n − 1) ÷ i]. For an annuity due, multiply that by (1 + i).

Understand Future Value of a Single Sum and Annuity

Money earns interest, so a rupee today can be worth more later. Future value (FV) tells you how much a sum or a series of sums will be worth at a future date, if it earns a given rate of return.

With compound interest, interest is earned on the original sum and also on past interest. That is why growth speeds up over time. The factor (1 + i)^n is called the future value interest factor (FVIF) or compound value factor. Tables give it for common rates and periods, so you do not need to compute powers by hand.

An annuity is a series of equal payments made at equal intervals. In an ordinary annuity, each payment is made at the end of the period. In an annuity due, each payment is made at the start of the period. Because every payment in an annuity due earns one extra period of interest, its future value is higher.

The future value interest factor for an annuity (FVIFA) adds up the compound factors of all payments. Tables give FVIFA too. In the exam, the table values are usually supplied in the question, so your job is to pick the right factor and apply it correctly.

Key rules to remember

Future value of a single sum: FV = P × (1 + i)^n = P × FVIF(i, n); P is the amount today, i is the rate per period, n is the number of periods.
Future value of an ordinary annuity: FVA = A × [((1 + i)^n − 1) ÷ i] = A × FVIFA(i, n); A is the equal payment at the end of each period. Use when payments fall at period end.
Future value of an annuity due: FVA(due) = A × [((1 + i)^n − 1) ÷ i] × (1 + i); Each payment is at the start of the period, so the ordinary annuity value is multiplied by (1 + i).
Compounding more than once a year: FV = P × (1 + r ÷ m)^(m × t); r is the annual rate, m is compounding periods per year, t is years. Use i = r ÷ m and n = m × t.
Effective annual rate: EAR = (1 + r ÷ m)^m − 1; Use it to compare nominal rates with different compounding frequencies.

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

Use this method for any future value question, whether it is a lump sum, an annuity or a mix of both.

1Read the question and mark what is given: amount, rate, time and how often interest is compounded.
2Decide the type of cash flow: a single sum, an ordinary annuity or an annuity due. Check the words 'at the end of each year' or 'at the beginning of each year'.
3Convert the rate and time to the same period. If compounding is half-yearly, halve the rate and double the number of periods.
4Write the correct formula and the factor you need, for example FVIFA(10%, 5).
5Take the factor from the table given in the question, or calculate it if no table is given.
6Multiply by the amount. For an annuity due, also multiply by (1 + i).
7If there are two or more cash flows, find each future value separately and add them.
8State the final answer with the unit in rupees and a one-line conclusion.

Quickest way: Factor-first shortcut

When to use it: Use when the question gives a compound value table or small values of n, such as 3 to 5 years.

Write the factor name first, such as FVIF(8%, 4) or FVIFA(10%, 5).
Fill the factor from the table. If none is given, build it by compounding year by year.
Multiply by the amount in one line.
For annuity due, take the ordinary annuity answer and multiply by (1 + i). Do not rebuild the whole calculation.
Check that the answer is bigger than the total of the payments. If it is not, you have missed the interest.

Common mistakes in Future Value of a Single Sum and Annuity

Using the ordinary annuity formula for an annuity due.
Students miss the words 'at the beginning of each year' in the question.
Fix: Underline the timing of payments. If it is the start of the period, multiply the ordinary annuity result by (1 + i).
Using the annual rate with half-yearly or quarterly compounding without adjusting.
The rate is given per annum, so students plug it in directly.
Fix: Divide the rate by the number of compounding periods per year and multiply n by the same number.
Using the FVIF factor for an annuity, or the FVIFA factor for a single sum.
The two tables look alike and have similar headings.
Fix: Ask: is there one payment or many? One payment uses FVIF. Equal repeated payments use FVIFA.
Treating interest as simple interest.
Students carry over the P × r × t habit from earlier classes.
Fix: Write (1 + i)^n every time the question says compound or asks for future value.
Reading the wrong row or column of the table.
Rate and period are swapped under time pressure.
Fix: Check the table heading. Normally periods run down the rows and rates across the columns. Confirm with a quick check: FVIF for 1 period must equal 1 + i.
Adding payments without compounding when there are uneven cash flows.
Students try to use the annuity formula for payments that are not equal.
Fix: Use the annuity formula only for equal payments. Otherwise compound each amount separately and add.

Worked examples

Example 1

Mr. Sharma deposits ₹2,00,000 in a bank that pays 10% per annum compounded annually. What will the deposit be worth after 3 years? Then find the value if interest is compounded half-yearly.

Show the solution

Annual compounding: FV = P × (1 + i)^n with P = ₹2,00,000, i = 10%, n = 3.
(1.10)^3 = 1.10 × 1.10 × 1.10 = 1.331.
FV = 2,00,000 × 1.331 = ₹2,66,200.
Half-yearly compounding: i = 10% ÷ 2 = 5% per half-year, n = 3 × 2 = 6 periods.
(1.05)^6 = 1.340096 (rounded to six decimals).
FV = 2,00,000 × 1.340096 = ₹2,68,019 (rounded to the nearest rupee).

Answer: Value after 3 years is ₹2,66,200 with annual compounding and about ₹2,68,019 with half-yearly compounding. More frequent compounding gives a higher value.

Example 2

Ms. Iyer plans to invest ₹50,000 every year for 4 years at 10% per annum compounded annually. Find the future value at the end of year 4 (a) if each payment is made at the end of each year, and (b) if each payment is made at the beginning of each year.

Show the solution

Find the annuity factor: FVIFA(10%, 4) = ((1.10)^4 − 1) ÷ 0.10.
(1.10)^4 = 1.4641, so FVIFA = 0.4641 ÷ 0.10 = 4.641.
(a) Ordinary annuity: FVA = 50,000 × 4.641 = ₹2,32,050.
(b) Annuity due: multiply by (1 + i) = 1.10.
FVA(due) = 2,32,050 × 1.10 = ₹2,55,255.
Check: total paid is ₹2,00,000, and both answers are higher, which is correct because of interest.

Answer: (a) ₹2,32,050 for payments at the end of each year. (b) ₹2,55,255 for payments at the beginning of each year.

Exam tips

Always write the formula and the factor name before calculating. Marks are given for method even if the arithmetic slips.
If compound value tables are given, use them as supplied and show the factor you picked. Do not recompute unless asked.
Check the timing words carefully: 'end of each year' means ordinary annuity, 'beginning of each year' means annuity due.
Show the compounding adjustment clearly when interest is half-yearly or quarterly, by writing the new i and n.
Finish with a one-line conclusion in rupees, as ICSI answers are expected to end with a clear result.

Practice questions from Time Value of Money

Future 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.

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

What is the formula for the future value of an annuity?

For equal payments at the end of each period, FVA = A × [((1 + i)^n − 1) ÷ i]. Here A is the payment, i is the rate per period and n is the number of periods. The bracket is the FVIFA factor.

How do I use the FVIF table?

Find the row for the number of periods and the column for the rate per period. The number at the meeting point is the factor. Multiply it by the sum invested to get the future value.

What is the difference between an ordinary annuity and an annuity due?

In an ordinary annuity, payments are made at the end of each period. In an annuity due, they are made at the start. An annuity due is worth more at the end because each payment earns one extra period of interest.

How do I handle half-yearly or quarterly compounding?

Divide the annual rate by the number of compounding periods in a year, and multiply the number of years by the same number. Then apply the usual formula with the new i and n.