Future value
Also written FV · Future value (FV) · Compounded value
What a sum of money invested today will be worth at a future date once returns have been earned and reinvested — the compounding half of the time value of money.
In plain language
Rs 100 today is not the same thing as Rs 100 a year from now. Today's Rs 100 can be put in a deposit and will have grown by then; the Rs 100 arriving later cannot.
Future value puts a number on that difference going forwards. It takes an amount you hold now, a rate, and a length of time, and tells you what the amount becomes. Present value is the same question asked backwards — what a future amount is worth today.
Everything in goal planning runs on this one equation. A client who says "I want Rs 50 lakh for my daughter's education in twelve years" has stated a future value; the adviser's job is to turn it into a sum to invest today, or a rate that must be earned, or a longer horizon.
How it works
Two adjustments do most of the damage in exam questions, and both come from compounding frequency.
The rate r is the return for each compounding period, not for the year. An investment paying 8% a year compounded quarterly earns 8%/4 = 2% per period. And n counts compounding periods, not years — five years of quarterly compounding is 20 periods, not 5.
The greater the frequency of compounding, the more often interest is paid on interest, and the greater the return. The workbook's own illustrations are the Post Office Monthly Income Scheme, which pays every month, against the Senior Citizens Scheme, which pays every quarter.
The formula
FV = PV × (1 + r)^n
PV = present value — the amount invested today
r = rate of return for each compounding period
n = number of compounding periods
Rearranged, the same equation gives the other two answers a planner needs:
PV = FV ÷ (1 + r)^n
CAGR = ((End value ÷ Beginning value) ^ (1/n)) − 1
A worked example
Krishna invests Rs 5,00,000 in a five-year bank deposit paying 8%. What he earns depends entirely on what happens to the interest.
| Scenario | Computation | Maturity value | Interest earned |
|---|---|---|---|
| 1. Interest withdrawn each year (simple interest) | 5,00,000 × 8% × 5 | Rs 5,00,000 returned, interest taken out | Rs 2,00,000 |
| 2. Cumulative, compounded annually | 5,00,000 × (1.08)^5 | Rs 7,34,664 | Rs 2,34,664 |
| 3. Cumulative, compounded quarterly | 5,00,000 × (1.02)^20 | Rs 7,42,974 | Rs 2,42,974 |
Scenario 2 beats Scenario 1 by Rs 34,664. That gap is the entire compounding benefit, and it comes from nothing except leaving the interest where it was. Scenario 3 beats Scenario 2 by a further Rs 8,310 for no extra rupee invested and not one extra day of waiting — the only change is that interest starts earning interest four times a year instead of once.
Now the planning use. Suppose Krishna actually needs Rs 12,00,000 in five years. The deposit gets him to Rs 7,42,974, a shortfall of Rs 4,57,026. Solve the same equation for the rate instead:
required rate = (12,00,000 ÷ 5,00,000) ^ (1/5) − 1
= (2.4) ^ 0.2 − 1
= 19.1% a year
No bank deposit in India pays 19%. The arithmetic has just told the adviser that the goal amount must come down, the horizon must go out, or Krishna must add to the investment — and it has said so with a number rather than an opinion.
Why NISM asks about it
Chapter 2 (Time Value of Money), section 2.2.2, with present value at 2.2.1 and rate of return / CAGR at 2.2.3. This is the most heavily computed chapter in the paper. Expect: given PV, r and n, find FV; given quarterly or monthly compounding, adjust both r and n correctly; and back out CAGR from a beginning and an end value, including over a fractional period.
Common exam traps
- Adjust the rate and the period count together. 8% a year compounded quarterly is 2% for 20 periods over five years — never 8% for 20 periods, and never 2% for 5.
- Simple interest is not slow compounding. It is the case where interest is taken out and spent. Scenario 1 above earns Rs 2,00,000 however you restate it, because nothing was ever reinvested.
- More frequent compounding always raises the future value, never lowers it. If your answer moves the other way, you divided the rate but forgot to multiply the periods.
- Future value and present value are one equation, rearranged. A question that hands you a future amount and asks what to set aside today is a present value question wearing a disguise.
- CAGR handles fractional periods in the exponent. The workbook's own example — Rs 11 to Rs 13.50 over 450 days — is ((13.5/11) ^ (365/450)) − 1 = 18.07%, not an annual approximation.
- CAGR is the accepted standard measure of return in financial markets except for periods of less than one year.
- The workbook's own FV example contradicts itself, and it does not change the answer. On page 36 the call is written FV (0.08,5,,-5000,,0) - type 0, with a stray extra comma - while the sentence under it says that type is 1, assuming the cash flow happens at the beginning of the period. Both give Rs 7,346.64, because type only shifts the timing of a recurring pmt, and this example has none. Type matters when you compound a stream of payments, not a single lump sum.
Check yourself
1.The formula for future value is:
- a)FV = PV / (1 + r)^n
- b)FV = PV x (1 + r)^n
- c)FV = PV x r x n
- d)FV = PV + (PV x r)
Show the answer
Answer: (b) FV = PV x (1 + r)^n
FV = PV x (1 + r)^n, where r is the rate of return for each compounding period and n is the number of compounding periods. FV = PV / (1 + r)^n is the present value formula rearranged, and PV x r x n would give simple interest rather than a compounded future value.
2.Rs 1,000 will be received after 3 years. The chapter says it is worth Rs 864 today when discounted at 5 per cent. What is it worth today at 8 per cent, and which principle does this show?
- a)Rs 864 — the discount rate does not change the present value of a fixed future sum
- b)Rs 760 — 8 per cent a year for 3 years is 24 per cent, taken off Rs 1,000
- c)Rs 794 — the higher the discount rate, the lower the present value of future cash flows
- d)Rs 1,260 — a higher rate raises the value of the cash flow
Show the answer
Answer: (c) Rs 794 — the higher the discount rate, the lower the present value of future cash flows
The chapter's own example: "Rs 1,000 received after 3 years when discounted at 5 per cent is worth Rs 864 today but the same amount discount at 8 per cent is worth only Rs 794."
Working: PV = FV/(1+r)^n = 1,000/(1.08)^3 = 1,000/1.259712 = Rs 793.83, about Rs 794. At 5 per cent: 1,000/1.157625 = Rs 863.84, about Rs 864.
The principle is one of the four points the chapter lists: "The higher the discount rate, the lower the present value of future cash flows. A higher rate means that investors have to forego more returns by opting to receive the money today instead of future cash flows."
Option A is wrong because the example shows the present value falling from Rs 864 to Rs 794 as the rate rises.
Option B is wrong because it takes simple interest off the future sum. Discounting divides by (1+r)^n — it does not subtract r × n.
Option D is wrong because Rs 1,260 is roughly the future value of Rs 1,000 compounded at 8 per cent for 3 years (1,000 × 1.2597). Future inflows are discounted to reach their present value, not compounded.
3.Rs 100 invested grows to Rs 120 in 2 years. Which Excel entry does the chapter use to find the CAGR, and what result does it give?
- a)=RATE(2,,-100,120), giving 0.0954 or 9.54 per cent
- b)=PMT(2,-100,120), giving 9.54 per cent
- c)=FV(2,,-100,120), giving 120
- d)=RATE(120,,-100,2), giving 0.10 or 10 per cent
Show the answer
Answer: (a) =RATE(2,,-100,120), giving 0.0954 or 9.54 per cent
The chapter: "In Excel the CAGR can be found out by the Rate Formula — CAGR = RATE(Years,,-PV,FV)". For this example, NPER (years) = 2, PV = -100, FV = 120, so "=RATE(2,,-100,120) = 0.0954 or 9.54%".
By the formula method the same answer is ((120/100)^(1/2)) − 1 = 1.0954 − 1, which the chapter rounds to 9.5%.
Option B is wrong because PMT gives the periodic payment, such as an EMI, from the rate, number of periods and present value. It does not solve for a rate.
Option C is wrong because FV gives a future value from a known rate. Here the future value (Rs 120) is already known and the rate is the unknown.
Option D is wrong because the arguments are in the wrong order. The syntax puts the number of years first, then the present value (negative) and then the future value. Getting 10 per cent is also the trap of dividing the 20 per cent gain by 2 years, which ignores compounding.
Where this is taught
Free preparation for NISM Series X-ARelated terms
- CAGRThe single smoothed annual rate at which a starting value would have to grow, compounding each year, to reach the ending value over a given period.
- PerpetuityA constant cash flow received every period with no end date. Its present value is simply the cash flow divided by the discount rate: PV = C / r.
- AnnuityAn insurance contract in which a corpus paid as a lump sum or in instalments buys a periodic payment from the insurer, at a rate fixed at purchase and guaranteed for life or the contracted term.
- Present valueWhat a sum of money due in the future is worth today, found by discounting it back at the rate the money could earn: PV = FV ÷ (1 + r)^n.
- Simple interestInterest earned only on the amount you first put in (the principal). The interest is not added back, so it never earns interest of its own, unlike compound interest.
- NPER functionThe Excel function that returns how many periods a loan takes to clear, given the rate, the instalment and the amount outstanding — the workbook's tool for the period of the loan.
- Ordinary annuityA series of equal payments falling at the end of each period — the default form of annuity, and the one every standard present value and EMI calculation assumes.
- Compound interestInterest earned on the original amount and also on the interest already earned. Rs 1,000 at 9% grows to Rs 31,409 in 40 years compounded, against Rs 4,600 with simple interest.
- PV functionThe spreadsheet function that converts a stream of future payments into the single lump sum needed today — the tool the workbook uses to size a retirement corpus and a life insurance cover.
- PMT functionThe spreadsheet function that converts a target corpus into the fixed periodic saving needed to reach it — the arithmetic behind every "how much should I invest a month" answer.
- Time value of moneyThe principle that the same sum of money is worth different amounts at different points on a timeline, because money held today can be invested and because inflation erodes what it will buy.
- Real rate of returnThe return on an investment after the effect of inflation has been removed — what the money actually buys more of, as against the nominal percentage the product advertises.
- Years to retirementThe period from now to the retirement date, or retirement age minus current age. It drives both how much inflation raises future expenses and how long savings have to compound.
- Two-step education fundingThe workbook's method for funding a multi-year education goal from one set-aside: inflate each year's fee to the year it falls due, then discount the later years back to the set-aside date.
- Annuity dueAn annuity whose payments fall at the beginning of each period rather than the end, which makes it worth more than an otherwise identical ordinary annuity.
- Effective yieldThe annual rate that would leave an investor with the same amount after one year as the quoted nominal rate does once its more frequent interest payments are taken into account.
- RestA lender's interest-charging cycle — how often the outstanding loan is recomputed for interest — which can differ from the repayment cycle and raise the true cost of a loan.
- Annuity tableA pre-computed table that gives the present-value multiple, or annuity factor, for a given interest rate and number of years, letting you value a regular cash flow without solving the formula each time.
- Compound interest rateThe rate at which a present sum of money grows into the future once its interest is reinvested and starts earning interest of its own, the forward-looking twin of the discount rate.
- Compounding periodThe interval, such as monthly, quarterly or annually, over which interest is calculated and added to an investment — the more often it happens within a year, the faster the investment grows.
- Frequency of compoundingHow often interest is calculated and added within a year — monthly, quarterly or annually — and the reason the same annual rate delivers a different actual return depending on the frequency chosen.