Future value

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.

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.

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

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 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.

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.

• 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.