Aptitude · Compound Interest · Model 2

Half the rate, twice the periods

Change how often interest is added and the amount changes, even though the advertised yearly rate does not. The adjustment is one line, and the two halves of it must both be made — making only one is the error the whole model tests.

Switch the compounding frequency and watch both adjustments →

Compounded k times a year: rate per period = R/k and periods = kT. Both changes, every time.

01 The idea

The same rate, added more often

A bank advertises 10% per annum. If it adds interest once a year, ₹1,000 becomes ₹1,100. If it adds interest every six months, it adds 5% twice — and because the second 5% is charged on ₹1,050 rather than on ₹1,000, you end the year with ₹1,102.50.

That ₹2.50 is the whole model. Nothing about the advertised rate changed. What changed is how many times the interest got the chance to start earning interest itself, and more chances means more money.

The adjustment is mechanical: divide the rate by the number of periods in a year and multiply the time by the same number. Half-yearly means 5% for 2T periods; quarterly means 2.5% for 4T periods; monthly means R/12 for 12T periods.

Both halves are essential and students routinely make only one. Halving the rate without doubling the periods understates the answer badly; doubling the periods without halving the rate overstates it wildly. The safest habit is to write the rate per period and the period count down explicitly before touching the formula.

Divide the rate by k and multiply the time by k. Do both, or the answer is not merely inaccurate — it is answering a different question.

Nominal rateThe advertised yearly rate, R. It is not what any single period actually charges unless compounding is annual.

Rate per periodR/k where k is the number of compoundings a year. This is the figure that actually goes into the formula.

Effective annual rateThe single annual rate that would give the same yearly growth. For 10% half-yearly it is 10.25%, because ₹1,000 becomes ₹1,102.50 in a year.

02 Worked example

₹1,000 at 10% for 2 years, half-yearly

The lesson’s sum, recompounded. Find the amount on ₹1,000 at 10% per annum for 2 years, compounded half-yearly, and compare it with annual compounding.

Halve the rateTen per cent a year, added twice a year, means five per cent each time.rate per period = 10/2 = 5%

Double the periodsTwo years of half-yearly compounding is four periods, not two.periods = 2 × 2 = 4

Apply the formulaFour periods of multiplying by 1.05.A = 1000 × (1.05)⁴ = 1000 × 1.21550625 = ₹1,215.51

Compare with annual compoundingThe same nominal rate over the same two years, added once a year.annual: 1000 × (1.1)² = ₹1,210

Name the gainCompounding twice as often is worth ₹5.51 over two years on this principal.1,215.51 − 1,210 = ₹5.51

Watch what happens to the two halves of the adjustment if you get them wrong. Halving the rate but keeping two periods gives 1000 × 1.05² = ₹1,102.50 — the answer to a one-year question. Doubling the periods but keeping 10% gives 1000 × 1.1⁴ = ₹1,464.10, which is a four-year answer. Both are plausible-looking and both are answers to questions nobody asked.

03 The method

The four frequencies, and the effective rate

One pattern covers all of them. The effective annual rate is the way to compare two different frequencies fairly.

A = P(1 + R/100k)^kT for k compoundings a year. So annually k = 1, half-yearly k = 2, quarterly k = 4, monthly k = 12.

To compare frequencies, use the effective annual rate: (1 + R/100k)^k − 1. A nominal 10% is effectively 10% annually, 10.25% half-yearly, 10.38% quarterly and 10.47% monthly. Note the gains shrink as k rises — going from annual to half-yearly buys far more than going from quarterly to monthly.

Compounded	On ₹1,000 at 10%, 1 year	Effective annual rate
Annually	`1,100.00`	`10.00%`
Half-yearly	`1,102.50`	`10.25%`
Quarterly	`1,103.81`	`10.38%`
Monthly	`1,104.71`	`10.47%`
Halving R, not doubling T	`1,102.50 for 2 yr`	answers a 1-year question
Doubling T, not halving R	`1,464.10 for 2 yr`	answers a 4-year question

05 Cheat sheet

Model 2 on one page

One pattern, four instances, and the two half-done adjustments that produce wrong answers.

Compounded	Rate per period	Periods in T years
Annually	`R`	`T`
Half-yearly	`R/2`	`2T`
Quarterly	`R/4`	`4T`
Monthly	`R/12`	`12T`
Effective annual rate	`(1+R/100k)^k − 1`	compare frequencies
Rate halved only	`wrong`	gives a shorter-term answer
Periods doubled only	`wrong`	gives a longer-term answer

Write both numbers down firstState the rate per period and the period count explicitly before using the formula. That one habit eliminates the only real error in this model.

More frequent always earns moreBut with diminishing returns. Annual to half-yearly gains ₹2.50 on ₹1,000 a year; quarterly to monthly gains under a rupee.

1.5 years half-yearly is three periodsNot one and a half. A fractional number of years can still be a whole number of periods, which is often why the question chose that time.

06 Where & why

Where Model 2 shows up

Compounding frequency is the most common way to make a compound interest question harder without making it longer.

Bank PO · SSC CGL

Half-yearly and quarterly variants

The standard form. Both adjustments needed, and the trap answers correspond to making only one of them.

TCS NQT · Infosys

“Find the difference between annual and half-yearly”

Two calculations and a subtraction. Small numbers, so precision matters more than usual.

Real banking

Quarterly deposits and monthly credit cards

Indian fixed deposits typically compound quarterly and credit cards monthly, which is why the effective rate on a card is well above its advertised figure.

Effective rate comparisons

Which of two offers is better

A nominal 10.2% compounded annually against a nominal 10% compounded quarterly — the effective rates settle it, and the lower nominal rate can win.

The whole model is one line of adjustment, applied completely. When you see “compounded half-yearly”, write “5%, 4 periods” on your page before you do anything else.

07 Interview questions

What gets asked

Ten, and the third one is the mistake this model exists to catch.

How do you adjust for half-yearly compounding?

Halve the rate and double the number of periods: A = P(1 + R/200)^(2T). Both changes are required. On ₹1,000 at 10% for 2 years that is 1000 × 1.05⁴ = ₹1,215.51.

And quarterly?

Quarter the rate and quadruple the periods: P(1 + R/400)^(4T). The general form for k compoundings a year is P(1 + R/100k)^(kT), which is worth learning instead of four separate formulas.

What happens if you halve the rate but forget to double the periods?

You answer a different question. On ₹1,000 at 10% for 2 years you would compute 1000 × 1.05² = ₹1,102.50, which is the one-year half-yearly answer. It looks like a plausible result, which is exactly why it appears in the options.

Why does more frequent compounding give more money?

Because interest is credited sooner and therefore starts earning interest sooner. The same nominal 10% gives ₹1,100 annually, ₹1,102.50 half-yearly and ₹1,103.81 quarterly over one year on ₹1,000.

What is the effective annual rate for 10% compounded half-yearly?

10.25%. Two periods of 5% give a multiplier of 1.05² = 1.1025, so a year’s growth is 10.25%. The effective rate is the honest basis for comparing offers with different compounding frequencies.

Is the gain from compounding more often unlimited?

No, it shrinks quickly. For a nominal 10%, the effective rate is 10% annually, 10.25% half-yearly, 10.38% quarterly and 10.47% monthly. The step from annual to half-yearly is worth more than every subsequent step combined, which is why the difference between quarterly and monthly rarely matters.

How many periods is 1½ years compounded half-yearly?

Three. A fractional number of years can be a whole number of periods, and that is usually why the question picked that time. At 10% nominal, ₹8,000 becomes 8000 × 1.05³ = ₹9,261.

Find the difference between annual and half-yearly compounding on ₹10,000 at 8% for 1 year.

₹16. Annually it is ₹10,800; half-yearly it is 10000 × 1.04² = ₹10,816. Differences in these questions are small, so carry the decimals rather than rounding early.

A card advertises 24% per annum compounded monthly. What is the real annual rate?

Hard

₹1,000 is invested at a nominal 12% per annum. (a) Find the amount after one year compounded annually, half-yearly, quarterly and monthly. (b) Compute the gain from each step up in frequency and comment on the pattern. (c) Without computing it exactly, argue whether the amount could exceed ₹1,130 for any compounding frequency however large, and say what quantity the sequence is approaching.

Follow-up

Part (c) is the interesting one. The amounts rise but are bounded, approaching ₹1,000 × e^(0.12) ≈ ₹1,127.50 — continuous compounding. Recognising that the gains converge rather than growing without limit is worth far more than another frequency calculation.

Show the hint

For (c), look at how fast the gains in part (b) are shrinking and ask what they sum to.