AFCAT Numerical Ability · Simple and Compound Interest

Compounding Half-Yearly, Quarterly or Monthly

Compounded k times a year, a rate of R% becomes R ÷ k per period and n years become k × n periods; the effective yearly rate is what one year of that compounding really adds.

All Simple and Compound Interest notes

Why this matters

Two of AFCAT's past-year questions are here, from 2015 and 2018, both compounded half-yearly. Two changes come before anything else: divide the rate and multiply the number of periods. Forgetting either one is the usual error, and the rest of the page turns a quoted rate into its effective yearly rate and back.

Concept 1 of 4: Rate per period and number of periods

8% a year compounded half-yearly is 4% every six months. Two years is four half-years, so the factor is

1.04^4

Definition

Half-yearly: rate $\tfrac R2$ , periods $2n$ . Quarterly: rate $\tfrac R4$ , periods $4n$ . Monthly: rate $\tfrac{R}{12}$ , periods $12n$ .
9 months compounded quarterly is 3 quarters; 18 months compounded half-yearly is 3 half-years.
Time from an amount: recognise the factor, count the periods, then turn periods back into years.
Present value: divide by the factor, as with annual compounding.

Compounded k times a year

A = P\left(1 + \dfrac{R}{100k}\right)^{kn}

Worked example

Find the compound interest on ₹20,000 for

1\tfrac12

years at 10% a year compounded half-yearly.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 04-203Moderate

Example 1 · Simple and Compound Interest · Compounding Half-Yearly or Quarterly

A bank gives 16% interest per annum compounded semi-annually. What interest will a man get on an amount of ₹10000 in 2 yr?

Change both the rate and the count

For 2 years at 12% compounded half-yearly the factor is

1.06^4

. Halving the rate but keeping 2 periods, or keeping 12% for 4 periods, are both offered as options.

Drill 18 more on rate per period and number of periods

Concept 2 of 4: Annual against half-yearly, and CI against SI

More frequent compounding earns a little more, because interest starts earning sooner. The gap between two ways of compounding is the difference of their factors times the sum.

Definition

Gap = $P \times (\text{factor}_1 - \text{factor}_2)$ . Half-yearly against annual for $n$ years: $P\left[\left(1 + \tfrac{R}{200}\right)^{2n} - \left(1 + \tfrac{R}{100}\right)^n\right]$ .
Compound minus simple interest when the compounding is half-yearly: work out each in full. The shortcut $P(R/100)^2$ holds only for annual compounding.
Two schemes: work out each amount; the better scheme is the larger amount.

Gap between two compoundings

\text{gap} = P \times (\text{factor}_1 - \text{factor}_2)

Worked example

₹10,000 is lent for 1 year at 10%. How much more does it earn compounded half-yearly than compounded annually?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 2 · Simple and Compound Interest · Compounding Half-Yearly or Quarterly

A sum lent for

2

years at

20

% compound interest would earn Rs.

482

more if interest is compounded half-yearly instead of annually. Find the sum.

The two-year shortcut is annual only

P(R/100)^2

is the gap between compound and simple interest for 2 years compounded annually. With half-yearly compounding, work out both interests in full.

Drill 4 more on annual against half-yearly, and ci against si

Concept 3 of 4: Effective annual rate

A rate compounded several times a year is worth more than it says. 10% compounded half-yearly turns ₹100 into ₹110.25 in a year, so it is worth 10.25% a year.

Definition

Effective annual rate $= \left(1 + \dfrac{R}{100k}\right)^k - 1$ .
1% a month compounded monthly is $1.01^{12} - 1 \approx 12.68\%$ ; 3% a quarter is $1.03^4 - 1 \approx 12.55\%$ .
The same number is the simple interest rate that gives the same amount in one year.

Effective annual rate

E = \left(1 + \dfrac{R}{100k}\right)^k - 1

Worked example

Find the effective annual rate of 12% a year compounded half-yearly.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Simple and Compound Interest · Compounding Half-Yearly or Quarterly

What is the effective annual rate of interest if the nominal rate is

8

% compounded quarterly?

Never below the quoted rate

12% compounded monthly is about 12.68% a year. An option below 12% cannot be its effective rate.

Drill 11 more on effective annual rate

Concept 4 of 4: From an effective rate back to a period rate

Going the other way takes a root. A yearly rate of 21% is the same as 10% each half-year compounded, because

1.1^2 = 1.21

Definition

Rate per period $= \sqrt[k]{1 + \tfrac{E}{100}} - 1$ ; the nominal yearly rate is $k$ times that.
Dividing by $k$ is wrong: 12% a year compounded annually is about 2.87% a quarter compounded, not 3%.
Growth over a longer time works the same way: a sum that doubles in 3 years grows by $\sqrt[3]{2}$ each year.
Continuous compounding, if it appears, is $e^{r}$ a year, so the rate is $\ln\left(1 + \tfrac{E}{100}\right)$ .

Period rate from the effective rate

\text{rate per period} = \sqrt[k]{1 + \dfrac{E}{100}} - 1

Worked example

What nominal yearly rate, compounded half-yearly, is the same as 44% a year compounded annually?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Simple and Compound Interest · Compounding Half-Yearly or Quarterly

What is the nominal annual rate if the effective annual rate is

16\%

with semi-annual compounding?

Take the root, do not divide

21% a year is 10% a half-year compounded, not 10.5%. Halving a compound rate is the simple interest habit.

Drill 10 more on from an effective rate back to a period rate

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Rate per period and number of periods
Compounded k times a year
$A = P\left(1 + \dfrac{R}{100k}\right)^{kn}$
Annual against half-yearly, and CI against SI
Gap between two compoundings
$\text{gap} = P \times (\text{factor}_1 - \text{factor}_2)$
Effective annual rate
Effective annual rate
$E = \left(1 + \dfrac{R}{100k}\right)^k - 1$
From an effective rate back to a period rate
Period rate from the effective rate
$\text{rate per period} = \sqrt[k]{1 + \dfrac{E}{100}} - 1$

Watch out for (4)

Change both the rate and the count
→ Rate per period and number of periods
The two-year shortcut is annual only
→ Annual against half-yearly, and CI against SI
Never below the quoted rate
→ Effective annual rate
Take the root, do not divide
→ From an effective rate back to a period rate

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