Compound Interest
& Annuity

The power of compounding and the mathematics of regular investments. Essential tools for financial planning.

Compound Interest (CI)

The Amount Formula

Unlike Simple Interest, CI is calculated on the principal and the interest accumulated from previous periods.

Amount (A)

A = P(1 + \frac{r}{100})^n

Total Interest

CI = P[(1 + \frac{r}{100})^n - 1]

Present Value (P)

P = \frac{A}{(1 + i)^n} \text{ where } i = r/100

⚠️

Frequencies of Compounding

Interest can be compounded semi-annually (k=2), quarterly (k=4), or monthly (k=12). The formula becomes:

A = P(1 + r/100k)^{nk}

Compound Interest Growth

Visualize how money grows exponentially compared to simple interest.

Present Value (P)

₹

Annual Rate (%)

1%10%30%

Time (Years)

Total Amount (A)₹25,937

Interest Gap₹5,937

Annuities & Cash Flows

What is an Annuity?

Annuity

A sequence of equal payments made at regular intervals of time.

Annuity Regular (Ordinary)

An annuity where payments are made at the **end** of each period.

Annuity Due

An annuity where payments are made at the **beginning** of each period.

Annuity Timeline

Visualize timing differences between Regular and Due annuities.

Start

₹ R

Period 1

₹ R

Period 2

₹ R

Period 3

₹ R

Period 4

₹ R

End (N)

Annuity Regular (End of Period)

Payments are made at the end of each interval. T1 is the first payment. Value at T0 is the Present Value (P).

Ordinary Annuity (Regular)

Amount (A)

A = R \left[ \frac{(1 + i)^n - 1}{i} \right]

Present Value (P)

P = R \left[ \frac{1 - (1 + i)^{-n}}{i} \right]

Annuity Due

Amount (A)

A(\text{due}) = R \left[ \frac{(1 + i)^n - 1}{i} \right] (1 + i)

Present Value (P)

P(\text{due}) = R \left[ \frac{1 - (1 + i)^{-n}}{i} \right] (1 + i)

Deferred Annuity (m + n Case)

When payments start after a "waiting period" of

m

periods. This is common in student loans or project finance where interest accumulates but repayment starts later.

Amount at

T_{m + n}

A = R \left[ \frac{(1 + i)^n - 1}{i} \right]

(Value at

T_{m+n}

)

Present Value at

T_0

P = R \left[ \frac{1 - (1 + i)^{-n}}{i} \right] (1 + i)^{-m}

M+N LogicThe "Amount" remains the same as an

n

-period annuity at the end of the term, but the "Present Value" is discounted back for the additional

m

periods.

Specialized Concepts

Notation ($R=₹1$)

S_{n|i} = \frac{(1 + i)^n - 1}{i} \text{ when } R=1

a_{n|i} = \frac{1 - (1 + i)^{-n}}{i} \text{ when } R=1

Perpetuity

An annuity that never ends. Valuable for valuing dividend-paying stocks or long-term endowments.

Simple Present Value (P)

P = \frac{R}{i}

Growing Present Value (P)

P = \frac{R}{i - g}

Case Studies & Problems

Real-world applications of compound interest and annuities.

EMI CalculationsSinking Funds

Practice Laboratory

Calculate the effective rate of interest corresponding to a nominal rate of 6% per annum compounded semi-annually.

✓ Show Solution

r = 6, k = 2

. Rate per period

i = 6/2 = 3% = 0.03

r_{eff} = (1 + i)^k - 1 = (1.03)^2 - 1 = 1.0609 - 1 = 0.0609

or 6.09%

EASY

A man deposits ₹2000 at the end of every 3 months in a bank which pays 8% interest p.a. compounded quarterly. How much will be to his credit at the end of 5 years?

✓ Show Solution

R = 2000, i = 8/4 = 2% = 0.02, n = 5 \times 4 = 20

S_{20} = 2000 [ (1.02)^{20} - 1 ] / 0.02

3. Using

(1.02)^{20} \approx 1.4859

S_{20} = 2000 [ 0.4859 ] / 0.02 = 100000 \times 0.4859 = \text{₹}48,590

MEDIUM

What is the present value of an annuity of ₹1200 payable at the end of each 6 months for 4 years when interest is 10% per annum compounded semi-annually?

✓ Show Solution

R = 1200, i = 10/2 = 5% = 0.05, n = 4 \times 2 = 8

P = 1200 [ 1 - (1.05)^{-8} ] / 0.05

3. Using

(1.05)^{-8} \approx 0.6768

P = 1200 [ 1 - 0.6768 ] / 0.05 = 1200 [ 0.3232 ] / 0.05 = \text{₹}7,756.80

MEDIUM

CBSE & Competitive Corner

CBSE PYQFIN-Q#fin_exam_1

A machine costs ₹5,00,000. It is estimated to have a useful life of 10 years at the end of which its scrap value is ₹50,000. How much should be set aside at the end of each year in a sinking fund earning 9% p.a. to replace the machine?

Formal Solution1. Amount needed

S = 5,00,000 - 50,000 = 4,50,000

n = 10, i = 0.09

S_n = R [ (1+i)^n - 1 ] / i \implies 450000 = R [ (1.09)^{10} - 1 ] / 0.09

(1.09)^{10} \approx 2.367

R = 450000 \times 0.09 / 1.367 \approx \text{₹}29,627

ConceptualFIN-Q#fin_exam_2

Distinguish between Annuity Regular and Annuity Due with an example.

Formal Solution1. **Annuity Regular**: Payments at the end (e.g., Monthly Salary, Rent at month end). 2. **Annuity Due**: Payments at the beginning (e.g., Insurance Premium, SIP at the start of the month).

Loan AnalysisFIN-Q#fin_exam_3

Find the present value of a deferred annuity of ₹20,000 p.a. for 5 years, the first payment being made at the end of 4 years, if money is worth 10% p.a. compounded annually.

Formal Solution1. Here

R = 20000, n = 5, i = 0.10

. The first payment is at end of 4 years, so

m = 3

(gap periods). 2.

P = R [ 1 - (1+i)^{-n} ] / i \times (1+i)^{-m}

P = 20000 [ 1 - (1.1)^{-5} ] / 0.1 \times (1.1)^{-3}

P = 20000 [ 3.7908 ] \times 0.7513 \approx \text{₹}56,960

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Compound Interest
& Annuity

The power of compounding and the mathematics of regular investments. Essential tools for financial planning.

Compound Interest (CI)

The Amount Formula

Unlike Simple Interest, CI is calculated on the principal and the interest accumulated from previous periods.

Amount (A)

A = P(1 + \frac{r}{100})^n

Total Interest

CI = P[(1 + \frac{r}{100})^n - 1]

Present Value (P)

P = \frac{A}{(1 + i)^n} \text{ where } i = r/100

⚠️

Frequencies of Compounding

Interest can be compounded semi-annually (k=2), quarterly (k=4), or monthly (k=12). The formula becomes:

A = P(1 + r/100k)^{nk}

Compound Interest Growth

Visualize how money grows exponentially compared to simple interest.

Present Value (P)

₹

Annual Rate (%)

1%10%30%

Time (Years)

Total Amount (A)₹25,937

Interest Gap₹5,937

Annuities & Cash Flows

What is an Annuity?

Annuity

A sequence of equal payments made at regular intervals of time.

Annuity Regular (Ordinary)

An annuity where payments are made at the **end** of each period.

Annuity Due

An annuity where payments are made at the **beginning** of each period.

Annuity Timeline

Visualize timing differences between Regular and Due annuities.

Start

₹ R

Period 1

₹ R

Period 2

₹ R

Period 3

₹ R

Period 4

₹ R

End (N)

Annuity Regular (End of Period)

Payments are made at the end of each interval. T1 is the first payment. Value at T0 is the Present Value (P).

Ordinary Annuity (Regular)

Amount (A)

A = R \left[ \frac{(1 + i)^n - 1}{i} \right]

Present Value (P)

P = R \left[ \frac{1 - (1 + i)^{-n}}{i} \right]

Annuity Due

Amount (A)

A(\text{due}) = R \left[ \frac{(1 + i)^n - 1}{i} \right] (1 + i)

Present Value (P)

P(\text{due}) = R \left[ \frac{1 - (1 + i)^{-n}}{i} \right] (1 + i)

Deferred Annuity (m + n Case)

When payments start after a "waiting period" of

m

periods. This is common in student loans or project finance where interest accumulates but repayment starts later.

Amount at

T_{m + n}

A = R \left[ \frac{(1 + i)^n - 1}{i} \right]

(Value at

T_{m+n}

)

Present Value at

T_0

P = R \left[ \frac{1 - (1 + i)^{-n}}{i} \right] (1 + i)^{-m}

M+N LogicThe "Amount" remains the same as an

n

-period annuity at the end of the term, but the "Present Value" is discounted back for the additional

m

periods.

Specialized Concepts

Notation ($R=₹1$)

S_{n|i} = \frac{(1 + i)^n - 1}{i} \text{ when } R=1

a_{n|i} = \frac{1 - (1 + i)^{-n}}{i} \text{ when } R=1

Perpetuity

An annuity that never ends. Valuable for valuing dividend-paying stocks or long-term endowments.

Simple Present Value (P)

P = \frac{R}{i}

Growing Present Value (P)

P = \frac{R}{i - g}

Case Studies & Problems

Real-world applications of compound interest and annuities.

EMI CalculationsSinking Funds

Practice Laboratory

Calculate the effective rate of interest corresponding to a nominal rate of 6% per annum compounded semi-annually.

✓ Show Solution

r = 6, k = 2

. Rate per period

i = 6/2 = 3% = 0.03

r_{eff} = (1 + i)^k - 1 = (1.03)^2 - 1 = 1.0609 - 1 = 0.0609

or 6.09%

EASY

A man deposits ₹2000 at the end of every 3 months in a bank which pays 8% interest p.a. compounded quarterly. How much will be to his credit at the end of 5 years?

✓ Show Solution

R = 2000, i = 8/4 = 2% = 0.02, n = 5 \times 4 = 20

S_{20} = 2000 [ (1.02)^{20} - 1 ] / 0.02

3. Using

(1.02)^{20} \approx 1.4859

S_{20} = 2000 [ 0.4859 ] / 0.02 = 100000 \times 0.4859 = \text{₹}48,590

MEDIUM

What is the present value of an annuity of ₹1200 payable at the end of each 6 months for 4 years when interest is 10% per annum compounded semi-annually?

✓ Show Solution

R = 1200, i = 10/2 = 5% = 0.05, n = 4 \times 2 = 8

P = 1200 [ 1 - (1.05)^{-8} ] / 0.05

3. Using

(1.05)^{-8} \approx 0.6768

P = 1200 [ 1 - 0.6768 ] / 0.05 = 1200 [ 0.3232 ] / 0.05 = \text{₹}7,756.80

MEDIUM

CBSE & Competitive Corner

CBSE PYQFIN-Q#fin_exam_1

Formal Solution1. Amount needed

S = 5,00,000 - 50,000 = 4,50,000

n = 10, i = 0.09

S_n = R [ (1+i)^n - 1 ] / i \implies 450000 = R [ (1.09)^{10} - 1 ] / 0.09

(1.09)^{10} \approx 2.367

R = 450000 \times 0.09 / 1.367 \approx \text{₹}29,627

ConceptualFIN-Q#fin_exam_2

Distinguish between Annuity Regular and Annuity Due with an example.

Loan AnalysisFIN-Q#fin_exam_3

Find the present value of a deferred annuity of ₹20,000 p.a. for 5 years, the first payment being made at the end of 4 years, if money is worth 10% p.a. compounded annually.

Formal Solution1. Here

R = 20000, n = 5, i = 0.10

. The first payment is at end of 4 years, so

m = 3

(gap periods). 2.

P = R [ 1 - (1+i)^{-n} ] / i \times (1+i)^{-m}

P = 20000 [ 1 - (1.1)^{-5} ] / 0.1 \times (1.1)^{-3}

P = 20000 [ 3.7908 ] \times 0.7513 \approx \text{₹}56,960