Financial Mathematics and Time Value of Money – Key Concepts

1. Time Value of Money – Core Idea

The time value of money (TVM) states that a rupee today is worth more than a rupee in the future because of its potential earning capacity.

• Present Value (PV): the current worth of a future cash flow.
• Future Value (FV): the amount a present cash flow will grow to after a period.
• Discount Rate (i): the rate at which future cash flows are discounted back to present value.

The basic relationship for a single cash flow is

$$PV=\frac{FV}{(1+i{)}^n}$$

and conversely

$$FV=PV\times (1+i{)}^n$$

where $n$ is the number of periods.

Figure: Present Value Factor vs Discount Rate

The curve shows that higher discount rates reduce the present value of future cash flows.

2. Present Value and Future Value of a Lump Sum

Formulas

• Present Value of a lump sum:

  $$PV=\frac{FV}{(1+i{)}^n}$$

• Future Value of a lump sum:

  $$FV=PV\times (1+i{)}^n$$

Worked Example – Bank Loan

A student takes a loan of Rs 1,000,000 from a bank at an annual interest rate of 10 % for 5 years.
Compute the present value of the loan (i.e., the amount the bank expects to receive in today’s terms).

Calculation

$$PV=\frac{1,000,000}{(1+0.10{)}^5}=\frac{1,000,000}{1.61051}\approx 620,921$$

So the loan’s present value is Rs 620,921.

Real‑world tie‑in: This calculation is exactly what banks perform when they issue a loan; they discount the future repayments to determine the loan’s worth today.

3. Annuities

An annuity is a series of equal payments made at regular intervals.

• Ordinary annuity: payments at the end of each period.
• Annuity due: payments at the beginning of each period.

Present Value Factor (PVAF) for an ordinary annuity:

$$PVAF=\frac{1-(1+i{)}^{-n}}{i}$$

Future Value Factor (FVAF):

$$FVAF=\frac{(1+i{)}^n-1}{i}$$

Worked Example – Monthly Installment Plan

Daraz offers a monthly installment plan of Rs 10,000 for 36 months at an annual discount rate of 6 % (monthly rate $i=0.06/12=0.005$).

Present Value

$$PV=10,000\times \frac{1-(1+0.005{)}^{-36}}{0.005}\approx 10,000\times 32.79\approx 327,900$$

Thus, the present value of the 36 monthly payments is Rs 327,900.

Future Value

$$FV=10,000\times \frac{(1+0.005{)}^{36}-1}{0.005}\approx 10,000\times 39.54\approx 395,400$$

The future value of the annuity is Rs 395,400.

4. Perpetuities

A perpetuity is an annuity that continues forever.
The present value of a perpetuity paying $C$ per period at discount rate $i$ is

$$PV=\frac{C}{i}$$

Worked Example – Dividend Discount Model

A company pays a dividend of Rs 50,000 annually and the required return is 5 %.
Present value of the dividend stream:

$$PV=\frac{50,000}{0.05}=1,000,000$$

So the stock’s intrinsic value based on dividends is Rs 1,000,000.

5. Net Present Value (NPV) and Internal Rate of Return (IRR)

Net Present Value (NPV) measures the profitability of a project:

$$NPV=\sum _{t=0}^n\frac{C_t}{(1+i{)}^t}$$

where $C_t$ is the cash flow at time $t$.

Internal Rate of Return (IRR) is the discount rate that makes $NPV=0$.

Worked Example – Project Evaluation

A firm considers a project with an initial outlay of Rs 5,000,000 and expected annual cash inflows of Rs 1,200,000 for 5 years.
Discount rate $i=12$.

Since NPV > 0, the project is acceptable.

IRR can be found by trial‑and‑error or using a financial calculator; it turns out to be about 15 %, which exceeds the required 12 %.

6. Loan Amortization

A loan amortization schedule shows how each payment is split into interest and principal.

Monthly payment formula for a loan of principal $P$, annual rate $r$, and $n$ months:

$$PMT=P\times \frac{r/12}{1-(1+r/12{)}^{-n}}$$

Worked Example – Home Loan

A borrower takes a loan of Rs 2,000,000 at an annual rate of 8 % for 10 years.

Monthly rate $i=0.08/12=0.006667$.
Number of payments $n=120$.

$$PMT=2,000,000\times \frac{0.006667}{1-(1+0.006667{)}^{-120}}\approx 2,000,000\times 0.0123\approx 24,600$$

So the monthly payment is Rs 24,600.

Amortization schedule (first 3 months):

Mermaid diagram – Loan Payment Process

7. Comparison Tables

Simple vs Compound Interest

Ordinary Annuity vs Annuity Due

Lump Sum vs Annuity

8. Advantages and Disadvantages

9. In the real world

• eSewa: Offers micro‑loans of Rs 50,000 at 12 % annual interest for 6 months. The borrower’s monthly payment is calculated using the loan amortization formula above.
• Daraz: Provides a 12‑month installment plan of Rs 10,000 per month at 6 % discount rate. The present value of the payments is Rs 327,900, which Daraz uses to price the product.
• Ncell: Sells mobile plans with monthly installments of Rs 5,000 for 24 months at 5 % annual rate. The company uses the annuity due formula to determine the upfront cost to the customer.

These examples illustrate how TVM concepts are embedded in everyday financial products.

10. Exam tip

• Identify the type of cash flow: lump sum, ordinary annuity, annuity due, perpetuity, or project cash flows.
• Use the correct formula: PV, FV, PVAF, FVAF, NPV, IRR, loan payment.
• Show all steps: write the formula, substitute numbers, compute intermediate values, and state the final answer.
• Check units: rates should be in the same period as the cash flow (e.g., monthly rate for monthly payments).
• Practice with real‑world numbers: e.g., bank loans, installment plans, and investment projects.

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