Introduction to Financial Management — Time Value of Money and Present Value

1. What Is Financial Management?

Financial Management: The decision-making process by which firms or individuals efficiently raise and deploy capital.

The Three Core Decisions of Financial Management:

1. Investment Decision: What assets to invest in (asset composition)
2. Financing Decision: How to raise capital (debt vs. equity ratio)
3. Dividend Decision: How to distribute earnings

Goal of Corporate Finance: Maximize shareholder value (firm value).

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2. The Time Value of Money

Time Value of Money: A dollar today is worth more than a dollar in the future.

Reasons:

1. Investment opportunity: today’s money can be invested to generate returns
2. Inflation: purchasing power erodes over time
3. Uncertainty: future cash flows carry risk

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3. Future Value (FV)

$$\begin{aligned} \text{Simple interest}: FV &= PV \times (1 + r \times n) \\ \text{Compound interest}: FV &= PV \times (1 + r)^n \\ PV &= \text{Present Value} \\ r &= \text{Interest rate} (\text{per period}) \\ n &= \text{Number of periods} \end{aligned}$$

Example:

$$\begin{aligned} PV &= \$10,000, r = 5\%, n = 3 \text{years} \\ FV &= \$10,000 \times (1.05)^3 = \$10,000 \times 1.1576 \approx \$11,576 \end{aligned}$$

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4. Present Value (PV)

$$\begin{aligned} PV &= \frac{FV}{1 + r}^n \\ &= FV \times [\frac{1}{1 + r}^n] \\ &\text{Discount rate} (r): \text{the rate used to convert future cash flows into present value} \end{aligned}$$

Example:

$$\begin{aligned} &\$10,000 \text{to be received} 3 \text{years from now}, \text{discount rate} 10\% \\ PV &= \$\frac{10,000}{1.10}^3 = \$\frac{10,000}{1.331} \approx \$7,513 \end{aligned}$$

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5. Present Value of Annuities

(1) Ordinary Annuity (End-of-Period Payments)

Equal payments (C) received at the end of each period.

$$\begin{aligned} PV of annuity &= C \times [1 - (1+r)^(-n)] / r \\ &= C \times PVIFA(r, n) \\ PVIFA(r, n) &= Present Value Interest Factor of Annuity \end{aligned}$$

Example:

$$\begin{aligned} \$2,000 received at year-end, r &= 5\%, 5 years \\ PV &= \$2,000 \times [1 - (1.05)^(-5)] / 0.05 \\ &= \$2,000 \times 4.3295 \\ &= \$8,659 \end{aligned}$$

(2) Perpetuity

An annuity that continues indefinitely.

$$\begin{aligned} \text{PV of perpetuity} &= \frac{C}{r} \\ \text{Example}: \$1,000 \text{per year}, r &= 4\% \\ PV &= \$\frac{1,000}{0.04} = \$25,000 \end{aligned}$$

(3) Growing Perpetuity

A perpetuity whose payments grow at rate g per period.

$$\begin{aligned} \text{PV of growing perpetuity} &= \frac{C}{r - g} (\text{requires} r > g) \\ \text{Example}: \text{First}-\text{year payment} \$1,000, \text{growing} 2\% \text{per year}, r &= 6\% \\ PV &= \$1,\frac{000}{0.06 - 0.02} = \$\frac{1,000}{0.04} = \$25,000 \end{aligned}$$

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6. DCF (Discounted Cash Flow) Analysis

DCF: A method of valuing an investment by discounting all future cash flows to the present using an appropriate discount rate.

$$\text{Firm Value} = \sum \frac{FCF_t}{1 + WACC}^t$$

• FCF: Free Cash Flow
• WACC: Weighted Average Cost of Capital

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7. Key Concept Cards

Compound Interest Calculation ★★★★★ : FV = PV × (1+r)^n. Compound interest earns interest on interest, producing exponential growth. The difference from simple interest becomes dramatic over long horizons. Memory tip: FV = PV × (1+r)^n

Present Value ★★★★★ : PV = FV / (1+r)^n. The higher the discount rate, the lower the present value of future cash flows. Memory tip: PV = FV ÷ (1+r)^n

Growing Perpetuity ★★★★☆ : PV = C / (r−g). Identical to the Gordon Growth Model (Dividend Discount Model). Memory tip: C ÷ (r−g) = growing perpetuity

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8. Practice Quiz

Q. You deposit $10,000 in a bank at 6% annual compound interest for 5 years. What is the future value?

FV = $10,000 × (1.06)^5 = $10,000 × 1.3382 ≈ $13,382.

Q. A firm is expected to pay a perpetual annual dividend of $500. If the required return is 8%, what is the firm’s value?

Perpetuity: PV = $500 / 0.08 = $6,250.