Ch1. Introduction to Financial Management — Time Value of Money and Present Value

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).

---

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

---

Future Value (FV)

Simple interest: FV = PV × (1 + r × n)

Compound interest: FV = PV × (1 + r)^n

PV = Present Value
r  = Interest rate (per period)
n  = Number of periods

Example:

PV = $10,000, r = 5%, n = 3 years
FV = $10,000 × (1.05)^3 = $10,000 × 1.1576 ≈ $11,576

---

Present Value (PV)

PV = FV / (1 + r)^n
   = FV × [1 / (1 + r)^n]

Discount rate (r): the rate used to convert future cash flows into present value

Example:

$10,000 to be received 3 years from now, discount rate 10%
PV = $10,000 / (1.10)^3 = $10,000 / 1.331 ≈ $7,513

---

Present Value of Annuities

Ordinary Annuity (End-of-Period Payments)

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

PV of annuity = C × [1 - (1+r)^(-n)] / r
              = C × PVIFA(r, n)

PVIFA(r, n) = Present Value Interest Factor of Annuity

Example:

$2,000 received at year-end, r = 5%, 5 years
PV = $2,000 × [1 - (1.05)^(-5)] / 0.05
   = $2,000 × 4.3295
   = $8,659

Perpetuity

An annuity that continues indefinitely.

PV of perpetuity = C / r

Example: $1,000 per year, r = 4%
PV = $1,000 / 0.04 = $25,000

Growing Perpetuity

A perpetuity whose payments grow at rate g per period.

PV of growing perpetuity = C / (r - g)    (requires r > g)

Example: First-year payment $1,000, growing 2% per year, r = 6%
PV = $1,000 / (0.06 - 0.02) = $1,000 / 0.04 = $25,000

---

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.

Firm Value = Σ FCF_t / (1 + WACC)^t

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

---

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

---

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.