Module 2 · Quantitative Methods

The Time Value of Money in Finance

EN: Discounting and compounding cash flows — present value, future value, annuities, perpetuities, and effective rate conversions.
VN: Giá trị thời gian của tiền — chiết khấu/lãi kép, niên kim, dòng tiền vĩnh viễn, quy đổi lãi suất.

In this module / Trong module này

FV / PV of a single cash flow
FV / PV of an ordinary annuity
Annuity due
Perpetuity
Stated vs Effective Annual Rate (EAR)
Continuous compounding
Bond price (cash-flow additivity)

1. Future / Present Value of a Single Cash Flow Core

EN: Compounding moves a single cash flow forward; discounting moves it back.
VN: Lãi kép đẩy 1 dòng tiền về tương lai; chiết khấu kéo nó về hiện tại.

\[ FV_N = PV\,(1 + r)^{N} \qquad PV = \frac{FV_N}{(1 + r)^{N}} \]

Components / Thành phần

$PV$ Present value / Giá trị hiện tại.
$FV_N$ Future value at the end of $N$ periods / Giá trị tương lai sau $N$ kỳ.
$r$ Periodic interest rate / Lãi suất mỗi kỳ.
$N$ Number of compounding periods / Số kỳ ghép lãi.

Practice problem

$1,000 is deposited at 6% compounded annually. What is its value after 5 years?

Show solution

$FV = 1{,}000\,(1.06)^{5} = 1{,}000 \times 1.3382$

≈ $1,338.23

2. FV / PV of an Ordinary Annuity Core

EN: An ordinary annuity pays a constant amount $A$ at the end of each period.
VN: Niên kim thường — trả số tiền cố định $A$ vào cuối mỗi kỳ.

\[ PV_{\text{ord}} = A \cdot \frac{1 - (1+r)^{-N}}{r} \] \[ FV_{\text{ord}} = A \cdot \frac{(1+r)^{N} - 1}{r} \]

Components / Thành phần

$A$ Periodic payment / Khoản thanh toán mỗi kỳ.
$r$ Interest rate per period / Lãi suất một kỳ.
$N$ Number of payments / Số kỳ thanh toán.

Practice problem

A retirement plan pays $10,000 at the end of each year for 20 years. The discount rate is 7%. What is the present value?

Show solution

$PV = 10{,}000 \cdot \dfrac{1 - (1.07)^{-20}}{0.07} = 10{,}000 \times 10.5940$

≈ $105,940

3. Annuity Due Core

EN: Payments occur at the beginning of each period (e.g. rent). Each cash flow gets one extra period of compounding.
VN: Niên kim đầu kỳ — dòng tiền phát sinh đầu mỗi kỳ.

\[ PV_{\text{due}} = PV_{\text{ord}} \cdot (1 + r) \qquad FV_{\text{due}} = FV_{\text{ord}} \cdot (1 + r) \]

Components / Thành phần

$PV_{ord}$ Ordinary annuity PV (formula 2).
$(1+r)$ Multiplier reflecting one extra period of compounding / hệ số ghép lãi thêm 1 kỳ.

Calculator tip: set [BGN] mode on the BA II Plus.

Practice problem

A lease requires $1,200 paid at the beginning of each month for 24 months. The monthly discount rate is 0.5%. What is the present value of the lease?

Show solution

$PV_{ord} = 1{,}200 \cdot \dfrac{1 - (1.005)^{-24}}{0.005} = 1{,}200 \times 22.5629$

$PV_{due} = 27{,}075.5 \times 1.005$

≈ $27,210.84

4. Perpetuity Core

EN: An annuity that pays forever. Used for preferred stock and growing-perpetuity equity valuation (Gordon).
VN: Dòng tiền vĩnh viễn — dùng định giá cổ phiếu ưu đãi, công thức Gordon.

\[ PV_{\text{perp}} = \frac{A}{r} \quad \text{(level)} \qquad PV = \frac{D_1}{r - g} \quad \text{(growing, } r > g\text{)} \]

Components / Thành phần

$A$ Constant periodic payment.
$D_1$ First-period cash flow (next year).
$g$ Constant growth rate, $g < r$.

Practice problem

A preferred stock pays a $5 annual dividend in perpetuity. Investors require 8%. What is its fair value?

Show solution

$PV = \dfrac{5}{0.08}$

= $62.50

5. Stated Annual Rate vs Effective Annual Rate (EAR) Core

EN: The EAR converts a nominal/stated rate compounded $m$ times per year into the equivalent annual rate.
VN: EAR đổi lãi suất danh nghĩa ghép $m$ lần/năm sang lãi suất hiệu dụng năm.

\[ EAR = \left(1 + \frac{r_s}{m}\right)^{m} - 1 \]

Components / Thành phần

$r_s$ Stated annual rate / Lãi suất danh nghĩa năm.
$m$ Number of compounding periods per year (12 = monthly, 4 = quarterly, 365 = daily).
$EAR$ Effective annual rate.

Practice problem

A credit card has a 18% stated annual rate compounded monthly. What is the effective annual rate?

Show solution

$EAR = (1 + 0.18/12)^{12} - 1 = (1.015)^{12} - 1$

≈ 19.56%

6. Continuous Compounding Core

EN: Limit as $m \to \infty$. Used in option-pricing models (Black-Scholes).
VN: Giới hạn khi $m \to \infty$ — dùng trong mô hình định giá option.

\[ EAR_{cc} = e^{r_s} - 1 \qquad FV = PV \cdot e^{r_s \cdot N} \]

Components / Thành phần

$e$ ≈ 2.71828, base of natural log.
$r_s$ Continuously compounded stated rate.
$N$ Time in years (can be fractional).

Practice problem

A stated annual rate of 8% is continuously compounded. What is the EAR?

Show solution

$EAR = e^{0.08} - 1 = 1.0833 - 1$

≈ 8.33%

7. Bond Price (Cash-Flow Additivity) Core

EN: A bond's price is the sum of the PV of every coupon and the par redemption — direct application of the additivity principle.
VN: Giá trái phiếu = tổng PV các coupon + PV mệnh giá đáo hạn.

\[ P_0 = \sum_{t=1}^{N} \frac{C}{(1 + r)^{t}} + \frac{F}{(1 + r)^{N}} \]

Components / Thành phần

$C$ Periodic coupon = coupon rate × face.
$F$ Face / par value paid at maturity.
$r$ Periodic yield-to-maturity (YTM).
$N$ Number of coupon periods to maturity.

Practice problem

A 3-year, 5% annual-coupon bond with face $1,000 is priced to yield 4%. What is its price?

Show solution

$P = \dfrac{50}{1.04} + \dfrac{50}{1.04^{2}} + \dfrac{1{,}050}{1.04^{3}}$

$= 48.08 + 46.23 + 933.45$

≈ $1,027.75 (premium because coupon > YTM)