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Home/Financial Mathematics/Course 2

Course 2

Foundation

5-6 hours

Financial Markets

Time value of money, bond valuation, stock valuation, and market instruments

Learning Objectives

•Master the time value of money concepts and derive compounding formulas
•Understand bond pricing, yield to maturity, duration, and convexity
•Learn stock valuation models including dividend discount and free cash flow approaches
•Apply these concepts to value real-world securities

1. Time Value of Money

The fundamental principle: a dollar today is worth more than a dollar tomorrow. This reflects opportunity cost - money today can be invested to earn returns. Time value of money (TVM) is central to all financial valuation.

Definition 1.1: Future Value

The future value (FV) of an amount $PV$ invested today at interest rate $r$ for $n$ periods with annual compounding is:

FV = PV \times (1 + r)^n

The factor $(1+r)^n$ is called the future value interest factor (FVIF).

Theorem 1.1: Derivation of Compound Interest Formula

Starting with principal $P$ at rate $r$ per period:

After 1 period: $P_1 = P(1+r)$
After 2 periods: $P_2 = P_1(1+r) = P(1+r)^2$
After 3 periods: $P_3 = P_2(1+r) = P(1+r)^3$

By induction, after $n$ periods:

FV_n = P(1+r)^n

Definition 1.2: Present Value

The present value (PV) is the current worth of a future cash flow $FV$ received in $n$ periods at discount rate $r$ :

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

This is simply the inverse of the FV formula. The factor $(1+r)^{-n}$ is the present value interest factor (PVIF).

Proposition 1.1: Continuous Compounding

As compounding frequency $m$ increases, the future value approaches:

FV = \lim_{m \to \infty} PV \left(1 + \frac{r}{m}\right)^{mn} = PV \cdot e^{rn}

where $e = \lim_{m \to \infty}(1 + 1/m)^m \approx 2.71828$ is Euler's number.

Proof of Continuous Compounding Formula:

Let $k = m/r$ . Then $m = kr$ and:

\lim_{m \to \infty} \left(1 + \frac{r}{m}\right)^{mn} = \lim_{k \to \infty} \left(1 + \frac{1}{k}\right)^{krn}

= \lim_{k \to \infty} \left[\left(1 + \frac{1}{k}\right)^{k}\right]^{rn} = e^{rn}

using the definition $e = \lim_{k \to \infty}(1 + 1/k)^k$ .

∎

Definition 1.3: Effective Annual Rate

The effective annual rate (EAR) is the actual annual rate earned after accounting for compounding within the year:

EAR = \left(1 + \frac{r_{\text{nominal}}}{m}\right)^m - 1

where $r_{\text{nominal}}$ is the stated annual rate and $m$ is the compounding frequency per year.

Example 1.1: Comparing Interest Rates

Compare these three investments:

Option A: 12% APR, annual compounding
Option B: 11.8% APR, monthly compounding
Option C: 11.6% APR, continuous compounding

Calculate EARs:

EAR_A = (1 + 0.12)^1 - 1 = 12.00\%

EAR_B = (1 + 0.118/12)^{12} - 1 = (1.009833)^{12} - 1 = 12.48\%

EAR_C = e^{0.116} - 1 = 1.1230 - 1 = 12.30\%

Ranking: Option B (12.48%) > Option C (12.30%) > Option A (12.00%). More frequent compounding increases the effective rate.

Definition 1.4: Annuities

An annuity is a sequence of equal cash flows $C$ at regular intervals. For an ordinary annuity (payments at end of period), the present value is:

PV_{annuity} = C \times \frac{1 - (1+r)^{-n}}{r}

Proof of Annuity Formula:

An $n$ -period annuity with payment $C$ has present value:

PV = \frac{C}{1+r} + \frac{C}{(1+r)^2} + \cdots + \frac{C}{(1+r)^n} = C \sum_{t=1}^n (1+r)^{-t}

This is a geometric series with first term $a = (1+r)^{-1}$ and ratio $q = (1+r)^{-1}$ :

\sum_{t=1}^n q^t = q \cdot \frac{1 - q^n}{1 - q} = \frac{(1+r)^{-1}[1 - (1+r)^{-n}]}{1 - (1+r)^{-1}}

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

Therefore: $PV = C \times \frac{1 - (1+r)^{-n}}{r}$

∎

Definition 1.5: Perpetuity

A perpetuity is an annuity that continues forever. Taking the limit as $n \to \infty$ :

PV_{perpetuity} = \lim_{n \to \infty} C \times \frac{1 - (1+r)^{-n}}{r} = \frac{C}{r}

since $\lim_{n \to \infty} (1+r)^{-n} = 0$ for $r > 0$ .

Corollary 1.1: Growing Perpetuity

If payments grow at rate $g$ per period ( $C_1, C_1(1+g), C_1(1+g)^2, \ldots$ ), the present value is:

PV = \frac{C_1}{r - g}, \quad \text{provided } r > g

This formula is fundamental to the Gordon Growth Model for stock valuation.

2. Bond Valuation

Definition 2.1: Bond Pricing

A bond with face value $F$ , coupon rate $c$ , maturity $n$ periods, and yield $y$ has price:

P = \sum_{t=1}^n \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n}

where $C = c \times F$ is the coupon payment per period.

Theorem 2.1: Bond Pricing Formula - Closed Form

The bond price can be expressed as:

P = C \times \frac{1-(1+y)^{-n}}{y} + F(1+y)^{-n}

The first term values the coupon annuity; the second term values the face value payment.

Example 2.1: Bond Trading at Par, Premium, Discount

Consider a bond with $F = \$1,000$ , annual coupon rate $c = 6\%$ , maturity $n = 5$ years.

Case 1: Trading at par (yield = coupon rate = 6%)

P = 60 \times \frac{1-(1.06)^{-5}}{0.06} + 1000(1.06)^{-5} = 252.74 + 747.26 = \$1,000

Case 2: Trading at premium (yield = 5% < coupon rate)

P = 60 \times \frac{1-(1.05)^{-5}}{0.05} + 1000(1.05)^{-5} = 259.77 + 783.53 = \$1,043.30

Case 3: Trading at discount (yield = 7% > coupon rate)

P = 60 \times \frac{1-(1.07)^{-5}}{0.07} + 1000(1.07)^{-5} = 246.01 + 712.99 = \$959.00

Rule: When $y > c$ , bond trades at discount. When $y < c$ , bond trades at premium.

Definition 2.2: Yield to Maturity (YTM)

The yield to maturity is the discount rate $y$ that makes the bond's present value equal to its market price:

P_{market} = \sum_{t=1}^n \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n}

YTM must be solved numerically (e.g., Newton-Raphson method) as it's a polynomial equation of degree $n$ .

Definition 2.3: Macaulay Duration

The Macaulay duration is the weighted average time to receive cash flows:

D = \frac{\sum_{t=1}^n t \cdot PV(CF_t)}{P}

where $PV(CF_t)$ is the present value of the cash flow at time $t$ .

Theorem 2.2: Duration and Price Sensitivity

The modified duration measures the percentage price change for a small yield change:

D^* = \frac{D}{1+y}

For small yield changes:

\frac{dP}{P} \approx -D^* \cdot dy

Proof of Duration Formula:

Differentiate the bond pricing equation with respect to yield:

\frac{dP}{dy} = \sum_{t=1}^n C \cdot (-t)(1+y)^{-t-1} + F \cdot (-n)(1+y)^{-n-1}

= -\frac{1}{1+y}\left[\sum_{t=1}^n t \cdot C(1+y)^{-t} + n \cdot F(1+y)^{-n}\right]

Therefore:

\frac{dP}{dy} = -\frac{1}{1+y} \sum_{t=1}^n t \cdot PV(CF_t) = -\frac{D \cdot P}{1+y}

Rearranging:

\frac{dP}{P} = -\frac{D}{1+y} dy = -D^* \cdot dy

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Definition 2.4: Convexity

Convexity measures the curvature of the price-yield relationship (second-order effect):

C = \frac{1}{P} \sum_{t=1}^n \frac{t(t+1) \cdot CF_t}{(1+y)^{t+2}}

A more accurate price change formula includes convexity:

\frac{\Delta P}{P} \approx -D^* \cdot \Delta y + \frac{1}{2}C \cdot (\Delta y)^2

3. Stock Valuation

Definition 3.1: Dividend Discount Model (DDM)

The value of a stock equals the present value of all future dividends:

V_0 = \sum_{t=1}^\infty \frac{D_t}{(1+r)^t}

where $D_t$ is the dividend at time $t$ and $r$ is the required return (discount rate).

Theorem 3.1: Gordon Growth Model

If dividends grow at constant rate $g$ forever, with $D_t = D_0(1+g)^t$ , then:

V_0 = \frac{D_1}{r-g} = \frac{D_0(1+g)}{r-g}

This requires $r > g$ for convergence.

Proof of Gordon Growth Model:

Starting with the DDM:

V_0 = \sum_{t=1}^\infty \frac{D_0(1+g)^t}{(1+r)^t} = D_0 \sum_{t=1}^\infty \left(\frac{1+g}{1+r}\right)^t

This is a geometric series with first term $a = \frac{1+g}{1+r}$ and ratio $q = \frac{1+g}{1+r}$ :

\sum_{t=1}^\infty q^t = \frac{q}{1-q} = \frac{\frac{1+g}{1+r}}{1 - \frac{1+g}{1+r}} = \frac{1+g}{r-g}

Therefore:

V_0 = D_0 \cdot \frac{1+g}{r-g} = \frac{D_0(1+g)}{r-g} = \frac{D_1}{r-g}

∎

Example 3.1: Two-Stage Dividend Growth Model

A stock currently pays $D_0 = \$2$ . Dividends grow at 15% for 3 years, then 5% forever. Required return is 12%.

High-growth phase dividends:

D_1 = 2(1.15) = 2.30, \quad D_2 = 2.30(1.15) = 2.645, \quad D_3 = 2.645(1.15) = 3.042

Terminal value at end of year 3:

V_3 = \frac{D_4}{r-g} = \frac{3.042(1.05)}{0.12-0.05} = \frac{3.194}{0.07} = 45.63

Total present value:

V_0 = \frac{2.30}{1.12} + \frac{2.645}{1.12^2} + \frac{3.042}{1.12^3} + \frac{45.63}{1.12^3}

= 2.054 + 2.108 + 2.166 + 32.479 = \$38.81

Proposition 3.1: Relationship Between Growth and Value

From the Gordon model, we can derive:

r = \frac{D_1}{P_0} + g

This shows that required return equals dividend yield plus growth rate. Rearranging:

P_0 = \frac{D_1}{r-g}

Higher growth $g$ increases value, while higher required return $r$ decreases value.

Definition 3.2: Price-Earnings Ratio

The P/E ratio compares stock price to earnings per share:

\text{P/E} = \frac{P_0}{EPS_1}

Using the Gordon model with payout ratio $b = D_1/EPS_1$ :

\text{P/E} = \frac{b}{r-g}

where $g = ROE \times (1-b)$ is the sustainable growth rate.

Remark 3.1: Limitations of DDM

The dividend discount model has several limitations:

Many companies don't pay dividends (growth stocks)
Dividend policy can be unstable
Very sensitive to growth rate assumption
Assumes constant growth (unrealistic for many firms)

Alternative models include Free Cash Flow to Equity (FCFE) and discounted cash flow (DCF) analysis.

Key Takeaways

1.Time value of money reflects opportunity cost and compounds exponentially
2.Bond prices move inversely with interest rates; duration measures sensitivity
3.Stock value equals present value of expected future dividends or free cash flows

Frequently Asked Questions

What is the difference between simple and compound interest?

Simple interest is calculated only on the principal: I = Prt. Compound interest is calculated on principal plus accumulated interest: FV = P(1+r)^t. Compound interest grows exponentially while simple interest grows linearly.

Why does continuous compounding use e^rt?

Continuous compounding is the limit as compounding frequency approaches infinity. Taking lim(n→∞) of (1 + r/n)^(nt) gives e^(rt), where e is Euler's number ≈ 2.71828.

What is the relationship between bond prices and interest rates?

Bond prices and interest rates are inversely related. When market interest rates rise, existing bonds with lower coupon rates become less valuable, causing their prices to fall. This inverse relationship is fundamental to fixed-income investing.

How is duration different from maturity?

Maturity is the time until the bond's final payment. Duration is the weighted average time to receive all cash flows, where weights are the present values of each payment. Duration measures interest rate sensitivity.

Why might a stock's value differ from its price?

Intrinsic value (from models like DDM) represents fundamental worth based on expected cash flows. Market price reflects supply and demand, which can be influenced by sentiment, information asymmetry, and behavioral factors. Value ≠ Price creates trading opportunities.

Practice Quiz

Financial Markets

Questions

Correct

Accuracy

You invest $1,000 at 6% annual interest compounded semi-annually for 3 years. What is the future value?

Easy

Not attempted

What is the present value of $5,000 received in 10 years with continuous compounding at 8%?

Medium

Not attempted

A bond with face value $1,000, 5% coupon (annual), 5 years to maturity trades at 6% YTM. What is its price?

Medium

Not attempted

A stock pays a dividend of $3 today. Dividends grow at 4% forever. Required return is 10%. What is the stock value?

Easy

Not attempted

What is the effective annual rate (EAR) for 12% APR compounded monthly?

Easy

Not attempted

A perpetuity pays $100 annually with first payment in 1 year. If discount rate is 8%, what is its value?

Easy

Not attempted

A 3-year bond has Macaulay duration of 2.8 years. If yield increases from 5% to 5.5%, what is the approximate percentage price change?

Hard

Not attempted

A stock has EPS = $5, payout ratio = 40%, ROE = 15%, required return = 12%. Using constant growth DDM, what is its value?

Hard

Not attempted

You will receive $1,000 at end of each year for 5 years. Discount rate is 6%. What is the present value?

Medium

Not attempted

A zero-coupon bond matures in 8 years with face value $1,000. If YTM is 7%, what is its price?

Easy

Not attempted

A bond has a duration of 6 years and convexity of 80. If yield decreases by 1%, what is the total percentage price change?

Hard

Not attempted

A stock pays dividends of

2 in year 1,

2.20 in year 2, then grows at 5% forever. Required return is 10%. What is its value?

Hard

Not attempted