Summary and Formulas

Zero-Coupon Bonds

Let:

$F$ : the face value of the bond
$T$ : the time to maturity
$y$ : the yield
$n$ : the number of compounding periods per year

Then the price of a zero-coupon bond is given by:

Discrete Compounding

P V = \frac{F}{( 1 + \frac{y}{n} ) ^{n T}}

Continuous Compounding

P V = F e^{- y T}

Continuous Compounding with Non-Constant Yield

P V = F e^{- \int_{0}^{T} y (t) d t}

Effective annual yield

Consider we want to know the actual amount of interest we will receive in a year with nominal rate $y$ and compounding $n$ times per year:

y_{e} = (1 + \frac{y}{n})^{n} - 1

Observe that Euler’s number appears:

n \to \infty lim (1 + \frac{y}{n})^{n} = e^{y}

Thus given a nominal yield $y$ (and no knowledge of $n$ ) we can construct an upper bound on effective annual yield $y_{e}$ as such:

y_{e} \leq e^{y} - 1

As the maximum possible yield will be when we have continuous compounding.

Annuities

Annuity : a sequence of equally spaced and equally sized cashflows. If paid yearly then:

P V = t = 1 \sum T \frac{C}{( 1 + y ) ^{t}} = C \cdot [\frac{1 - ( 1 + y ) ^{- T}}{y}]

if paid $n$ times a year with $\frac{C}{n}$ each time then:

P V = t = 1 \sum T \frac{C / n}{( 1 + \frac{y}{n} ) ^{n t}} = \frac{C}{n} \cdot [\frac{1 - ( 1 + \frac{y}{n} ) ^{- n T}}{\frac{y}{n}}]

this simplifies to the yearly formula if $n = 1$ . Future value given by:

F V = t = 1 \sum T \frac{C}{n} \cdot (1 + \frac{y}{n})^{n (T - t)} = \frac{C}{n} \cdot [\frac{( 1 + \frac{y}{n} ) ^{n T} - 1}{\frac{y}{n}}]

Coupon-Paying Bond

A (coupon-paying) bond is characterised by a promise to pay

$(i)$ an annuity cashflow of coupon payments C,
$(ii)$ the face value F at maturity.

Compounding is typically semiannual, i.e. $n = 2$ : the value $B$ is

B = (i) Annuity \frac{C}{n} \cdot [\frac{1 - ( 1 + \frac{y}{n} ) ^{- n T}}{\frac{y}{n}}] + (ii) Face Value Zero-Coupon \frac{F}{( 1 + \frac{y}{n} ) ^{n T}}

Perpetuity

Consider the case $T = \infty$
This is called a perpetuity/preference share
It pays an annuity forever
Assume n = 1. Then we have:

P V = t = 1 \sum T \frac{C}{( 1 + y ) ^{t}} = C \cdot [\frac{1 - ( 1 + y ) ^{- T}}{y}] T \to \infty \frac{C}{y}

n doesn’t matter it will cancel out anyway.

Dividend Discount Model (DDM)

Pricing shares creates a perpetuity as we don’t expect the business to dissolve anytime soon:

P V = t = 1 \sum \infty \frac{D _{t}}{( 1 + k ) ^{t}} = \frac{D}{k}

where $k$ is the required rate of return and

$D_{t}$ is the dividend payment at time $t$

Now suppose you will sell the share in T years, then you will also receive the selling price at time T
Then the selling price would be:

P_{T} = \frac{D _{T + 1}}{1 + k} + \frac{D _{T + 2}}{( 1 + k ) ^{2}} + \dots

which can be factorised like this

= (1 + k)^{T} \cdot (\frac{D _{T + 1}}{( 1 + k ) ^{T + 1}} + \frac{D _{T + 2}}{( 1 + k ) ^{T + 2}} + \dots)

Therefore we can rewrite the PV (assuming we are selling at time $T$ ) as:

P V = \frac{D _{1}}{1 + k} + \dots + \frac{D _{T} + P _{T}}{( 1 + k ) ^{T}}

Noting that this is the case where dividends are paid annually, need to adjust for semiannual or arbitrary $n$ values.

Constant Growth DDM Model

let the constant growth rate be $g$

D_{1} = D_{0} (1 + g), D_{2} = D_{0} (1 + g)^{2}, \dots, D_{t} = D_{0} (1 + g)^{t}

if $g < k$ then by geometric series we have

P V = t = 1 \sum \infty \frac{D _{0} ( 1 + g ) ^{t}}{( 1 + k ) ^{t}} = D_{0} t = 1 \sum \infty \frac{( 1 + g ) ^{t}}{( 1 + k ) ^{t}} = D_{0} t = 1 \sum \infty (\frac{1 + g}{1 + k})^{t} = D_{0} \cdot \frac{1 + g}{k - g}

Net Present Value (NPV) Project Evaluation

Some notation:

$I O :$ the initial outlay or project cost. Can come from Debt and Equity markets
$k :$ the required rate of return or cost of capital.
$T :$ the forecast timespan of the project in years.
$C_{1}, \dots, C_{T}$ is the discrete net cashflow stream at times $1, \dots, T$ .
$C (t)$ : the continuously received net cashflow stream.

As the company will have to make bank loan repayments, bond coupon payments, dividends etc there must be a minimum value we need to earn before we make profit. We can compute required rate of return (i.e. cost of capital) $k$ as the Weighted Average cost of the finance sources.

i \in sources \sum y_{i} \cdot amount_{i}

in practice other factors will need to be accounted for like inflation, risk-premium etc.

Net present value rule is:

NP V = P V - I O

Invest in the project if NPV >0.
Don’t invest if NPV <0.

DCF Valuation: Risky Cashflows

The expected discrete cashflow at time t is:

E [C_{t}] = m = 1 \sum M_{t} p_{t m} C_{t m}

Cyrus

Explorer