CHAPTER 1
Economical and financial background
1
CHAPTER 2
Probability background and Modern Portfolio
Theory
1. Randomness and Probability
The way randomness is typically viewed in science and engineering is as a
mathematical way of modelling complexity or our lack of information. What is for
example random in the most archetypical of randomness, namely when flipping a
coin? In principle, we decide to hit the coin with our thumb with a certain force and
at a certain angle, after which the flight path and spin of the coin is well described
by Newton’s equations of motion, and fully deterministic (not random). The reason
we think that the outcome is random is that this is in practice, too complicated for
us to compute or predict: there is uncertainty in how hard we end up striking the
coin, and this translates into uncertainty in how the coin will fly.
There are endless examples of problems in science and engineering as well as
everyday life where we have lack of information or too complicated a system to
fully understand (weather, financial markets, how the Covid-19 virus will spread,
...), but we still have to make some kind of decisions on how to behave (should
I carry an umbrella today, should I invest in a company, should society go into
lockdown or not, ...). Ideally, we would like some kind of scientifically justifiable
method to guide us when making decisions about such matters. Probability theory
provides a mathematically precise way of modeling lack of information, and then
quantitatively estimating risk and likelihood. One should always remember though,
the mathematical theory is a simplification and idealization. When we look at
the weather forecast and it says that in two weeks, with a chance of 70%, the
temperature is between -3 and 3 degrees Celsius, this is an educated guess based
on a model that meteorologists have come up with, based on their beliefs and
experience, it is not some universal truth.
In this section we introduce probability. To this end we work with a finite set
Ω which we call the set of all possible outcomes. We introduce the notion of a
probability, random variable, expectation and variance.
1.1. Experiments and set of possible outcomes.
Probability theory deals with mathematical models of situations depending
on chance. We shall call such a situation an experiment. Thus we use the word
experiment as a technical term, with a meaning different from the everyday use
of the word. For instance, in our sense, observing how long one has to wait for
the departure of an airplane is an experiment. We call everything an experiment
which may have different outcomes or scenarios (i.e. mutually exclusive results),
and which of them occurs depends on chance.
3
4 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
Definition 1.1: The set of possible outcomes Ω
To every experiment there corresponds a nonempty set Ω = {ω1, . . . , ωn},
n ≥ 2, the set of possible outcomes of the experiment, and its elements, ωi,
will be called outcomes of the experiment.
Example 1.1
Let an experiment consist in throwing a dice. Then, the set Ω of outcomes
of this experiment is Ω = {1, 2, 3, 4, 5, 6}.
Definition 1.2: Event
An event is a set A ⊆ Ω.
If Φ is a statement which is correct for some ω ∈ Ω, then we define the set
A := {ω ∈ Ω : Φ(ω) is true}
of those outcomes where the statement is true, A is an event.
Example 1.2
Let us return to the experiment of throwing a dice. Getting the number 2
is an event, it is the set {2}. Getting an odd number is an event, it is the
set A := {1, 3, 5}.
Definition 1.3
Let Ω = {ω1, . . . , ωn}, n ≥ 2, be a finite set and P(Ω) the power set of Ω
(the set of all subsets of Ω). A probability is an application
P : P(Ω) −→ [0, 1]
A 7→ P (A)
that satisfies:
(i) P (Ω) = 1,
(ii) For any sequence of pairwise disjoint (Ai ∩ Aj = ∅ for i ̸= j)
events (Ai)ni=1
P (∪ni=1Ai) =
n∑
i=1
P (Ai).
Example 1.3: Example of P(Ω)
Let Ω = {1, 2, 3}, then P(Ω) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.
2. RISK AND RETURN 5
Exercise 1.1
Let P be a probability. Prove (using (ii) in the previous definition) that for
any A ⊆ Ω,
P (A) =
∑
ω∈A
P ({w}).
1.2. Random variables, expectation and variance.
Let Ω be a finite set and P be a probability (on P(Ω)).
Definition 1.4
A random variable is simply any function X : Ω→ R.
Definition 1.5
Let X be a random variable on Ω. Then X takes on only a finite number
of distinct values x1, . . . , xm (because Ω is a finite set).
1) The expectation of X is
E[X] :=
m∑
k=1
xkP (X = xk),
where P (X = xk) = P ({ω ∈ Ω : X(ω) = xk}).
2) The variance of X is
Var[X] := E[(X − E[X])2] =
m∑
k=1
(xk − E[X])2P (X = xk).
Proposition 1.1
Let X,Y be two random variables on Ω, then
1) ∀a, b ∈ R, E[aX + bY ] = aE[X] + bE[Y ] (Linearity).
2) Let a, b ∈ R, Var[aX + b] = a2Var[X].
Proof
The proof is left as an exercise to be solved in class.
Remark 1.1
Let c ∈ R, then E[c] = c.
2. Risk and return
The basic premise of an investor is that they like returns and dislike risk. An
investor will purchase a financial asset because they want to increase their wealth,
6 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
i.e., to earn a positive rate of return on their investment. Due to uncertainty, they
do not know what rate of return their investment will bring.
In finance, people will invest in risky assets only if they expect to receive higher
returns. When evaluating potential investments in financial assets, there are two
dimensions of the decision making process namely expected rate of return and risk.
2.1. Investment returns.
Returns. The concept of return gives to investors an appropriate way of
expressing the financial performance of an investment. There are two concepts of
returns:
(1) The currency (GBP, USD, EURO,...) return which is given by
Currency return = Amount received - Amount invested.
(2) Rates of return, or percentage returns which is given by
Rate of return = Amount received − Amount investedAmount invested × 100%
= Currency returnAmount invested × 100%.
Investment in shares. Shares, also known as equities, provide you with part-
ownership of a company so when you invest in shares; you are buying ’a share’ of
that business. Companies issue shares to raise money and investors buy shares in
a business because they believe the company will do well and they want to ’share’
in its success.
Dividends. Shareholders can be financially rewarded in two primary ways;
firstly through the appreciation of value in a share’s price and, secondly, through
dividends. Dividends are amounts of monies given to shareholders from company
profits.
The rate of return r from an investment in a share over time T whose beginning
and ending share price are S(0) and S(T ) respectively, which paid a dividend of D
at T , is
r = S(T )− S(0) +D
S(0) × 100%.
In this case, the value S(T )−S(0) represents the currency return in the investment
through appreciation in the share price. The amount D is comparable to interest
received from a bank account. The rate of return for the dividend income only is
D/S(0)× 100% over time T .
Example 2.1
Suppose you buy 10 shares of a stock for £100. At the end of one year, you
sell the 10 shares for £110 after receiving a dividend of £3. What is the
currency return on your £100 investment? What is the rate of return?
Solution
(1) The currency return is: £110−£100 = £10.
(2) The rate of return is:
£110−£100 +£3
£100 × 100% = 13%.
Mathematical interpretation
2. RISK AND RETURN 7
(1) A negative rate of return indicates the original investment was not recov-
ered.
(2) Rates of return are a better measure of relative returns. A £10 return
on a £100 investment is a good return, but a £30 return on a £3,000
investment is poor.
(3) Taking into account the timing of the return, a £50 return from a £100
investment is a quite good return if it occurs after one year, however, the
same return after 20 years would not be so good.
2.2. Measuring individual asset return.
The return on an investment will depend on the outcome of a series of future
events. For example, the returns on banking shares will depend on economic factors
such as individual and commercial activity, political aspects, global markets and so
on. The outcome of key future events will determine individual returns. We use
the concepts of events and probability distributions to define and measure risks and
rewards.
Definition 2.1
Given a probability distribution of rate of returns, the expected rate of
return on the individual stock or investment is defined as:
rˆ = E[r] = P1r1 + . . .+ Pnrn =
n∑
i=1
Piri,
where
• rˆ = E[r] = the expected rate of return on the stock,
• n = the number of states (events) used to determine rates of return,
• Pi = the probability of state i occurring, and
• ri = the rate of return on the investment in state i.
This weighted average is viewed as the expected reward for investing in an
investment.
Example 2.2
The table below provides a probability distribution for the rates of return
on stocks A,B,C and D. What is the expected rate of return (r.o.r.) on
stocks A,B,C and D?
State Probability r.o.r. on A r.o.r. on B r.o.r. on C r.o.r. on D
1 0.10 15% 10% 25% 25%
2 0.20 20% 40% 15% 50%
3 0.40 5% 30% 5% 20%
4 0.30 10% −10% 10% 65.4%
In this example of probability distribution, there are four possible states
of the world. For example, each state may represent the behaviour of the
economy in the UK.
8 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
Solution
The expected rate of return on stock A, stock B, stock C and stock D are
given by:
rˆA = E[rA] = P1r1 + P2r2 + P3r3 + P4r4
= 0.10× 15% + 0.20× 20% + 0.40× 5% + 0.30× 10%
= 10.5%
and similarly we get rˆB = E[rB ] = 18%, rˆC = 10.5% and rˆD = 40.12%.
Decision making: Stock D offers a higher expected rate of return than stocks
A, B and C which makes it the most attractive from a return’s point of view.
What about the risk of each stock?
Probability distributions for returns on stocks A, B.
2.3. Measuring individual asset risk.
The phrase ”dispersion of outcomes around the expected value” could be sub-
stituted for the word risk. The word riskier simply means ”more dispersion of
outcomes around the expected value.” In general, the tighter the probability distri-
bution of expected future rates of return, the smaller the risk of a given investment.
The mathematics terms variance and standard deviation measure dispersion of out-
comes around the expected value.
2. RISK AND RETURN 9
Definition 2.2
Given an asset’s probability distribution for its rate of return. The risk
associated to the rate of return r is measured by the standard deviation σ
which is described by the following equation:
σ2 = Var(r) =
n∑
i=1
Pi(ri − rˆ)2, (1)
where
• n = the number of states,
• Pi = the probability of state i,
• ri = the rate of return on the stock in state i, and
• rˆ = the expected rate of return on the stock.
Example 2.3
Consider stocks A, B, C and D given in Example 2.2.
1) Find the variance and the standard deviation on stock A, stock B, stock
C and stock D.
2) Using the knowledge of the standard deviations associated to the rates of
return along with the expected rate of return of the investments, how does
the investor choose between the investments?
Solution
1) The risks associated to stocks A, B, C and D are as follows.
σ2A = 0.1(0.15− 0.105)2 + 0.2(0.2− 0.105)2 + 0.4(0.05− 0.105)2
+0.3(0.1− 0.105)2 = 0.0032.
So, σA = 0.0566 (or 5.66%).
Similarly, we get that σB = 19.9%, σC = 6.1% and σD = 19.9%.
2) Interpreting these values:
• If the investments have the same expected rate of return then the
investor should choose the investment with lower risk (standard
deviation).
• If the investments have the same standard deviation, then the in-
vestor should choose the investment with the higher expected rate
of return.
2.4. Coefficient of Variation.
The coefficient of variation allows investors to take a balanced viewed by fac-
toring in risk relative to reward. It allows investor to decide whether risks are
sufficiently covered by rewards.
10 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
Definition 2.3
In probability theory and statistics, the coefficient of variation is defined as
a normalized measure of dispersion of a probability distribution. It shows
the risk per unit of rate of return. Mathematically, this is defined by
CV = σ
rˆ
(2)
Example 2.4
1) Find the coefficient of variation of stock A, stock B, stock C and
stock D in Example 2.2.
2) What conclusions can you draw?
Solution
It follows from Equation (2) that
1) CVA = 5.6610.5 = 0.5390, CVB =
19.9
18 = 1.1055, CVC =
6.1
10.5 =
0.5809, CVD = 19.940.12 = 0.4960.
2) From these computations, we see that stock A has more risk per
unit of rate of return than stock D in spite of the fact that stock
D’s standard deviation is larger. Hence, it is possible that some
investors would consider stock D as less risky.
2.5. Rate of return in a portfolio.
In the preceding section, we considered the risk of individual assets. Most
investors do not hold stocks in isolation, they choose to hold a portfolio of several
stocks. In this section, we will analyse the risk of assets held in portfolios and
see that, an asset held as part of a portfolio is less risky than the same asset held
individually.
Consider an investment in a portfolio with n assets with total amount invested
V . If Vi is the amount of money invested in asset i, then its weight in the overall
portfolio is ωi = Vi/V . By viewing the mix of assets in the portfolio in this way,
we note ω1 + . . .+ ωn = 1.
Proposition 2.1
The expected rate of return on a portfolio, is the weighted average of the ex-
pected rate of returns on the individual assets which comprise the portfolio.
This can be expressed as follows:
rˆp =
n∑
i=1
ωirˆi = ω1rˆ1 + . . .+ ωnrˆn, (3)
where
• rˆp = the expected rate of return on the portfolio,
• n = the number of assets in the portfolio,
• ωi = the monetary proportion of the portfolio invested in asset i,
• rˆi = the expected rate of return on stock i.
2. RISK AND RETURN 11
Proof
The proof is left as an exercise to be solved in class.
Exercise 2.1
Consider stocks A and B from Example 2.2. Let p1 be the portfolio that
consists of 60% stock A and 40% stock B. Calculate rˆp1 .
2.6. Portfolio Risk.
There are two measures for studying the simultaneous changes in the rate of
returns of two assets: the covariance and the correlation coefficient.
Definition 2.4: Covariance
LetX,Y be two random variables, their covariance is defined as the expected
value of the product of their deviations from their individual expected values.
Cov(X,Y ) = σX,Y := E[(X − E(X))(Y − E(Y ))] = E[XY ]− E(X)E(Y ).
The covariance measures simultaneous changes in the random variables X and
Y . The units of measurement of the covariance are those of X times those of Y.
Exercise 2.2
Let a, b be two constants. From the previous definition prove that:
• Cov(X, a) = 0, Cov(X,X) = V ar(X), Cov(X,Y ) = Cov(Y,X),
• Cov(aX, bY ) = a bCov(X,Y ), Cov(a+X, b+ Y ) = Cov(X,Y ).
Definition 2.5: Correlation coefficient
Given a pair of random variables X and Y the correlation coefficient is
defined by
Corr(X,Y ) = ρX,Y =
Cov(X,Y )
σXσY
,
where Cov is the covariance and σX , σY are the standard deviations of X
and Y respectively.
In the context of modern portfolio theory, the covariance between the rate of
returns on a pair of assets r1 and r2, each viewed as a (discrete) random variable,
can be expressed as follows:
Cov(r1, r2) = σ12 =
n∑
i=1
Pi(r1i − rˆ1)(r2i − rˆ2), (4)
where
• σ12 = the covariance between the rate of returns on assets 1 and 2,
• n = the number of states,
• Pi = the probability of state i,
• r1i and r2i are the rate of return on assets 1 and 2 respectively in state i,
• rˆ1 and rˆ2 are the expected rate of returns on assets 1 and 2 respectively.
12 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
The covariance measures simultaneous changes in the rate of returns of assets.
From the point of view of modern portfolio theory, the correlation coefficient
between the rate of returns on a pair of assets can be expressed as follows:
Corr(ri, rj) = ρij =
σij
σiσj
, (5)
where
• σij = the covariance between the rate of returns on assets i and j,
• σi, σj = the standard deviations on the rate of returns on assets i and j
respectively.
Interpretation of the correlation
• −1 ≤ ρij ≤ +1.
• The correlation coefficient indicates the degree of linear dependence be-
tween the rates of returns.
• If ρij > 0, ri and rj tend to move in the same direction while if ρij < 0
they tend to move in opposite directions.
• Two assets are perfectly positively correlated if ρij = 1. In this case, the
rates of returns will have a linear relationship with positive gradient.
• Two assets are perfectly negatively correlated if the correlation coefficient
ρij = −1. In this case, the rates of returns will have a linear relationship
with a negative gradient.
• A correlation coefficient equal to zero means that the two rates of re-
turns do not have a linear relationship, we say that the two assets are
uncorrelated.
2.7. Portfolio variance.
Proposition 2.2
The variance of the portfolio is given by:
σ2p =
n∑
i=1
n∑
j=1
wiwjCov (ri, rj) (6)
where
• σ2p = the variance on the portfolio,
• n = the number of assets in the portfolio,
• wi = the proportion of the portfolio invested in asset i, and
• Cov (ri, rj) = the covariance between the rate of returns on assets
i and j.
Proof
Consider a portfolio with two assets. We know the observed portfolio rate
of return, viewed as a random variable, is rp = w1r1 + w2r2 and we know
from (3) that the expected portfolio rate of return is rˆp = w1rˆ1 + w2rˆ2.
Using these we compute the variance of returns on the portfolio using the
definition.
2. RISK AND RETURN 13
Proof : Proof continued
σ2p = Var (rp)
= E
[
(rp − rˆp)2
]
(using the definition)
= E
[
(w1 (r1 − rˆ1) + w2 (r2 − rˆ2))2
]
(substituting in)
= E
[
w21 (r1 − rˆ1)2 + 2w1w2 (r1 − rˆ1) (r2 − rˆ2) + w22 (r2 − rˆ2)2
]
(expanding)
= w21E
[
(r1 − rˆ1)2
]
+ 2w1w2E [(r1 − rˆ1) (r2 − rˆ2)]
+ w22 E
[
(r2 − rˆ2)2
]
(using the linearity)
= w21σ21 + 2w1w2Cov (r1, r2) + w22σ22
In the general case with n assets, we know the rate of return of the portfolio
and the expected rate of return are
rp =
n∑
i=1
wiri, rˆp =
n∑
i=1
wirˆi.
Substituting these into the definition for the portfolio variance along with
the algebraic identity
(rp − rˆp)2 =
(
n∑
i=1
wi (ri − rˆi)
)2
=
n∑
i=1
n∑
j=1
wiwj (ri − rˆi) (rj − rˆj) ,
therefore we deduce that
σ2p = Var (rp) = E
[
(rp − rˆp)2
]
= E
[( n∑
i=1
wi (ri − rˆi)
)2 ]
= E
[ n∑
i=1
n∑
j=1
wiwj (ri − rˆi) (rj − rˆj)
]
=
n∑
i=1
n∑
j=1
wiwjE
[
(ri − rˆi) (rj − rˆj)
]
=
n∑
i=1
n∑
j=1
wiwjCov(ri, rj),
which proves (6).
Understanding portfolio risk. Consider a portfolio with n assets. To com-
pute the portfolio variance we need to know the covariance Cov (ri, rj) of rate of
returns between all pairs of assets. The variance can be calculated by multiplying
the weights in the rows and columns by the entries in the covariance matrix (given
below) and then summing all terms.
Weights w1 . . . wj . . . wn
w1 Cov (r1, r1) . . . Cov (r1, rj) . . . Cov (r1, rn)
...
... . . .
...
...
wi Cov (ri, r1) . . . Cov (ri, rj) . . . Cov (ri, rn)
...
...
... . . .
...
wn Cov (rn, r1) . . . Cov (rn, rj) . . . Cov (rn, rn)
14 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
Example 2.5
Consider three portfolios of stocks A and B given in Example 2.2: The first
portfolio p1 consisting of 50% stock A and 50% stock B, the second portfolio
p2 consisting of 60% stock A and 40% stock B and the third portfolio p3
consisting of 95% stock A and 5% stock B. What is the variance on each
portfolio? What conclusions can you make?
Solution
For a holding of two assets, the portfolio variance is
σ2p = w2Aσ21 + 2wAwBCov (rA, rB) + w2Bσ22
Representing rate of returns using decimals we have
Cov (rA, rB) = 0.1(0.15− 0.105)(0.1− 0.18) + 0.2(0.2− 0.105)(0.4− 0.18)
+ 0.4(0.05− 0.105)(0.3− 0.18) + 0.3(0.1− 0.105)(−0.1− 0.18)
= 0.0016.
Using this we can compute the correlation coefficient
ρAB =
Cov (rA, rB)
σAσB
= 0.00160.0566× 0.199
= 0.1421,
where σA and σB are computed in Example 2.3. This suggests there is a
weak positive correlation between the rate of returns on assets A and B.
The standard deviations on the three portfolios are as follows.
(1) Portfolio consisting of 50% stock A and 50% stock B
σ2p1 = 0.5
2 × 0.0032 + 0.52 × 0.0396 + 2× 0.5× 0.5× 0.0016 = 0.0115
=⇒ σp1 = 0.1072 (or 10.72%)
(2) Similarly, we get from portfolio consisting of 60% stock A and 40%
stock B
σ2p2 = 0.00826
σp2 = 9.09%
(3) Portfolio consisting of 95% stock A and 5% stock B, then we get
σ2p3 = 0.00314
σp3 = 5.61%
using (3), we have also that r̂p3 = 10.88%
Conclusions
• The portfolio consisting of 95% in stock A and 5% in stock B
has a lower standard deviation than either stocks A or B and the
portfolio has a higher expected rate of return than stock A.
• Hence, by forming a portfolio of assets, some of the risk inherent
to the individual assets can be reduced.
2.8. Minimising the portfolio variance.
Graphically representing σ2p as a function of weights. Consider a port-
folio of two assets A and B with known fixed risks and expected rate of returns.
Given wB = 1− wA, we can view the portfolio variance,
σ2p = w2Aσ2A + 2wAwBCov (rA, rB) + w2Bσ2B
2. RISK AND RETURN 15
as a function of wA. A simple substitution exercise gives,
σ2p (wA) = w2A
[
σ2A + σ2B − 2ρA,BσAσB
]
+ wA
[
2ρA,BσAσB − 2σ2B
]
+ σ2B .
The following graph represents y = σ2p(x) as a function of investment weight in
stock A, x = wA.
From the sketch we are able to estimate that the weight corresponding to the
minimum variance is at approximately
(
wA, σ
2
p
)
= (0.95, 0.003) or wA = 0.95 and
σp = 0.05(5%).
Proposition 2.3
For a portfolio consisting of two assets A and B with weights wA and wB re-
spectively, then the weight wA(min) giving the minimum-variance portfolio
is computed as follows:
wA(min) =
σ2B − Cov (rA, rB)
σ2A + σ2B − 2Cov (rA, rB)
,
where
• Cov (rA, rB) = the covariance between the rate of returns on assets
A and B,
• σ2A = the variance on stock A, and
• σ2B = the variance on stock B.
And we have wB(min) = 1− wA(min).
16 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
Proof
The portfolio variance is
σ2p = (wA)
2
σ2A + (wB)
2
σ2B + 2wAwBρABσAσB ,
with wB = 1− wA. Substituting wB into the preceding equality leads to:
σ2p = (wA)
2
σ2A + (1− wA)2 σ2B + 2wA (1− wA) ρABσAσB .
Our purpose is to minimize the variance (as a function of wA). Let us
compute the derivative of σ2p,(
σ2p
)′ (wA) = 2wAσ2A+2(−1) (1− wA)σ2B+2( (1− wA)+(−1)wA)ρABσAσB ,
The minimum-variance weight wA(min) must satisfy
(
σ2p
)′ (
wA(min)
)
= 0
and hence we get
2wA(min)σ2A + 2(−1) (1− wA(min))σ2B + 2
(
(1− wA(min)) +
(−1)wA(min)
)
ρABσAσB = 0.
Therefore,
wA(min)
(
σ2A + σ2B − 2Cov (rA, rB)
)
= σ2B − Cov (rA, rB)
where Cov (rA, rB) = ρABσAσB . Thus, we get
wA(min) =
σ2B − Cov (rA, rB)
σ2A + σ2B − 2Cov (rA, rB)
.
Example 2.6
Consider three stocks A, B and C. Assume that these stocks have the same
expected rate of return and standard deviation. The following table shows
the correlation between the rate of returns on these stocks.
Stock A Stock B Stock C
Stock A +1.0
Stock B +0.9 +1.0
Stock C +1.0 −0.4 +1.0
Consider the following portfolios constructed from these stocks:
Pa: Equally invested in stocks A and B.
Pb: Equally invested in stocks A and C.
Pc: Equally invested in stocks C and B.
Pd: Totally invested in stock C.
Find the portfolio with the lowest risk.
3. DIVERSIFICATION 17
Solution
From hypothesis, we have that rA = rB = rC = r and σA = σB = σC = σ.
Pa: Equally invested in stocks A and B means that wA = wB = 50%.
We have ρAB = 0.9.
σ2Pa = (wA)
2
σ2A + (wB)
2
σ2B + 2wAwBρABσAσB
= 0.52 × σ2 + 0.52 × σ2 + 2× 0.5× 0.5× 0.9× σ2
= 0.95× σ2.
Hence σPa = 0.97σ.
Pb: Equally invested in stocks A and C means that wA = wC = 50%.
We have ρAC = 1.
σ2Pb = 0.52 × σ2 + 0.52 × σ2 + 2× 0.5× 0.5× 1× σ2
= σ2.
Hence σPb = σ.
Pc: Equally invested in stocks C and B means that wC = wB = 50%.
We have ρCB = −0.4.
σ2Pc = 0.52 × σ2 + 0.52 × σ2 + 2× 0.5× 0.5× (−0.4)× σ2
= 0.3× σ2
Hence σPc = 0.55σ.
Pd: Totally invested in stock C means σPd = σ.
The portfolio with the lowest variance is portfolio Pc. This is of no surprise
from the correlation coefficient matrix since the negative value decreases the
third term of σ2Pc, hence reducing the portfolio variance.
3. Diversification
In this section, we will investigate the effect of diversification in terms of re-
ducing the risk. In fact, we will see that, this reduction depends on the covariance
between the rates of returns on the assets in the portfolio. Let’s consider portfo-
lios with two assets A and B with known fixed risks and expected rates of return.
Studying the changes in the expected rate of return on the portfolio. Using our
definitions r̂p = w1rˆ1 + w2rˆ2 and w2 = 1− w1, we have the linear relationship
r̂p = r̂p (w1) = rˆ2 + w1 (rˆ1 − rˆ2) . (7)
between the weight of stock A (i.e. w1) and the expected rate of return on the
portfolio. Studying the changes in the portfolio standard deviation. This time
using
σ2p = w21σ21 + 2w1w2Cov (r1, r2) + w22σ22 , w2 = 1− w1,
we arrive at the (w1, σp) relationships described by the relation
σp (w1) =
√
w21σ
2
1 + 2w1 (1− w1) Cov (r1, r2) + (1− w1)2 σ22 . (8)
18 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
Example 3.1
Consider a portfolio consisting of
• Stock A. rˆ1 = 10% and σ1 = 12%.
• Stock B. rˆ2 = 15% and σ2 = 20%.
• Correlation between rates of return ρ. (We will consider different
values of ρ).
We illustrate the concept of diversification by forming portfolios under var-
ious (five) different correlation coefficients between the rate of returns on
stocks A and B. Table 2 below shows standard deviations for different port-
folio weights and different values of ρ.
Table 2. Expected rate of return and standard deviation with varying weights
and values of ρ
w1 w2 r̂p σp (if ρ = −1 ) σp (if ρ = −0.5 ) σp (if ρ = 0 ) σp (if ρ = 0.4 ) σp (if ρ = 1 )
0 1 15% 20% 20% 20% 20% 20%
0.1 0.9 14.5% 16.8% 17.43% 18.04% 18.51% 19.2%
0.2 0.8 14% 13.5% 14.95% 16.18% 17.1% 18.4%
0.3 0.7 13.5% 10.4% 12.59% 14.46% 15.79% 17.6%
0.4 0.6 13% 7.2% 10.46% 12.92% 14.6% 16.8%
0.5 0.5 12.5% 4% 8.72% 11.66% 13.56% 16%
0.6 0.4 12% 0.8% 7.63% 10.76% 12.72% 15.2%
0.7 0.3 11.5% 2.4% 7.49% 10.32% 12.12% 14.4%
0.8 0.2 11% 5.6% 8.35% 10.4% 11.78% 13.6%
0.9 0.1 10.5% 8.8% 9.95% 10.98% 11.74% 12.8%
1 0 10% 12% 12% 12% 12% 12%
Studying the relationship between r̂p and σp.
Combining Equations (7) and (8) we are able to describe an explicit relationship
between the portfolio standard deviation and its expected rate of return. In our
Example 3.1, the formulation is left as a exercise to be completed in tutorials. Using
the data from our Example 3.1 we get the following sketch.
3. DIVERSIFICATION 19
Remark 3.1
In the vertical axis of the graph above we must have r̂p instead of ”Return
rp”.
Definition 3.1
1) The portfolio opportunity set describes the set of risk-return choices
that can be achieved by forming a portfolio of stocks A and B in the
(σ, rˆ)-plane.
2) The efficient set (or efficient frontier) is the positively sloped portion
of the opportunity set. It is the set of risk return choices which offer the
highest expected rate of return for a given level of risk.
Observations from the graph:
• When the two stocks are perfectly correlated (i.e., ρ = 1), σp ≥ σ1,
therefore the risk of the individual stocks cannot be eliminated by diver-
sification.
• When the two stocks are correlated and the ρ decreases from 1 to −1, the
risk can be reduced via diversification and the investor can benefit from
it.
• When the two stocks are perfectly inversely correlated (i.e., ρ = −1), all
risk can be eliminated (for σp = 0).
Concluding remarks:
• The investor might make profit from diversification, when the correlation
is less than 1.
20 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
• The lower the correlation, the greater the potential benefit from the di-
versification.
• For correlation equal to −1, it is possible to have perfect hedging oppor-
tunity (for σp = 0).
• The selection of optimal portfolio from the opportunity set will depend
on the risk aversion of the investor, i.e. risk averse investors will invest
according to a lower σp.
4. Capital Asset Pricing Model
4.1. Making investment decisions: The Capital Allocation Line.
To control risk investors divide their investments into two asset classes, namely,
risk-free market securities like US Treasury bills or UK Government Bonds, and
risky assets such as in the equity markets. How much should an investor place in
risk-free money market securities versus other risky asset classes? Let P denote the
portfolio of risky assets and F the risk-free asset.
Consider a complete portfolio C where a proportion wp is invested in P , and
hence a proportion wf = 1 − wp is invested in F . Let rp denote the risky rate of
return on the portfolio P with expected rate of return E (rp) and standard deviation
σp. Let the rate of return on the risk-free asset be rf (which is a constant). As
an investor, suppose the risk in C is determined to be σC = x, now the question
remains, what is the expected rate of return y = E(rC)?
The rate of return on the complete portfolio C is given by
rC = wprp + (1− wp) rf .
Taking expectations of this portfolio’s rate of return gives
y = E(rC) = wpE (rp) + (1− wp) rf ⇐⇒ y = rf + wp (E (rp)− rf ) .
This tells us that the expected rate of return has a base rate of rf plus a proportion
wp of the risk premium E (rp) − rf . Consider the standard deviation, since rf is
constant, we find
σC = wpσp. (9)
This tells us that the risk associated to the complete portfolio C is proportional to
the portion wp invested in the risky asset. Re-arranging for wp and substituting
into the equation for the expected rate of return on the complete portfolio gives
y = rf + x
(
E (rp)− rf
σp
)
. (10)
The relationship between y = E(rC) and x = σC is clearly linear. This straight
line is called the Capital Allocation Line (CAL).
4. CAPITAL ASSET PRICING MODEL 21
The CAL depicts all the risk-return combinations available to investors. Any
portfolio formed from the risk-free asset F and the portfolio P will lie on a straight
line connecting those two points (F and P on the figure above), the exact location
depending on the weights wf and wp:
1) If wp = 0, then from equations (9) and (10) it can be seen that σC = 0 and
E(rC) = rf . The resulting portfolio is the risk-free asset itself.
2) Similarly, if wf = 0, then σC = σp and E(rC) = E(rp). The resulting portfolio
is the risky portfolio P .
3) Assuming that rf < E(rp), if 0 < wp < 1, then 0 < σC < σp and rf < E(rC) <
E(rp). Depositing some money in the bond market and some more money in the
risky portfolio P results in a portfolio C located somewhere on the solid line segment
between points F and P in the figure above. This line segment could be labeled
lending, because wf > 0, where wf = 1− wp.
4) Next, assume rf < E(rp), if wp > 1, then σC > σp and E(rC) > E(rp). This
corresponds to a point located somewhere on the line segment above point P in the
figure above, which could be labeled borrowing, because wf < 0 (this due to the
fact that wf = 1 − wp). The more borrowing that is done, the farther out on this
line segment the investor’s portfolio will lie.
The slope of the CAL which is denoted S (for Sharp ratio) is a measure of the
risk premium per unit of standard derivation.
Definition 4.1
Given a portfolio of risky assets and a risk-free asset, Sharp’s ratio is
defined as
S = E(rp)− rf
σp
where
• E (rp) is the expected rate of return on the portfolio of risky assets
• σp is the risk associated to the portfolio of risky assets
• rf is the rate of return on the risk-free asset.
This is referred to as the risk-to-volatility ratio.
22 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
Example 4.1
Consider the two stocks A and B given in Example 3.1 that is Stock A has
an expected rate of return of 10% and a standard deviation of 12% and
Stock B has an expected rate of return of 15% and a standard deviation
of 20%. Suppose that the correlation coefficient between the two stocks is
ρ = 0.4 and the risk-free interest rate is 9% on the risk-free asset. What is
the equation of the CAL where the portfolio P is made of 90% in stock A
and 10% in stock B.
Solution
We know from Table 2 that σp = 11.74%, r̂p = 10.5%. Hence, the sharp
ratio is given by
S = 10.5− 911.74 = 0.128,
from which it follows that the equation of the CAL is
y = 0.09 + 0.128x.
4.2. Portfolios of risky assets. Our next goal, an optimization problem, is
to find the portfolio of risky assets for which Sharp’s ratio is maximized. During
this module, we will study the case with two risky assets, i.e. we assume that the
market portfolio contains risky assets A and B only. The ideas can be extended to
the general case with n assets, however the calculations become complicated and
would require the use of software packages.
Definition 4.2
The optimal risky portfolio is the portfolio of risky assets with the high-
est return-to-risk combination measured by Sharp’s ratio.
Consider a portfolio of two risky assets A and B. In order to get the optimal
risky portfolio, we will find the fraction of wealth to invest in stock A that will
result in the risky portfolio with the maximum reward to volatility ratio S.
max
wA
Sp(wA) = max
wA
(
r̂p(wA)− rf
σp(wA)
)
= max
wA
 wAr̂A + (1− wA) r̂B − rf[
w2Aσ
2
A + (1− wA)2 σ2B + 2wA (1− wA) Cov (rA, rB)
] 1
2
 .
We will need to apply some results from calculus to find the value of wA which
maximizes this function, written w∗A, the result of which is summarized next.
4. CAPITAL ASSET PRICING MODEL 23
Proposition 4.1
The weights of stock A and B in the optimal risky portfolio are given by:
w∗A =
RAσ
2
B −RB Cov (rA, rB)
RAσ2B +RBσ2A − (RA +RB) Cov (rA, rB)
, w∗B = 1− w∗A,
where
• RA = r̂A − rf is the excess rate of return of stock A, and
• RB = r̂B − rf is the excess rate of return of stock B.
Example 4.2
Consider a portfolio of two stocks A and B. Stock A has an expected rate
of return of 10% and a standard deviation of 12%. Stock B has an expected
rate of return of 15% and a standard deviation of 20%. The correlation
coefficient between the two stocks is ρ = 0.4 (it is the correlation between
the rates of return of stock A and stock B) and the risk-free interest rate is
9% on the risk-free asset. What proportion of stock A should be invested
in the optimal risky portfolio? What is the expected rate of return and the
standard deviation on the optimal risky portfolio? What is its Sharpe ratio?
Solution
We are given rf = 9%, r̂A = 10%, r̂B = 15%, RA = 10 − 9 = 1%, RB =
15− 9 = 6%, σA = 12%, σB = 20%, ρAB = ρ = 0.4.
We aim at finding: w∗A, r̂op, σop and Sop.
It follows from Proposition 4.1 that
w∗A =
0.01× 0.22 − 0.06× 0.4× 0.12× 0.2
0.01× 0.22 + 0.06× 0.122 − (0.01 + 0.06)× 0.4× 0.12× 0.2
= −0.29
r̂op = w∗Ar̂A + (1− w∗A) r̂B = 16.45%
σop =
√
(w∗A)
2
σ2A + 2w∗A (1− w∗A) ρABσAσB + (1− w∗A)2 σ2B = 24.61%
Sop =
r̂op − rf
σop
= 0.302.
Remark 4.1: Remark about the last example
The optimal Capital Allocation Line (CALop) has the following equation:
y = rf + xSop = 0.09 + 0.302x.
This line is tangent to the opportunity set as you can see below. The slop of
this line is Sop and the point of tangency corresponds to the optimal risky
portfolio.
24 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
4.3. The optimal complete portfolios. As previously discussed, investors
manage risk through investment in risky (stocks) and risk-free (Government bonds)
assets; see the subsection on the Capital Allocation Line. We consider portfolios
with two risky assets A, B and the risk-free asset F .
An optimal complete portfolio is formed by taking the optimal risky portfolio
with some weight wp and the risk-free asset with some weight wf (wp + wf = 1).
Using Proposition 4.1, we know that in the optimal risky portfolio, the assets A and
B are weighted w∗A and w∗B respectively. Hence, in the optimal complete portfolio,
the proportion of the investor’s wealth to be invested in A and B is wAp = wpw∗A
and wBp = wpw∗B respectively.
4.4. The Capital Market Line (CML).
In reality, the market will contain n stocks, which will vary as new companies are
registered and old ones become extinct, along with a risk-free asset.
Definition 4.3: The market portfolio
Imagine a capital market in equilibrium. In this equilibrium all investors’ op-
timal portfolio choices have been aggregated into one huge market portfolio,
denoted m, and, supply equals demand for every asset. After aggregating
all investors’ optimal portfolio choices, the market portfolio must contain
every marketable asset in the proportion wi, where
wi =
total market value of asset i
total market value of all assets in the market
In this equilibrium, rf must be the interest rate that equates the supply and
demand for loanable funds.
The market portfolio is an unanimously desirable risky portfolio containing
all risky securities in the proportions in which they are supplied in equilibrium.
Accordingly, the return on the market portfolio is the weighted average of the
returns of all securities in the market. Unfortunately, the market portfolio cannot
actually be observed because it does not really exist. Thus, commonly used stock
market indexes (like the S&P 500 index) are imperfect substitutes for the theoretical
market portfolio.
The CML is a special case of the CAL that represents optimal complete port-
folios, the slop of the CML is Sop where op is the optimal risky portfolio; the points
4. CAPITAL ASSET PRICING MODEL 25
of the CML offer the highest expected rate of return for a defined level of risk, or
the lowest risk for a given level of expected rate of return. The point m in the
figure below corresponds to the market portfolio.
Remark 4.2
In fact, the market portfolio m is the optimal risky portfolio containing all
risky assets.
The figure above shows the combination of the risk-free asset with various risky
portfolios in the opportunity set. Combinations of portfolio B with the risk-free
asset along the line rfB are more desirable than combinations of portfolio A with
the risk-free asset along the line rfA, because portfolios along the line rfB have
the higher expected returns in the same risk class than portfolios along the line
rfA. However, combinations of portfolio B with the risk-free asset along the line
rfB are dominated by combinations of portfolio m with the risk-free asset along
the line rfmH. The portfolios along the line rfmH have the highest returns at
every level of risk. Within the figure above, any ray out of the riskless rate, rf ,
that has a smaller slope than the tangency ray (line rfmH) will be dominated by
the tangency ray because it represents a less favorable risk-return trade-off than
the ray rfmH.
Remark 4.3
In our examples and exercises, we assume that the market portfolio contains
only two risky assets denoted (for example) A and B. The weight of stock A
in this portfolio is w∗A and the weight of stock B is w∗B , where w∗A, w∗B are
given in Proposition 4.1.
26 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
Example 4.3
Let A and B be the stocks considered in Example 4.2. And, following this
example, suppose that the risk-free interest rate is 9% on the risk-free asset.
Determine the equation of the CML.
Solution
The equation of the CML is:
y = rf + xSop = 0.09 + 0.302x,
where Sop is calculated in Example 4.2. In fact, the CML is the CALop in
Remark 4.1.
Example 4.4
Let F be the risk-free asset, we suppose that the risk-free interest rate is
rf = 5%. Let A and B be two stocks. The table below provides a probability
distribution for the rates of return (r.o.r.) on stocks A,B.
State Probability r.o.r. on A r.o.r. on B
1 0.10 25% 30%
2 0.20 20% 10%
3 0.70 5% 15%
Determine the equation of the CML.
Solution
The equation of the CML is
y = rf + xSop = rf + x
( r̂op − rf
σop
)
= 0.05 + x
( r̂op − 0.05
σop
)
,
where op is the optimal risky portfolio. Let r1 and r2 denote the r.o.r. on
stocks A and B respectively. Observe that
r̂op = w∗Ar̂1 + w∗B r̂2.
σ2op = (w∗A)2σ2A + 2w∗Aw∗BCov(r1, r2) + (w∗B)2σ2B ,
where w∗A and w∗B are given in Proposition 4.1. So, we need to calculate:
r̂1, r̂2, σ
2
A, σ
2
B , Cov(r1, r2), w∗A, w∗B .
4. CAPITAL ASSET PRICING MODEL 27
Solution : continued
r̂1 = E[r1] = 0.1× 0.25 + 0.2× 0.2 + 0.7× 0.05 = 0.1
r̂2 = E[r2] = 0.1× 0.3 + 0.2× 0.1 + 0.7× 0.15 = 0.155
σ2A = Var(r1) =
3∑
i=1
Pi(r1,i − r̂1)2 = 0.1× (0.25− 0.1)2 + 0.2× (0.2− 0.1)2
+0.7× (0.05− 0.1)2 = 6× 10−3
σ2B = Var(r2) =
3∑
i=1
Pi(r2,i − r̂2)2 = 0.1× (0.3− 0.155)2
+0.2× (0.1− 0.155)2 + 0.7× (0.15− 0.155)2
= 2.725× 10−3
Cov(r1, r2) =
3∑
i=1
Pi(r1,i − r̂1)(r2,i − r̂2) = 0.1× (0.25− 0.1)(0.3− 0.155)
+0.2× (0.2− 0.1)(0.1− 0.155)
+0.7× (0.05− 0.1)(0.15− 0.155) = 1.25× 10−3.
w∗A =
RAσ
2
B −RBCov (r1, r2)
RAσ2B +RBσ2A − (RA +RB) Cov (r1, r2)
, w∗B = 1− w∗A,
where RA = r̂1−rf = 0.1−0.05 = 0.05, RB = r̂2−rf = 0.155−0.05 = 0.105.
We know from the previous computations above the values of σ2A, σ2B , and
Cov(r1, r2). Thus, by a simple substitution we deduce that
w∗A = 0.00873, w∗B = 0.99127.
Using the previous computed values, we have
r̂op = w∗Ar̂1 + w∗B r̂2
= 0.00873× 0.1 + 0.99127× 0.155 = 0.15452
σ2op = (w∗A)2σ2A + 2w∗Aw∗BCov(r1, r2) + (w∗B)2σ2B
= (0.00873)2 × 6× 10−3 + 2(0.00873)(0.99127)× 1.25× 10−3
+(0.99127)2 × 2.725× 10−3
= 2.69972× 10−3.
Thus, σop = 0.05196. So, the equation of the CML is:
y = 0.05 + x
(0.15452− 0.05
0.05196
)
= 0.05 + 2.01155x.
4.5. The Capital Asset Pricing Model (CAPM).
The CAPM is also called the security market line (SML), it is one of the modern
centrepieces of financial economics. It was published in 1964. The model gives us
a precise prediction of the relationship that we should observe between the risk
and expected rate of return of an asset. Using the CAPM we are able to perform
two vital calculations. Firstly, we are able to quantify a benchmark expected rate
of return associated to an investment which serves in determining whether, for a
given level of risk, the expected rate of return is more or less than its ”fair value”.
Secondly, the models allows us to quantify the expected rate of return on assets
that have not yet been traded in the marketplace, for example, during an Initial
Public Offering (IPO) in which new shares enter the marketplace.
28 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
4.5.1. Understanding risks.
An asset’s total risk consists of both systematic and unsystematic risk.
1. Systematic risk: Also known as market risk or non-diversifiable risk. This
is the portion of an asset’s risk that cannot be removed through diversi-
fication. Interest rates, recessions and catastrophes are examples of sys-
tematic risks. In the CAPM, the measure of systematic risk is called beta
(β).
2. Unsystematic risk: Also known as specific risk or diversifiable risk. This
is the portion of an asset’s risk that can be eliminated by including the
security as part of a well diversified portfolio. It represents the component
of a stock’s rate of return that is not correlated with general market moves.
4.5.2. Beta factors.
It is important to understand how the rate of return ri on a given portfolio or a
single security will react to the trends affecting the whole market. We could plot
the values of ri for each market scenario against the rate of return rm of the market
portfolio and compute the line of best fit: For any given α and β the values of the
random variable α + βrm can be regarded as predictions for the rate of return on
the given portfolio or single security. The difference ϵ = ri − (α+ βrm) between
the actual rate of return ri and the predicted rate of return is called the residual
random variable. The condition defining the line of best fit yields optimal values
α = αi and β = βi given by
αi = E(ri)− βi × E (rm) , βi = Cov (ri, rm)
σ2m
.
Remark 4.4
We got αi and βi from the assumption that E(ϵ) = 0, and from minimizing
Var(ϵ) (in function of β).
Definition 4.4
The beta factor of a given portfolio or individual security A is defined by
βA =
Cov (rA, rm)
σ2m
where rm and σm are the market rate of return and market risk respectively.
The beta factor is an indicator of expected changes in the rate of return on a
particular security or portfolio, say A, in response to the behaviour of the market as
a whole. βA measures the contribution of security A to the variance of the market
portfolio as a fraction of the total variance of the market portfolio. Notice that by
definition, the beta of the market portfolio βm, is equal to one, since
βm =
Cov (rm, rm)
σ2m
= σ
2
m
σ2m
= 1.
4. CAPITAL ASSET PRICING MODEL 29
Proposition 4.2
For any portfolio p of n risky assets with beta values β1, . . . , βn, then the
beta value of the portfolio βp is given by
βp =
n∑
i=1
ωiβi
where ωi is the monetary proportion of the risky asset i, for i = 1, . . . , n, in
the overall portfolio.
Proof
By definition of the portfolio beta and using linearity of the covariance
βp =
Cov (w1r1 + . . .+ wnrn, rm)
σ2m
= w1
Cov (r1, rm)
σ2m
+ . . .+ wn
Cov (rn, rm)
σ2m
= w1β1 + . . .+ wnβn.
Measuring the total risk of ri:
We have
ri = (αi + βirm) + ϵ. (11)
The rate of return ri in equation (11) can be partitioned into two mutually
exclusive components: a systematic part and an unsystematic part: The systematic
part of ri, αi+βirm, is systematically explained by the explanatory variable, the rate
of return on the market portfolio (rm). The unsystematic part, ϵ, is the remaining
portion of ri that is left unexplained by the explanatory variable. Economic insights
can be gained from equation (11)’s return-generating function by partitioning its
total variance into two economically meaningful components.
Given that ϵ = ri − (αi + βirm), we find that
Var(ϵ) = Var (ri) + Var (αi + βirm)− 2Cov(ri, αi + βirm)
= Var (ri) + β2iVar (rm)− 2
(
Cov(ri, αi) + βiCov(ri, rm)
)
= σ2i + β2i σ2m − 2βiCov(ri, rm) = σ2i + β2i σ2m − 2β2i σ2m
= σ2i − β2i σ2m,
where in the first equality we used the following well known result: Var(X − Y ) =
Var(X) + Var(Y )− 2Cov(X,Y ). Hence the total risk of ri is
σ2i = β2i σ2m +Var(ϵ). (12)
The first term of the equation (12), β2i σ2m, measures systematic (or undiversifi-
able) risk, and the second term, Var(ϵ), measures the unsystematic (or diversifiable)
risk of asset i. Since all assets experience the same market variance, σ2m, the beta
emerges as an index of systematic (or diversifiable) risk.
The Var(ϵ) term is called the residual variance. It vanishes for the market
portfolio. The market portfolio involves only the systematic risk.
30 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
4.5.3. Underlying assumptions.
The CAPM is an equilibrium model that explains expected rates of return over
one holding period and is based on the assumptions underlying portfolio analysis,
because the theory is essentially an accumulation of the logical implications of port-
folio analysis. The initial portfolio theory assumptions are:
1. Investors in capital assets (defined as all terminal-wealth-producing assets)
are risk-averse one-period expected-utility-of-terminal-wealth maximizers. Equiva-
lently, investors are risk averse and maximize their expected utility of returns over
a one-period planning horizon.
2. Investors are expected to make their portfolio decisions solely in terms of the
means and standard deviations of the rates of return on their portfolios.
3. The means and standard deviations of rates of return on these portfolios are
finite numbers that exist and can be estimated or measured.
4. There are a collection of assumptions that underlie most economic theories. All
capital assets are infinitely divisible, meaning that fractions of shares can be bought
or sold. Investors are assumed to be price takers (instead of price setters). Finally,
taxes and transactions costs are assumed to be nonexistent.
Investors who conform to the preceding assumptions will prefer Markowitz
efficient portfolios over inefficient portfolios. Bearing these investors in mind, it is
possible to begin to discuss capital market theory. A fairly exhaustive list of the
additional assumptions necessary to generate the theory follows.
5. There is a single risk-free interest rate at which all borrowing and lending takes
place.
6. All assets, including human capital, are marketable.
7. Capital markets are perfect, meaning that (a) all information is freely and
instantly available to everyone (transparency prevails), (b) no margin requirements
exist, and (c) investors have unlimited opportunities to borrow, lend, or sell assets
short.
8. All investors have homogeneous expectations over the same one-period investment
horizon. First, they are assumed to define the relevant period in exactly the same
manner. Second, they are assumed to have identical expectations with respect to
the expected rates of return, the variance of rates of return, and the correlation
matrix representing the correlation structure between all pairs of stocks.
4.5.4. Construction of the CAPM.
A basic principle of equilibrium is that all investments should offer the same
reward-to-risk ratio:
E (ri)− rf
Cov (ri, rm)
= E (rm)− rfCov (rm, rm) ⇐⇒ E (ri)− rf =
Cov (ri, rm)
σ2m
(
E (rm)− rf
)
Observe that Cov(ri,rm)σ2m = βi. So, we arrive at the most familiar CAPM expression,
the expected rate of return-beta relationship
E (ri) = rf + βi (E (rm)− rf ) ,
also written as
rˆi = rf + βi (rˆm − rf ) . (13)
This last equation is the equation of the security market line (SML). Since
the market beta is 1 and the risk-free asset has beta zero, we know (0, rf ) and
4. CAPITAL ASSET PRICING MODEL 31
(1,E (rm)) lie on the SML. Hence its slope is E (rm) − rf ; also known as the risk
premium of the market portfolio or the Market Risk. The SML is the straight-line
graph in the (βi, rˆi)-plane given below.
Analysing the values of βi:
• The market portfolio has a beta of 1. Securities with betas greater than
1 are more risky than the market portfolio and securities with betas less
than 1 are less risky than the market portfolio. Those securities with
βi > 1 are called aggressive securities and those securities with βi < 1
are called defensive securities. A more risk averse investor will choose
securities with betas less than 1.
• If a stock has a high positive beta, then:
(1) It will have large price swings driven by the market;
(2) It will increase the risk of this stock;
(3) The investor will ask a high expected rate of return in compensation.
• If the stock has a negative beta, then:
(1) It will move ”against” the market;
(2) The investor will accept a lower expected rate of return.
Analysing the SML:
• A security plotted on the SML is fairly priced. That is, the expected
rate of return on the security is appropriate given the level of risk of
this security. A security that lies perfectly on the SML is said to be in
equilibrium.
• A security plotted above the SML is said to be underpriced. That is,
you are getting a greater expected rate of return given the level of risk.
Investors will want to take advantage of these types of securities.
• A security plotted below the SML is said to be overpriced. That is, you
are getting less expected rate of return given the level of risk. Investors
will want to avoid these types of securities.
32 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
The CAPM has security price implications illustrated in Figure 12.4. Points
between the SML and the vertical axis, such as point L, represent securities whose
current prices are lower than they would be in equilibrium. Thus, points such as
L represent securities with unusually high expected rates of return for the amount
of systematic risk they bear. Because they have unusually high expected rates of
return, there will be strong demand for them. This means that investors will bid
their purchase prices up until their equilibrium expected rate of return is driven
down onto the SML at L′.
For example, consider a stock whose estimated end-of-period price is E(P1).
Given its current price P0 and assuming that no dividends will be paid this period,
its initial expected rate of return E(r) is equal to [E(P1)−P0]/P0 = [E(P1)/P0]−1.
Having P0 be too low can be seen to be equivalent to having E(r) be too high,
meaning that E(r) > rf + [E(rm)− rf ]β. The resulting bidding up of P0 will lower
E(r) and will continue to lower E(r) until the following equilibrium is reached:
E(r) = rf + [E(rm)− rf ]β. In other words, all securities positioned above the SML
are underpriced.
Similarly, securities represented by points between the SML and the horizontal
axis represent securities whose prices are too high. This means that securities such
as H do not offer sufficient levels of expected rate of return to induce rational
investors to accept the amount of systematic risk they bear. As a result, H’s price
will fall due to a lack of demand. Furthermore, it will continue falling until it is low
enough so that the security’s expected rate of return is on the SML at H ′. More
generally, all securities positioned below the SML are overpriced. After considering
the effect the CAPM can have on the market prices of all assets, it is easy to see
why it is called an asset pricing model.
The whole concept of the CAPM is that rational investors will choose to invest
in securities that lie on or above the SML.
5. RISK MEASURES 33
5. Risk measures
Risk management is of paramount importance for the management of a firm.
The risks facing a firm can be generally classified under market risk (exposure to
potential loss due to changes in market prices and market conditions), credit risk
(risk of customers defaulting), and operational risk (any business risk that is not a
market nor credit risk). The risk management process should be a holistic process
covering the analysis of risk incidents, assessment of management control, reporting
procedures, and prediction of risk trends.
In this chapter we focus on measuring the risks of an insurance company. A
major risk an insurance company encounters is the loss arising from the insurance
policies. In other words, our focus is on the operational risk of the firm. We shall
discuss various measures that attempt to summarize the potential risks arising from
the possible claims of the insurance policies. Thus, the measures will be based on
the loss random variables.
Prior to introducing some premium-based risk measures, we first provide a
formal definition of a risk measure based on a random loss X, which is the aggregate
claim of a block of insurance policies.
Definition 5.1
A risk measure φ is a mapping from the set L of random losses to the set
of real numbers
φ : L −→ R
X 7→ φ(X)
As a loss random variable, X is nonnegative. Thus, the risk φ(X) may be
imposed to be nonnegative for the purpose of measuring insurance risks. However,
if the purpose is to measure the risks of a portfolio of assets, X may stand for the
change in portfolio value, which may be positive or negative. In such cases, the risk
φ(X) may be positive or negative.
5.1. Some premium-based risk measures.
We denote the mean and the variance of the random loss X by µX and σ2X , respec-
tively. The expected-value principle premium risk measure is defined as
φ(X) = (1 + θ)µX , (14)
where θ ≥ 0 is the premium loading factor. Thus, the loading in excess of the
mean loss µX is θµX . In the special case of θ = 0, φ(X) = µX , and the risk
measure is called the pure premium.
Note that in the expected-value premium risk measure, the risk depends only on
the mean µX and the loading factor θ. Thus, two loss variables with the same mean
and same loading will have the same risk, regardless of the higher-order moments
such as the variance. To differentiate such loss distributions, we may consider the
variance principle premium risk measure defined by
φ(X) = µX + ασ2X , (15)
or the standard-deviation principle premium risk measure defined by
φ(X) = µX + ασX , (16)
where α ≥ 0 in equations (15) and (16) is the loading factor. Under the variance
premium and standard-deviation premium risk measures, the loss distribution with
a larger dispersion will have a higher risk. This appears to be a reasonable property.
34 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
5.2. Value-at-Risk (VaR).
We recall that given a real-valued random variable X, the distribution function
(df) of X denoted FX(·) is defined as follows: for any x ∈ R, FX(x) = P (X ≤ x).
We now introduce a risk measure constructed for the purpose of evaluating
economic capital. Value-at-Risk (VaR) is probably one of the most widely used
measures of risk. Simply speaking, the VaR of a loss variable is the minimum value
of the distribution such that the probability of the loss larger than this value is not
more than a given probability. We now define VaR formally as follows.
Definition 5.2
Let X be a random variable of loss with df FX(·), and δ be a probability
level such that 0 < δ < 1, the Value-at-Risk at probability level δ, denoted
by VaRδ(X), is the δ-quantile of X. That is
VaRδ(X) = inf{x ∈ [0,∞) : FX(x) ≥ δ}. (17)
The probability level δ is usually taken to be close to 1 (say, 0.95 or 0.99), so
that the probability of loss X exceeding VaRδ(X) is not more than 1 − δ, and is
thus small. We shall write VaR at probability level δ as VaRδ(X) when the loss
variable is understood.
Example 5.1
Find VaRδ(X), for δ = 0.95, 0.96, 0.98, and 0.99, of the following discrete
loss distribution
X =

100, with prob 0.02
90, with prob 0.02
80, with prob 0.04
50, with prob 0.12
0, with prob 0.80
5. RISK MEASURES 35
Solution
Observe that
• for x < 0, FX(x) = P (X ≤ x) = 0;
• for 0 ≤ x < 50, FX(x) = P (X ≤ x) = P (X = 0) = 0.8;
• for 50 ≤ x < 80, FX(x) = P (X ≤ x) = P (X = 0) + P (X = 50) =
0.92;
• for 80 ≤ x < 90, FX(x) = P (X = 0) + P (X = 50) + P (X = 80) =
0.96;
• for 90 ≤ x < 100, FX(x) = P (X = 0) + P (X = 50) + P (X =
80) + P (X = 90) = 0.98;
• for x ≥ 100, FX(x) = P (X = 0) + P (X = 50) + P (X = 80) +
P (X = 90) + P (X = 100) = 1.
The df of X is plotted in the following Figure.
The dotted horizontal lines correspond to the probability levels
0.95, 0.96, 0.98, and 0.99.
Note that the df of X is a step function.
For VaRδ(X) we require the value ofX corresponding to the probability level
equal to or next-step higher than δ. Thus, VaRδ(X) for δ = 0.95, 0.96, 0.98,
and 0.99, are, respectively, 80, 80, 90, and 100.
5.3. Semivariance and semi-deviation.
The semivariance is a risk measure defined as the expected squared deviation
from the mean, calculated over those points that are not greater than the mean.
Its square root is the semi-deviation:
SD(X) =
(
E
[
(X − E[X])21{X≤E[X]}
]) 12
. (18)
36 2. PROBABILITY BACKGROUND AND MODERN PORTFOLIO THEORY
Example 5.2
Let X be a random variable that represents the return of some financial
asset. Suppose that X takes on the following possible values: 10, 20, 30, 50.
And P (X = 10) = 12 , P (X = 20) =
1
8 , P (X = 30) =
1
8 , and P (X = 50) =
1
4 .
Calculate SD(X)
Solution
We have
SD(X) =
(
E
[
(X − E[X])21{X≤E[X]}
]) 12
.
We can verify that E(X) = 23.75. Thus, we have
SD(X) =
(
E
[
(X − 23.75)21{X≤23.75}
]) 12
=
(
(10− 23.75)2 × P (X = 10) + (20− 23.75)2 × P (X = 20)
)1/2
=
(
(10− 23.75)2 × 12 + (20− 23.75)
2 × 18
)1/2
= 9.81.