MATH260. FINANCIAL MATHEMATICS
PART I:
DEPARTMENT OF MATHEMATICAL SCIENCES
DR SIMON A. FAIRFAX
1
2PART I: Modern Portfolio Theory
1. Time value of money
.The time value of money (TVM) is an economic principle that suggests present day money
is worth less than money in the future because of its earning power over time. Put simply a
pound today is worth more than a pound next year because money can be invested today and
earn interest. The time value of money relates to three basic parameters: inflation, opportunity
cost and risk.
.Inflation is reducing what is known as the purchasing power of money because it increases the
prices of goods and services. Therefore, over time the same amount of money can purchase fewer
goods and services. Opportunity cost refers to the potential gain or loss on an investment that
someone gives up by taking alternative action. Risk relates to the investment risk that investors
undertake when putting their money into investment assets. We will study investment risk and
returns in subsection 4.
1.1. Future Value.
.Money value fluctuates over time hence has different values in the future. This is because one
can invest today in an interest-bearing bank account or any other investment and that money will
grow/shrink due to the rate of return.
.To evaluate the real worthiness of an amount of money today after a given period of time,
economic agents compound the amount of money at a given interest rate. Compounding at the
risk-free interest rate would correspond to the minimum guaranteed future cash flow. If one wants
to compare their change in purchasing power, then they should use the real interest rate which
incorporates inflation.
.Future value (FV) is the amount to which a current investment will grow over time when placed
in an account that pays compound interest. The process of going from today’s values, known as
the present values (PV) to the future value (FVs) is called compounding.
Suppose you deposit £100 in a bank that pays 10% interest each year. What is the
future value of this deposit at the end of years 1, 2 and 3?
Example 1.1
Solution 1.1. At the end of each the first three years the investment will grow to £110, £121
and £132.10 respectively. This information is best displayed in a time-line.
Years
0 1 2 3
Initial deposit £100
Interest earnt £10 £11 £12.10
Total future value £110 £121 £132.10
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 3
Mathematical formulas for calculating future values
.In general, to find how much the investor earns at the end of year n we using the compounding
formula
FVn = PV (1 + r)
n.(1.1)
where
• FVn = the future value at the end of year n,
• PV = the initial investment or initial value of your account,
• r = interest rate paid by the bank in the account holder, and
• n = the number of years.
A company invests £2M to clear a tract of land and plants some palm trees. The trees
will mature in 5 years, at which time the farm will have a market value of £5M . What
is the expected annual rate of return for the company’s investment?
Example 1.2
Solution 1.2. We are given n = 5, FVn = 5M and PV = 2M . We aim at finding r we know
from Equation (1.1) that
FVn = PV (1 + r)
n.
Hence FVnPV = (1 + r)
n i.e., r = n
√
FVn
PV − 1.
r = 5
√
5
2 − 1 = 0.2011
.Care is needed with the units. The rate of interest should be represented as a decimal in the
compounding formula. The investment offers an opportunity to gain 20.11% per annum.
How long will it take to double a capital investment of £c attracting interest at 6%
compounded yearly?
Example 1.3
Solution 1.3. We are given PV = c, FV = 2c and r = 6%, where c is the initial capital. We
aim at finding n. We have that
FVn = PV (1 + r)
n,
i.e. ln
(
FVn
PV
)
= n ln(1 + r) and therefore
n =
ln
(
FVn
PV
)
ln(1 + r)
=
ln 2
ln(1 + r)
.
n = 11.89. Hence it would take 12 years.
1.2. Present Value.
.Present value, often called the discounted value, is a financial formula that calculates how much
a given amount of money received on a future date is worth in today’s dollars. In other words, it
computes the amount of money that must be invested today to equal the payment or amount of
cash received on a future date.
.The process of finding present values is called discounting and the interest rate used to calculate
present values is called the discount rate.
4 DR SIMON A. FAIRFAX
Consider a riskless investment opportunity that will pay £133.10 at the end of 3 years.
Suppose your local bank is currently offering 10 percent interest on 3-year Certificates
of Deposit (CDs), and you regard the security as being exactly as safe as a CD. How
much should you be willing to pay for investment? What conclusions can we draw?
Example 1.4
Solution 1.4.
Representing cash flows Let’s set up a time line for the cash flows.
Years
0 1 2 3
FV3 = £133.10PV =?
Conclusion From the future value example, an initial amount of £100 invested at 10% per
year would be worth
£100× 1.103 = £133.10
at the end of 3 years. Hence, £100 is defined as the present value of the £133.10 due in 3 years
when the opportunity cost rate is 10%.
Interpretation of the fair value Using this defined value, we can make several conclusions.
• If the cost of the investment was less than £100, you should buy it, because its price would
then be less than the £100 you would have to spend on a similar-risk alternative to end
up with £133.10 after 3 years.
• If the investment cost more than £100, you should not buy it, because you would have to
invest only £100 in a similar-risk alternative to end up with £133.10 after 3 years.
• If the cost was exactly £100, then you should be indifferent. Therefore, £100 is defined as
the fair value of the investment.
Mathematical formulas for calculating present values
In general, the present value of a future cash flow given the discount rate and number of years
in the future that the cash flow occurs can be computed using the following equation.
PV =
CFn
(1 + r)n
,(1.2)
where
• CFn = the future cash flow occurring at the end of year n,
• PV = the present value,
• r = the interest or discount rate, and
• n = the number of years.
Theresa will retire in 15 years. This year she wants to fund an amount of 18, 000 GBP
to become available in 15 years. How much does she have to deposit into a pension plan
earning 8% annually?
Example 1.5
Solution 1.5. We are given n = 15, CFn = 18000GBP and r = 8%. We aim at finding PV
We know from Equation (1.2) that
PV =
CFn
(1 + r)n
.
Hence PV = 18.000(1+0.08)15 = 5674.35
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 5
2. Interest rates
.An interest rate is a promised rate of return denominated in some unit of account (GBP, USD,
EURO, etc) over some time period (a month, a trimester, a semester, a year, 10 years, or longer).
The time period and the unit of the account are indicted.
2.1. Nominal rates.
.The nominal rate rnom is the rate that is quoted by banks, brokers, and other financial institutions.
However, to be meaningful, the quoted nominal rate must also include the number of compounding
periods per year. The quoted rate does not account for the effects of compounding.
.An interest rate in GBP terms might be risky when estimated as a function of inflation. Suppose
you deposit 100 GBP in a bank that pays 10% interest each year. In one year’s time, you are
guaranteed to collect 110 GBP in cash, but at this time costs will be higher than one year ago.
2.2. Real rates.
.The real return on your investment, studied by Fisher, will depend on what your money can buy
in one year relative to what it could buy today. Using the following notation
• R the real interest rate
• i the inflation rate in a period
• r the interest rate received in that period
then the relationship between R and r is described by compounding income using the rate r
(numerator) and then discounting outgoings using the rate of inflation (denominator):
1 +R =
1 + r
1 + i
⇐⇒ R = 1 + r
1 + i
− 1 = r − i
1 + i
.
Hence, the real rate of return is the rate of interest received above inflation discounted to remove
the effects of inflation over that period,
(2.1) R =
r − i
1 + i
.
Simple example Let i be 4% in a year. With 10% nominal interest rate paid yearly, after netting
out the 4% depreciation in purchasing power of money, you are left with a net growth in purchasing
power of approximately 6% in the future. Here 6%/(1 + 0.04) = 5.77% is called the real rate of
interest for an investment made today.
2.3. Periodic Rate.
.The periodic rate is the rate charged by a lender or paid by a borrower each period. It can be
a rate per day, per week, per three-month, per six-month period, per year, or per any other time
interval.
.We find the periodic rate as using
rPER =
rnom
m
,(2.2)
where m denote the number of compounding periods a year.
.The future value of an initial investment at a given interest rate compounded m times per year
at any point in the future can be found by applying the following equation:
FVn = PV (1 +
rnom
m
)nm ,(2.3)
where n is the number of years.
6 DR SIMON A. FAIRFAX
Suppose that you invest 100 GBP in an account that pays a nominal rate of 10%,
compounded monthly. How much would you be paid after 3 years?
Example 2.1
Solution 2.1. We are given n = 3, m = 12, PV = £100 and r = 10%. We know from (2.3) that
FVn = PV (1 +
rNom
m
)nm
Hence FV3 = 100(1 +
0.1
12 )
3×12 = £134.82.
Calculate the point in time at which some initial capital c has doubled, if interest is
compounded (a) monthly or (b) weekly, using an interest rate of r% per annum (p.a.).
In particular give a numerical answer to the above for r = 5.
Example 2.2
Solution 2.2. We have FVn = 2c, PV = c, m = 12 or 52. We aim at finding n given in terms
of the compounding periods. It follows from (2.3) that
FVn = PV (1 +
r
m
)nm.
Hence ln
(
FVn
PV
)
= mn ln(1 + r%m ) and therefore
n =
ln 2
m ln(1 + rm )
(a): In this case, m = 12 and hence
n =
ln 2
12 ln(1 + r%12 )
(b): In this case, m = 52 and hence
n =
ln 2
52 ln(1 + r%52 )
Numerical solutions for r = 5
(a): n = ln 2
12 ln(1+ 0.0512 )
= 13.9. The initial capital would have doubled after 13.9 years. But
if you are only allowed to take your earning at the end of each month then it would have
doubled (and even a little) after 13 years and 11 months. (Assuming 0.9 years corresponds
to 0.9× 12 = 10.8 months)
(b): n = ln 252 ln(1+ r52 )
= 13.87. The initial capital would have doubled after 13.87 years. But
if you are only allowed to take your earning at the end of each week then it would have
doubled (and even a little) after 13 years and 46 weeks. (Assuming 0.87 years corresponds
to 0.87× 52 = 45.24 weeks)
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 7
Assume that one year from now, you will deposit £1, 000 into a savings account that
pays 8%. How much will you have in your account five years from now (a) If the bank
compounds interest annually? (b) if the bank used quarterly compounding rather than
annual compounding?
Example 2.3
Solution 2.3. We simply need to know that the time to maturity is 4 years (5 years from now
minus 1 year from now)
(a): With annual compounding, the investor will have:
1000(1 + 0.08)4 = 1360.49.
(b): With quarterly compounding, the investor will have:
1000(1 +
0.08
4
)4×4 = 1372.79.
2.4. Effective annual rate.
.The Effective Annual Rate rEAR is the annual rate that produces the same result as if we had
compounded at a given periodic rate m times per year.
.To calculate the EAR we use
rEAR = (1 +
rnom
m
)m − 1.(2.4)
.The present value of a future cash flow given the discount rate (compounded m times per year)
and the n number of years in the future that the cash flow occurs can be computed as follows
PV = CFn(1 +
rNom
m
)−nm,(2.5)
.The number (1 + rNomm )
−nm is called the discount factor.
8 DR SIMON A. FAIRFAX
3. Continuous compounding
.The formula of the future value FVn at time n of a present value PV attracting interest at a
rate r > 0 compounded m times a year is given by
FVn = PV (1 +
r
m
)nm.
.Assume that the compounded period becomes shorter and shorter i.e., m becomes bigger and
bigger. Then in the limit as n goes to infinity, we obtain:
lim
m→∞FVn =PV limm→∞(1 +
r
m
)nm
=PV lim
m→∞
[
(1 +
r
m
)
m
r
]nr
=PV enr,
where we have used e = lim
x→∞(1 +
1
x )
x.
.This is known as the continuous compounding with corresponding growth factor enr.
3.1. Continuous rates of interest.
.The future value M(t) with a nominal rate r with continuous compounding and initial or present
value M(0) is given by
M(t) = M(0)ert.(3.1)
An investor receives £1100 in one year in return for an investment of £1000 now. Cal-
culate the percentage return per annum with
(a) Annual compounding
(b) Semi-annual compounding
(c) Monthly compounding
(d) Continuous compounding.
Example 3.1
Solution 3.1. (a): With the annual compounding, using Equation (1.1), we have that:
1000(1 + r) = 1100.
Then, the return is
r =
1100
1000
− 1 = 0.1
or 10% per annum.
(b): With the Semiannual compounding, using Equation (2.3), we have that:
1000(1 +
r
2
)2 = 1100.
Then, the return is
r = 2
(√1100
1000
− 1
)
= 0.0976
or 9.76% per annum.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 9
(c): With monthly compounding, using Equation (2.3), we get
1000(1 +
r
12
)12 = 1100.
Hence, the return is
r = 12
(
12
√
1100
1000
− 1
)
= 0.0957
or 9.57% per annum.
(d): With continuous compounding, using Equation (3.1), we get
1000er = 1100.
Hence, the return is
r = ln
1100
1000
= 0.0953
or 9.53% per annum.
3.2. Relation between continuous and nominal compounded interest rates.
.We want to give the relation between continuous and nominal compounded interest rates. Let rc
be the interest rate with continuous compounding and rnom be the equivalent nominal rate with
compounding m times per annum. We have that
FVn = PV (1 +
rNom
m
)nm = PV ercn,
i.e.,
(1 +
rNom
m
)nm = ercn.
This means that
rc = m ln
(
1 +
rNom
m
)
(3.2)
and
rNom = m
(
erc/m − 1
)
(3.3)
Consider an interest that is quoted as 10% per annum with semiannual compounding.
Find the equivalent rate with continuous compounding.
Example 3.2
Solution 3.2. We are given m = 2 and rNom = 0.1 We aim at finding rc.
We know from Equation (3.2) that
rc = m ln
(
1 +
rNom
m
)
.
Hence rc = 2 ln
(
1 + 0.12
)
= 0.09758 or rc = 9.758% per annum.
10 DR SIMON A. FAIRFAX
A bank quotes you an interest rate of 14% per annum with quarterly compounding. What
is the equivalent rate with (a) continuous compounding? (b) annual compounding?
Example 3.3
Solution 3.3. We have that m = 4 and rNom = 0.14 We aim at finding rc and rEAR.
(a): The rate with continuous compounding is given according to Equation (3.2) by:
4 ln
(
1 +
0.14
4
)
= 0.1376
or 13.76% per annum.
(b): The rate with annual compounding is given according to Equation (2.4) by(
1 +
0.14
4
)4
− 1 = 0.1475
or 14.75%
Paul has 1500 GBP and wishes to invest this money in order to use it in six years. What
advice would you give him considering that he has the following two possibilities?
A: Put the 1500 GBP in a bank that offers to pay interest (compounded annually)
of 4% in the first year, 4.4% in the second year and so on, increasing by 0.4%
each year and hence paying 6% in the sixth year.
B: Invest the 1500 GBP in an instant access account which pays a fixed continu-
ously compounded rate of interest 4.3%.
EXERCISE
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 11
4. Risk and return
.The basic premise of an investor is that he likes returns and dislikes risk. An investor will purchase
a financial asset because he wants to increase his wealth, i.e., to earn a positive rate of return on
his investments. Due to uncertainty, he does not know what rate of return his investments 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 return and risk.
4.1. 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 invested
Amount invested
× 100%
=
Currency return
Amount 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 (we will revisit valuation later in the course) 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 ∈ [0, T ] whose beginning and
ending share price are S(0) and S(T ) respectively, which paid a dividend of D at t = 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 .
Suppose you buy 10 shares of a stock for £100. At the end of one year, you sell the stock
for £110 after receiving a dividend of £3. What is the return on your £100 investment?
What is the rate of return?
Example 4.1
Solution 4.1.
• The currency return is £110−£100 + £3 = £13.
• The rate of return is:
£110−£100 + £3
£100
× 100% = 13%.
12 DR SIMON A. FAIRFAX
.Mathematical interpretation
(1) A negative rate of return indicates the original investment was not recovered.
(2) Rates of return are a better measure of relative returns. A £10 return on a £100 invest-
ment 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.
4.2. Measuring individual asset return.
.The return on an investment will depend 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 4.1. Given a probability distribution of rate of returns, the expected rate of return
on the individual stock or investment is defined as:
r̂ = E[r] =P1r1 + P2r2 + . . .+ Pnrn
r̂ =
n∑
i=1
Piri,
where
• r̂ = E[r] =the expected return on the stock,
• n = the number of events/states used to determine returns,
• Pi = the probability of state i occurring, and
• ri = the return on the investment in state i.
This weighted average is viewed as the expected reward for investing in an investment.
The table below provides a probability distribution for the returns on stocks A, B, C
and D. What is the expected rate of return stocks A, B, C and D?
State Probability Return on A Return on B Return on C Return 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.
Example 4.2
Solution 4.2. The expected rate of return on stock A, stock B, stock C and stock D are given by:
r̂A = E[rA] =P1r1 + P2r2 + P3r3 + P4r4
=0.10× 15% + 0.20× 20% + 0.40× 5% + 0.30× 10%
=10.5%
and similarly r̂B = E[rB ] = 18%, r̂B = E[rC ] = 10.5% and r̂B = E[rD] = 40.12%.
Decision making: Stock D offers a higher expected return than stocks A, B and C which makes it
the most attractive from a returns point of view. What about the risk of each stock?
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 13
Probability distributions for returns on stocks A,B,C and D.
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
0
0.05
0.1
0.15
0.20
0.25
0.3
0.35
0.4
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
0
0.05
0.1
0.15
0.20
0.25
0.3
0.35
0.4
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
0
0.05
0.1
0.15
0.20
0.25
0.3
0.35
0.4
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
0
0.05
0.1
0.15
0.20
0.25
0.3
0.35
0.4
14 DR SIMON A. FAIRFAX
4.3. Measuring individual asset risk.
.Risk can be viewed as the potential that a chosen action or activity, including the choice of
inaction, will lead to a gain or a loss, viewed as desirable or undesirable outcome. Risk has two
components: exposure (events affecting returns) and uncertainty (probabilities). Note that if
either is not present then there is no risk. In general, the tighter the probability distribution of
expected future returns, the smaller the risk of a given investment. The question will then be:
How do we measure the tightness of the probability distribution?
Definition 4.2. 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 − r̂)2,(4.1)
where
• n = the number of states,
• Pi = the probability of state i,
• ri = the return on the stock in state i, and
• r̂ = the expected return on the stock.
Consider stocks A, B C and D given in Example 4.2.
(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 returns along
with the expected return of the investments, how does the investor choose be-
tween the investments?
Example 4.3
Solution 4.3.
(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
=⇒ σA = 0.0566 (or 5.66%)
similarly 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 investor should choose
the investment with the higher expected rate of return.
4.4. Coefficient of Variation.
The coefficient of variation allows investors to take a balanced viewed by factoring in risk
relative to reward. It allows investor to decide whether risks are sufficiently covered by rewards.
Definition 4.3. 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 return. Mathematically, this is defined by
CV =
σ
r̂
.(4.2)
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 15
(1) Find the coefficient of variation of stock A, stock B, stock C and stock D.
(2) What conclusions can you draw?
Example 4.4
Solution 4.4. If follows from Equation (4.2) that
(1)
CVA =
5, 66
10.5
= 0.5390 CVB =
19.9
18
= 1.1055
CVC =
6.1
10.5
= 0.5809 CVD =
19.9
40.12
= 0.4960
(2) • From these computations, we see that stock A has more risk per unit 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.
• This is only a part of the picture because most investors choose to hold securities as
part of a diversified portfolio.
4.5. Returns 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 wi = Vi/V . By
viewing the mix of assets in the portfolio in this way, we note w1 + . . .+ wn = 1.
Definition 4.4. The expected return on a portfolio, is the weighted average of the expected
returns on the individual assets which comprise the portfolio. This can be expressed as
follows:
r̂p =
n∑
i=1
wir̂i = w1r̂1 + w2r̂2 + . . .+ wnr̂n,
(4.3)
where
• r̂p = E[r̂p] = the expected return on the portfolio,
• n = the number of assets in the portfolio,
• wi = the monetary proportion of the portfolio invested in asset i, and
• r̂i = the expected return on stock i.
Consider two portfolios of stock A and B given in Example 4.2. The first portfolio p1
consists of 50% Stock A and 50% Stock B and the second portfolio p2 consists of 60%
Stock A and 40% Stock B. What are the expected return on these portfolios?
Example 4.5
Solution 4.5. The expected return on these portfolios are given by:
r̂p1 = 0.5(10.5%) + 0.5(18%) = 14.25%
r̂p2 = 0.6(10.5%) + 0.4(18%) = 13.5%
16 DR SIMON A. FAIRFAX
4.6. Portfolio Risk.
.There are two measures for studying the simultaneous change in the returns of two assets: the
correlation coefficient and the covariance.
.Covariance between pairs of returns. In general, for two jointly distributed random variable
X and Y , the 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 units of measurement of the covariance are those of X times those of Y .
.In the context of modern portfolio theory, the covariance between the 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 − r̂1)(r2i − r̂2),(4.4)
where
• σ12 = the covariance between the returns on assets 1 and 2,
• n = the number of states,
• Pi = the probability of state i,
• r1i and r2i are the return on assets 1 and 2 respectively in state i,
• r̂1 and r̂2 are the expected returns on assets 1 and 2 respectively.
The covariance measures simultaneous changes in the returns of assets. Suppose X, Y are
random variables and a, b are constants. Here are some useful properties that are valid in general.
(1) From the definition we immediately observe
cov(X, a) = 0, cov(X,X) = var(X), cov(X,Y ) = cov(Y,X)
(2) We can deduce from the definition
cov(aX, bY ) = ab cov(X,Y ), cov(X + a, Y + b) = cov(X,Y )
.Correlation coefficient between the returns In general, given a pair of random variable 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. From
the point of view of modern portfolio theory, we will write the correlation coefficient between the
returns on a pair of assets can be expressed as follows:
Corr(ri, rj) = ρij =
σij
σiσj
,
where
• σij = the covariance between the returns on assets i and j,
• σi, σj = the standard deviations on stocks i and j
.Interpretation of corr
• In general, −1 ≤ ρij ≤ +1.
• Two assets are perfectly positively correlated if ρ12 = 1. In this case, returns will have a
linear relationship with positive gradient.
• Two assets are perfectly negatively correlated if the correlation coefficient ρ12 = −1. In
this case, returns will have a linear relationship with a negative gradient.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 17
• A correlation coefficient equal zero means that the two assets are not related to each other.
4.7. Portfolio variance.
Proposition 4.1. The variance of the portfolio is given by:
σ2p =
n∑
i=1
n∑
j=1
wiwjCov(ri, rj),(4.5)
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 returns on assets i and j.
Proof. Consider a portfolio with two assets. We know the observed portfolio return, viewed as
a random variable, is rp = w1r1 + w2r2 and the expected portfolio return is rˆp = w1r1 + w2r2.
Using these we compute the variance of returns on the portfolio using the definition.
σ2p = Var(rp) = E[(rp − r˜p)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)] + w22E[(r2 − rˆ2)2] (using linearity)
= w21σ
2
1 + 2w1w2Cov(r1, r2) + w
2
2σ
2
2 .
In the general case with n assets, we know the return of the portfolio and the expected 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)
gives the desired result.
.Understanding portfolio risk. Consider a portfolio with n assets. To compute the portfolio
variance we need to know the covariance Cov(ri, rj) of returns between all pairs of assets. The
variance can be calculated by multiplying the weights in the rows and columns by in the entries
in covariance matrix and then summing all terms.
Weights w1 . . . wj . . . wn
w1 Cov(r1, r1) . . . Cov(r1, ri) . . . Cov(r1, rn)
...
...
. . .
...
...
wi Cov(ri, r1) . . . Cov(ri, rj) . . . Cov(ri, rn)
...
...
...
. . .
...
wn Cov(rn, r1) . . . Cov(rn, rj) . . . Cov(rn, rn)
Table 1. Covariance matrix
18 DR SIMON A. FAIRFAX
Consider three portfolios of stocks A and B given in Example ??: 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?
Example 4.5
Solution 4.6. For a holding of two assets, the portfolio variance is
σ2p = w
2
Aσ
2
1 + 2wAwBcov(rA, rB) + w
2
Aσ
2
2
Representing 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
=
0.0016
0.0566× 0.199
=0.1421.
This suggests there is a weak positive correlation between the 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
σp1 =9.09%
(3) Portfolio consisting of 95% stock A and 5% stock B, then we get
r̂p3 =10.88
σ2p3 =0.00314
σp3 =5.61%
Conclusion
• 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 return than stock A.
• Hence, by forming a portfolio of assets, some of the risk inherent to the individual asset
can be reduced.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 19
4.8. Minimising the portfolio variance.
.Graphically representing σ2p as a function of weights Consider a portfolio of two assets
A and B with known fixed risks and rewards. Given wB = 1 − wA, we can view the portfolio
variance,
σ2p = w
2
Aσ
2
A + 2wAwBCov(rA, rB) + w
2
Bσ
2
B
as a function of wA. A simple substitution exercise gives,
σ2p(wA) = w
2
A[σ
2
A + σ
2
B − 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.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.01
0.02
0.03
0.04
ωA
σ2p
Portfolio of two assets A and B.
r̂A = 10.5%, σA = 5.55%,
r̂B = 18.0%, σB = 19.90%
and CovA,B = 0.0016
Sketch of σ2p = σ
2
p(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 4.2. For a portfolio consisting of two assets A and B with weights wA and
wB respectively, then the weight wA giving the minimum-variance portfolio is computed as
follows:
wA(min) =
σ2B − Cov(rA, rB)
σ2A + σ
2
B − 2Cov(rA, rB)
,
where
• Cov(rA, rB) = the covariance between the returns on assets A and B,
• σ2A = the variance on stock A, and
• σ2B = the variance on stock B,
with wB(min) = 1− wA(min).
Proof.
The portfolio variance is
σ2p =(wA)
2σ2A + (wB)
2σ2B + 2wAwBρABσAσB ,
with wB = 1− wA. Substituting wB into the preceding equality leads to:
σ2p = (wA)
2σ2A + (1− wA)2σ2B + 2wA(1− wA)ρABσAσB .
Minimizing the variance with respect to wA gives the first condition for a minimal wA(min).
(σ2p)
′ = 2wAσ2A + 2(−1)(1− wA)σ2B + 2
(
(1− wA) + (−1)wA
)
ρABσAσB .
20 DR SIMON A. FAIRFAX
The minimum-variance weight must satisfy (σ2p)
′ = 0 and hence we get
2wA(min)σ
2
A + 2(−1)(1− wA(min))σ2B + 2
(
(1− wA(min)) + (−1)wA(min)
)
ρABσAσB = 0.
Therefore
wA(min)
(
σ2A + σ
2
B − 2Cov(rA, rB)
)
= σ2B − Cov(rA, rB),
where Cov(rA, rB) = ρABσAσB .
Consider three stocks A, B and C. Assume that these stocks have the same expected
return and standard deviation. The following table shows the correlation between the
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
Given these correlations, the portfolio constructed from these stocks having the lowest
risk is a portfolio:
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.
Example 4.6
Solution 4.7. 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)
2σ2A + (wB)
2σ2B + 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 =0.5
2 × σ2 + 0.52 × σ2 + 2× 0.5× 0.5× 1× σ2
=σ2
Pc.: Equally invested in stocks C and B means that wB = wC = 50%. We have ρAC = −0.4.
σ2Pc =0.5
2 × 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 σPc, hence reducing the
portfolio variance.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 21
5. Diversification
In this section, we will investigate the effect of diversification in terms of reducing the risk.
In fact, we will see that, this reduction depends on the covariance between the returns on the
assets in the portfolio. Let’s consider portfolios with two assets with known fixed risk and rewards.
Studying the changes in the expected portfolio return
Using our definitions rˆp = w1rˆ1 + w2rˆ2 and w2 = 1− w1, we have the linear relationship
(5.1) rˆp = rˆp(w1) = rˆ2 + w1(rˆ1 − rˆ2).
between the weight of stock A and the portfolio return.
Studying the changes in the portfolio standard deviation
This time using
σ2p = w
2
1σ
2
1 + 2w1w2cov(r1, r2) + w
2
2σ
2
2 , w2 = 1− w1,
we arrive at the (w1, σp) relationships described by the relation
(5.2) σp(w1) =
√
w21σ
2
1 + 2w1(1− w1)cov(r1, r2) + (1− w1)2σ22
Worked example
Consider a portfolio consisting of
• Stock A. rˆA = 10% and σA = 12%.
• Stock B. rˆB = 15% and σB = 20%.
• Correlation between returns ρ variable.
We illustrate the concept of diversification by forming portfolios under various (five) different
correlation coefficients between the returns on stocks A and B. Table 2 below shows standard
deviations for different portfolio weights and different values of ρ.
Table 2. Expected return and standard deviation with varying weights
wA wB 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%
22 DR SIMON A. FAIRFAX
The following graph illustrates the relation between the portfolio standard deviation and the
stock weight for different values of ρ.
0.0 0.2 0.4 0.6 0.8 1.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
ωA
σ
ρ = 0
ρ = −0.5
ρ = 1
ρ = 0.4
ρ = −1
Portfolio of two assets A and B.
r̂A = 10.5%, σA = 5.55%,
r̂B = 18.0%, σB = 19.90%
and CovA,B = 0.0016
Sketch of σp = σp(wA) for different values of ρ.
For ρ = 0.4, the graph shows that the portfolio standard deviation decreases from 0.2 to a
minimal value (as a portfolio is concentrated in both stocks A and B with wA(min) = 0.86) and
then increases again to 0.12.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 23
Studying the relationship between rˆp and σp.
Combining Equations (5.1) and (5.2) we are able to describe an explicit relationship between
the portfolio standard deviation and its expected return. In our Worked Example, the formulation
is left as a exercise to be completed in tutorials. Use the data from our Worked Example we arrive
the following sketch.
0 0.05 0.10 0.15 0.20
0.10
0.11
0.12
0.13
0.14
0.15
ρ = −1
ρ = −0.5
ρ = −0.4
ρ = 0
ρ = 1
Risk σp
Return rp
Opportunity sets for various values of ρ.
Definition 5.1. 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. 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 return for a given level of risk.
Observations from the graph
(1) When the two stocks are perfectly correlated (i.e., ρ = 1), none of the risk of the individual
stocks can be eliminated by diversification.
(2) When the two stocks are correlated and the ρ decreases from 1 to −1, the risk can be
eliminated via diversification and the investor can benefit from it.
(3) When the two stocks are perfectly inversely correlated (i.e., ρ = −1), all risk can be
eliminated by investing a in the two stock (i.e., σp = 0). The opportunity set is linear and
offer a maximum advantage of diversification.
Concluding remarks
• The investor might make profit from diversification, when the correlation is less than 1.
• The lower the correlation, the greater the potential benefit from the diversification.
• For correlation equal to −1, it is possible to have perfect hedging opportunity (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.
• The benefits of diversification increase as more stocks are added to the portfolio.
24 DR SIMON A. FAIRFAX
6. Capital Asset Pricing Model
6.1. 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 portfolio C with risky assets and a risk-free asset. Let rp denote the risky rate of
return on assets in P with expected rate of return E(rp) and standard deviation σp. Let the
rate of return on the risk-free asset be rf . As an investor, suppose the acceptable risk in C is
determined to be σC = x, now the question remains, what is the expected return y = E(rC)?
.Now suppose a proportion wp is invested in P , hence wf = 1 − wp in invested in F . Now the
rate of return on the complete portfolio 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 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. This tells
us that the risk associated to the complete portfolio is proportional to the portion wp invested in
the risky asset. Re-arranging for wp and substituting into the equation for the expected return on
the complete portfolio gives
y = rf + x
(E(rp)− rf
σp
)
.
.The relationship between y = E(rC) and x = σC is clearly linear. This straight line is called the
Capital Allocation Line (CAL).
x = σC
y = E(rC)
E(rp) •P
•FE(rf )
σP
E(rp)− rf
Gradient S =
E(rp)−rf
σP
The Capital Allocation Line
.The CAL depicts all the risk-return combinations available to investors. The slope of the CAL
which is denoted S (for Sharp ratio) is a measure of the expected return of the complete portfolio
per unit of additional standard derivation.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 25
Definition 6.1. Given a portfolio of risky assets, Sharp’s ratio is defined as
S =
E(rp)− rf
σp
where
• E(rp) is the portfolio expected return
• σp the associated risk at the portfolio
• rf is the risk-free return.
This is referred to as the risk-to-volatility ratio.
Consider two stocks A and B such that Stock A has an expected return of 10% and a
standard deviation of 12% and Stock B has an expected 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 CALs of the minimum-variance portfolio (Portfolio 1)? What is the CAL of the
portfolio made off 90% in stock A and 10% in stock B (Portfolio 2).
Example 6.1
Solution 6.1. We know from Table 2 that σp2 = 11.74%, r̂p2 = 10.5% and it is easy to compute
σp1 = 11.72%, r̂p1 = 10.7%. Hence, the sharp ratios are given by
S1 =
10.7− 9
11.72
= 0.145, S2 =
10.5− 9
11.74
= 0.128,
from which it follows that
y1 = 0.145x+ 0.09,
y2 = 0.128x+ 0.09
The curve in the figure below represents the opportunity set of the portfolio consisting of asset A
and B.
Clearly S2 < S1, and the two portfolios intersect the opportunity set. Portfolio 1 and the risk
free asset dominates portfolio 2 and the risk free asset.
26 DR SIMON A. FAIRFAX
6.2. Portfolios of risky assets. Our next goal, an optimisation problem, is to find the portfolio
of risky assets for which Sharp’s ratio is maximised. During this module, we will study the case
with two risky assets, i.e., we assume 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 6.2. The optimal risky portfolio is the portfolio of risky assets with the
highest return-to-risk combination, measured by Sharp’s ratio, given the specific investor’s
tolerance for risk.
.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 =max
wA
( r̂p − rf
σp
)
= max
wA
( wAr̂A + wB r̂B − rf
[w2Aσ
2
A + w
2
Bσ
2
B + 2wAwBcov(rA, rB)]
1
2
)
=max
wA
( wAr̂A + (1− wA)r̂B − rf
[w2Aσ
2
A + (1− wA)2σ2B + 2wA(1− wA)cov(rA, rB)]
1
2
)
.
.We will need to apply some results from calculus (or read the recommended course book) to find
the value of wA which maximises this function, written w
∗
A, the result of which is summarised next.
Proposition. The weights of stock A and B in the optimal risky portfolio are given by:
w∗A =
RAσ
2
B −RBCov(RA, RB)
RAσ2B +RBσ
2
A − (RA +RB)Cov(RA, RB)
, w∗B = 1− w∗A.
where
• RA = rA − rf is the excess rate of return of stock A, and
• RB = rB − rf is the excess rate of return of stock B.
Consider a portfolios of two stocks A and B. Stock A has an expected return of 10% and
a standard deviation of 12%. Stock B has an expected return of 15% and a standard
deviation of 20%. 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 proportion of stock A should be
invested in the optimal risky portfolio? What is the expected return and the variance
on the optimal risky portfolio? What is its Sharpe ratio?
Example 6.2
Solution 6.2. We are given rf = 9%, RA = 10− 9 = 1, RB = 15− 9 = 6, ρAB = ρ = 0.4, σA =
12%, r̂A = 10%, σB = 20% and r̂B = 15%. We aim at finding: w
∗
A, σop, r̂op and Sop.
It follows from
w∗A =
0.01× 0.22 − 0.06× 0.0096
0.01× 0.22 + 0.06× 0.122 − (0.06 + 0.01)0.0096
=− 0.29
r̂op =w
∗
Ar̂A + (1− w∗A)r̂B = 16.48%
σop =(w
∗
A)
2σ2A + 2w
∗
A(1− w∗A)ρABσAσB + (1− w∗A)2σ2B = 24.73%
Sop =
r̂op − rf
σ̂op
= 0.302
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 27
We see the straight line CALop has a point of tangency with the opportunity set. The optimal
CAL is the line which is tangent to the opportunity set. In this case, the risky portfolio is called
the optimal risky portfolio.
6.3. 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 Captial Allocation Line. We consider
portfolios with two risky assets A, B and the risk-free asset F .
.An optimal complete porfolio is formed by taking the optimal risky portfolio with weight wp and
the risk free asset with weight wf . Using the above Propostion, we know in the risky portfolio
assets A and B are weighted w∗A and w
∗
B respectively. Hence in the optimal complete portfolio,
the proportion of the investors wealth to be invested in A and B is wAp = wpw
∗
A and w
B
p = wpw
∗
B .
Assuming the investor in Example 6.2 is required to have a efficient (optimal) complete
portfolio with a standard derivation of 12%. What is the expected return of the portfolio?
What is the proportion invested in the risk-free asset and each of the two risky funds?
Example 6.3
Solution 6.3. We aim to find rˆC , w
A
p and w
B
p in the optimal case. We are given w
∗
A = −0.29,
σC = 12% and σop = 24.73%. Using wop =
σC
σop
we find wop = 48.5% so wf = 51.5%. It follows,
rˆC = woprˆop + (1− wop)rf = 12.6%,
wAop = 48.5%× (−0.297) = −14.4%, wBop = 48.5%× (1− (−0.297)) = 62.9%
28 DR SIMON A. FAIRFAX
6.4. 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 6.3. The market portfolio M is a theoretical bundle of investments that in-
cludes every type of asset available in the investment universe, with each asset weighted in
proportion to its total presence in the market.
.The CML is a special case of the CAL where the risk portfolio is the market portfolio M . The
CML represents portfolios that optimally combine risk and return; the points of the line offer the
highest expected returns for a defined level of risk, or the lowest risk for a given level of expected
return. We consider the case whereby the market portfolio contains two risky stocks A and B.
.Understanding the key features of the Capital Market Line (red) in relation to the opportunity
set (blue).
Risk σp
Return rp
A•
B•
C•
•M
D•
Stock A•
Stock B•
The Optimal Capital Allocation Line (CML)
• Point A. This corresponds to an investment portfolio with zero risk, namely, all invest-
ments funds in the risk-free asset. The investor does not invest in the market portfolio.
• Point B. In this case, an investor is willing to take a risk, however their risk appetite is
lower than that of the market portfolio. To reduce the risk, funds are split between the
risk-free asset and the market portfolio.
• Point C. This is the minimum-variance portfolio relating to a portfolio with containing
stocks A and B only. This isn’t on the CML since low risks are not sufficient rewarded
with returns.
• Point M. This is the market portfolio also known as the tangent portfolio. All investors
are assumed to have arrived at this portfolio. An investor willing to take this amount of
risk would invest all their investment funds here.
• Point D. In this case, investors have an appetite for high risks which come with higher
expected returns. To form these portfolios the investor short sells the risk-free asset, i.e.,
borrows money at the risk-free rate, and invests the proceeds into the market portfolio.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 29
6.5. The Capital Asset Pricing Model (CAPM).
.CAPM is one of the modern centrepieces of financial economics. The model gives us a precise
prediction of the relationship that we should observe between the risk and expected return of
an asset. Using the CAPM we are able to perform two vital calculations. Firstly we are able
to quantify a benchmark rate of return associated to an investment which serves in determining
whether, for a given level of risk, the expected returns is more or less than its “fair value”.
Secondly, the models allows us to quantify the expected 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.
.Assumptions The CAPM is a set of predictions for the returns on risky assets. The foundation
of modern portfolio management started in 1952. The CAPM was published 1964. The models
begins with a collection of assumptions:
1. Individual behaviour
(a) Investors are rational mean-variance optimizers
(b) Their common planning horizon is a single period
(c) Homogeneous expectations. Investors all use identical information and all relevant
information is publicly available.
2. Market structure
(a) All assets are publicly held and trade on public exchannges
(b) Investors can borrow or lend at a common risk-free rate and they can take short
positions on traded securities
(c) No taxes nor transaction costs.
These are strong assumptions unlikely to stand up to direct observation or align with experi-
ence, but a good starting point. The CAPM presents a simple framework that gives a simple result.
.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 diversification. Interest rates, recessions and
catastrophes are examples of systematic 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 return that is not correlated with general market moves.
.Beta factors It is important to understand how the 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 returns rM of the market portfolio and compute the line of best fit.
30 DR SIMON A. FAIRFAX
For any given α and β the values of the random variable α+βrM can be regarded as predictions
for the return on the portfolio. The difference = ri − (α + βrM ) between the actual return ri
and the predicted return is called the residual random variable. The condition defining the line of
best fit yields optimal values α = αop and β = βop given by
αop = E(ri)− βop × E(rM ), βop = Cov(ri, rM )
σ2M
.
Definition 6.4. The beta factor of a given portfolio or individual security A is defined by
βA =
Cov(rA, rM )
σ2M
where rM and σ
2
M are the market return and market risk respectively.
The beta factor is an indicator of expected changes in the return on a particular security or
portfolio, say A, in response to the behaviour of the market as a whole. Notice that by definition,
the beta of the market portfolio βM , is equal to one, since
βM =
Cov(rM , rM )
σ2M
=
σ2M
σ2M
= 1.
Proposition 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
.

.Measuring total risk σ2: Given our interpretation ri = α + βrM + , we find the total risk
associated to the returns on an asset σ2 is
σ2 = V ar(ri) = V ar(α+ βrM + ) = β
2V ar(rM ) + V ar(),
hence
σ2i = V ar() + β
2
i σ
2
M .
The V ar() term is called the residual variance or diversifiable risk. It vanishes for the market
portfolio. The second term β2σ2M is called the systematic or undiversifiable risk. The market
portfolio involves only this kind of risk. The beta factor can be regarded as a measure of systematic
risk associated to an asset or a portfolio of assets.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 31
.Construction of the CAPM A basic principle of equilibrium is that all investments should
offer the same reward-to-risk ratio.
E(Ri)
Cov(Ri, RM )
=
E(RM )
σ2M
⇐⇒ E(Ri) = Cov(Ri, Rm)
σ2M
E(RM ).
The ratio Cov(Ri, Rm)/σ
2
M measures the contribution of stock i to the variance of the market
portfolio as a fraction of the total variance of the market portfolio. The ratio is called the beta
and is written βi. Arranging our equilibrium equation and using RA = rA − rf , we arrive at the
most familiar CAPM expression, the expected return - beta relationship
E(ri) = rf + βi(E(rM )− rf ) also written as rˆi = rf + βi(rˆM − rf ).
.The Security Market Line The straight-line graph in the (βi − rˆi) -plane is known as the
Security Market Line (SML). Since the market beta is 1 and the risk-free asset has beta zero, we
know (0, rf ) and (1,E(rM )) lie on the SML. Hence its gradient is E(rM )− rf ; also known as the
risk premium of the market portfolio or the Market Risk.
.Analysing Security Markets Line
• A security plotted on the SML is fairly priced. That is, the securities expected return is
appropriate given the securities level of risk. 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 return given the level of risk. Investors will want to take advantage these
types of securities.
• A security plotted below the SML is said to be overpriced. That is, you are getting less
expected return given the level of risk. Investors will want to avoid these types of securities.
• M depicts the market as a whole. The market has a beta of 1. Securities with betas
greater than 1 are more risky than the market and securities with betas less than 1 are
less risky than the market. A more risk adverse investor will choose securities with betas
of less than 1
The whole concept of the CAPM is that rational investors will choose to invest in securities
that lie on or above the SML.
32 DR SIMON A. FAIRFAX
.Analysing the values of β
• If an individual asset i (or portfolio) is chosen that is not efficient, then CML does not tell
anything about that asset. It seems useful to know, for example, how the expected excess
rate of return r̂i − rf , is related to M. The SML gives that relation.
• 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 the investor’s portfolio;
(3) The investor will ask a high expected return in compensation.
• If the stock has a negative beta, then:
(1) It will move “against” the market;
(2) It will decrease the risk of the market portfolio;
(3) The investor will accept a lower expected return.
Consider a portfolios of two stocks A and B. Stock A has an expected return of 10% and
a standard deviation of 12%. Stock B has an expected return of 15% and a standard
deviation of 20%. The correlation coefficient between the two stocks is ρ = 0.4 and the
risk free interest rate is 9% on the risk free asset. Find the betas on the stocks A and B.
Example 6.4
Solution 6.4. Using the SML, we get
βi =
r̂i − rf
r̂M − rf .
Hence
βA =
0.1− 0.09
0.1648− 0.09 = 0.13,
βB =
0.15− 0.09
0.1648− 0.09 = 0.8.
This means that stock A will rise by 1.3% if the market rises by 10% and fall by 1.3% if the market
falls by 10%.
MATH260. FINANCIAL MATHEMATICS PART I: DEPARTMENT OF MATHEMATICAL SCIENCES 33
You are interested in buying a Stock A. The current risk free rate is 9%. The market
return on this particular security is expected to be 16.48%. The beta of the security is
βA = 0.13.
• According to the CAPM, if you were told that the expected return in the next
one year on a stock A was to be 15%, would the security be overpriced or
underpriced?
• What happens in the case of an underpriced security? Does it always remain
underpriced?
Example 6.5
Solution 6.5. • According to the SML Equation, we get
r̂A =βA(r̂M − rf ) + rf
=10
The expected return, 15% is greater than the expected return given by the CAPM equa-
tion, 10%. Hence, investors are likely to receive higher returns than expected which makes
the security worth investing into. The security is therefore underpriced
• Once investors become aware that a security is underpriced they purchase the security
therefore raising its price. The price of the security will rise until the security is forced
downwards and it sits on the SML.
• If the security was overpriced, the investors will sell them as they are providing expected
returns less than that predicted by the CAPM equation. Investors will sell the security
until the price falls and the security is forced back up onto the SML
Email address: simonfairfax@liverpool.ac.uk