1

FRM Part I
Foundation of Risk Management (FRM)

Topic 5
Modern Portfolio Theory and The Capital Asset Pricing Model (CAPM)
Learning Objectives – Topic 5:
Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model (CAPM)

LO 5 a: Explain modern portfolio theory and interpret the Markowitz efficient frontier.

LO 5 b: Understand the derivation and components of the CAPM.

LO 5 c: Describe the assumptions underlying the CAPM.

LO 5 d: Interpret and compare the capital market line and the security market line.

LO 5 e: Apply the CAPM in calculating the expected return on an asset.

LO 5 f: Interpret beta and calculate the beta of a single asset or portfolio.

LO 5 g: Calculate, compare, and interpret the following performance measures: the Sharpe performance
index, the Treynor performance index, the Jensen performance index, the tracking error, information ratio,
and Sortino ratio.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 2

FRM Part I
Foundation of Risk Management (FRM)

Topic 5
Modern Portfolio Theory and The Capital Asset Pricing Model (CAPM)

LO 5 a: Explain modern portfolio theory and interpret the Markowitz efficient frontier.

PORTFOLIO - EXPECTED RETURN, VARIANCE, STANDARD DEVIATION, COVARIANCE AND

CORRELATION COEFFICIENT

1. Expected return on a portfolio

The expected return on a portfolio is really just the expected return on each individual asset within the
portfolio multiplied by its weighting in the portfolio. The weightings must obviously sum to 100%.

The equation for this can be expressed as follows:

E(Rp) = E(R1)W1 + E(R2)W2

Where:

E(Rp) = Expected return on the portfolio

E(R1) = Expected return on Asset 1
W1 = Weighting of Asset 1 in the portfolio
E(R2) = Expected return on Asset 2
W2 = Weighting of Asset 2 in the portfolio

Example - Expected return on a portfolio

Noodle Inc. is an investment holding company that has investments in two separate assets. The details
of the investments are as follows:

Asset 1 Asset 2

Initial investment $2m $3m

Expected return 20% 15%
Standard deviation 12% 10%
Correlation coefficient 0.6

Let us use the expected return formula to calculate the expected return on the portfolio:

E(Rp) = E(R1)W1 + E(R2)W2

$2m $3m
E(Rp) = (0.20)(($2m+$3m)) + (0.15)(($2m+$3m))

E(Rp) = (0.20)(0.40) + (0.15)(0.60)

E(Rp) = 0.17 or 17%

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 3

2. Variance of a portfolio

The variance of the portfolio is the sum of the variances of the individual assets in the portfolio taking
into account their correlation with one another.

The equation for this can be expressed as follows:

σ2 port = w12 σ12 + w22 σ22 + 2w1 w2 r1,2 σ1 σ2

Where:

σ2 port = Variance on the portfolio

σ12 = Variance on Asset 1
W1 = Weighting of Asset 1 in the portfolio
σ22 = Variance on Asset 2
W2 = Weighting of Asset 2 in the portfolio
r1,2 = Correlation coefficient – Asset 1 and 2
𝜎1 = Standard deviation on Asset 1
𝜎2 = Standard deviation on Asset 2

Example – Variance of a portfolio

Let us go back to our original example.

Let us use the portfolio variance calculation formula:

σ2 port = w12 σ12 + w22 σ22 + 2w1 w2 r1,2 σ1 σ2

σ2 port = (0.40)2(0.12)2 + (0.60)2(0.10)2 + 2(0.40)(0.60)(0.60)(0.12)(0.10)

σ2 port = 0.0023 + 0.0036 + 0.0034

σ2 port = 0.00936

3. Standard deviation of a portfolio

σport = √w12 σ2i + w22 σ22 + 2w1 w2 r1,2 σ1 σ2

Or
1
σport = [w12 σ2i + w22 σ22 + 2w1 w2 r1,2 σ1 σ2 ]2

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 4

Example – Standard deviation of a portfolio

The variance of the portfolio was calculated above as follows:

σ2 port = 0.00936

The standard deviation is merely the square root of the variance, which equals:

σport = √0.00936

σport = 0.0967 or 9.67%

4. Covariance calculation

Covariance is described as a measure of the degree to which two variables, such as rates of return on
investment assets, move together over time, relative to their individual average returns. When assessing
a portfolio, we are primarily concerned with the covariance of rates of return as opposed to the prices of
any variables.

A positive covariance means that the rates of return for two investments tend to move in the same
direction relative to their individual means during the same time period.

A negative covariance shows that the rates of return for two investments usually move in different
directions in relation to their means during precise time intervals over a period of time.

The magnitude of the covariance depends upon the variances of the individual investment’s returns and
the relationship between the investments.

For two assets, 1 and 2, the covariance of rates of return is defined as:

Cov1,2 = E[R1 - E(R1)][R2 - E(R2]

Where:

Cov1,2 = Covariance of returns between Asset 1 and 2

E(R1) = Expected return on Asset 1
R1 = Return on Asset 1
E(R2) = Expected return on Asset 2
R2 = Return on Asset 2

5. Correlation coefficient

The correlation coefficient is a standardized measure of the relationship between the returns of two
securities. The correlation coefficient ranges between -1.00 and +1.00.

A correlation coefficient of + 1 between two stocks indicates a positive linear relationship, that is the
two stocks move together in a completely linear manner. A negative correlation coefficient of -1.00
between two assets indicates a negative linear relationship, i.e. the two stocks move in opposite
directions.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 5

The correlation coefficient of the portfolio is calculated above as follows:

Cov1,2
r(1,2) =
(𝜎1 )(𝜎2 )

Where:

r(1,2) = Correlation coefficient of Asset 1 and 2

Cov1,2 = Covariance of returns between Asset 1 and 2
𝜎1 = Standard deviation on Asset 1
𝜎2 = Standard deviation on Asset 2

The correlation coefficient between the returns of securities 1 and 2, r(1,2) has no units since it is
simply a measure of the co-movement of the two stock’s returns.

MINIMUM-VARIANCE PORTFOLIOS

Minimum-variance portfolios represent that combination of portfolios that have the maximum rate of
return for every given level of risk, or the minimum risk for every level of return. This means that the
Minimum-variance frontier encompasses all of the best combinations of a portfolio.

However, before we touch upon the minimum variance portfolio, we would need to derive the portfolio
possibilities curve. This is the graph upon which all expected returns (y-axis) and standard deviations
(x-axis) of all possible asset combinations (infinite) are plotted. This derivation of this curve will lead us
to the minimum variance portfolio.

By way of example, let us work with our initial example.

Example - Expected return on a portfolio

Noodle Inc. is an investment holding company that has investments in two separate assets. The details
of the investments are as follows:

Asset 1 Asset 2

Initial investment $2m $3m

Expected return 20% 15%
Standard deviation 12% 10%
Correlation coefficient 0.6

We have worked on several different weightings in our example. However, keep in mind that in reality
there will be infinite possibilities for the weightings.

Weighting - Asset 1 100% 80% 60% 40% 30% 20% 0%

Weighting - Asset 2 0% 20% 40% 60% 70% 80% 100%

Expected return - port 20.00% 19.00% 18.00% 17.00% 16.50% 16.00% 15.00%
Standard deviation - port 12.00% 10.92% 10.12% 9.67% 9.60% 9.63% 10.00%

The expected return and standard deviation calculations for the portfolio have not been shown.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 6

The expected returns and standard deviations have been plotted in the following graph:
Expected
return

21,00%

100% of
20,00%

19,00%

18,00%

17,00%
Minimum Variance
Portfolio
16,00% 30% Asset 1
70% Asset 2

15,00%
100% of

14,00%
Standard
8,50% 9,00% 9,50% 10,00% 10,50% 11,00% 11,50% 12,00% 12,50% deviation

Points to consider:

• If we invested 100% in Asset 1 – then the portfolio return will reflect the expected return and
standard deviation of Asset 1.
• If we invested 100% in Asset 2 – then the portfolio return will reflect the expected return and
standard deviation of Asset 2.
• The minimum variance portfolio is the portfolio with the lowest standard deviation among all the
possibilities. We have marked this portfolio “minimum variance portfolio’. This portfolio reflects an
investment of 30% in Asset 1 and 70% in Asset 2.

DIVERSIFICATION AND CORRELATION

In the previous section we saw how to calculate the correlation coefficient, now let’s interpret it.

Essentially, as we explained earlier, the correlation coefficient is bounded by –1 and 1. This means that
the value can never exceed 1 and can never be lower than –1. The closer the value is to 1 or –1, the
stronger is the relationship between the variables, and the closer the value is to 0, the weaker the
relationship.

Thus,

R = 0 implies no linear relationship between the variables

R = 1 implies a perfect positive relationship between the variables
R = -1 implies a perfect negative relationship between the variables

Let us look at these various correlations:

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 7

No correlation

The original portfolio standard deviation calculation is as follows:

1
σport = [w12 σ2i + w22 σ22 + 2w1 w2 r1,2 σ1 σ2 ]2

When there is no correlation between the two assets, then the third term in the equation disappears
(since the correlation is zero). Based on this, the portfolio standard deviation calculation is as follows:
1
σport = [w12 σ2i + w22 σ22 ]2

In this case the standard deviation reduces to a non-linear equation and the portfolio possibilities curve
will also be non-linear.

The weightings for the portfolio can be solved as follows:

σ2
W1 = σ2 +σ
2
2
1 2

W2 = 1 – W1

Going back to our previous example, let us change the correlation coefficient to 0 and see the results on
the portfolio possibilities curve.

Expected
return

Correlation coefficient = 0
21,00%

20,00%

19,00%

18,00%

17,00%

16,00%

15,00%

14,00% Standard
7,00% 8,00% 9,00% 10,00% 11,00% 12,00% 13,00% deviation

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 8

Perfect positive correlation

The original portfolio standard deviation calculation is as follows:

1
σport = [w12 σ2i + w22 σ22 + 2w1 w2 r1,2 σ1 σ2 ]2

When there is perfect positive correlation between the two assets, in other words a correlation
coefficient of 1, then the portfolio standard deviation calculation is reduced as follows:

σport = w1 σi + w2 σ2

In this case the expected return is linear (the sum of the asset returns multiplied by their weightings)
and the standard deviation is linear the portfolio possibilities curve will also be linear.

Keep in mind that no diversification is achieved when the correlation coefficient is equal to 1.

Going back to our previous example, let us change the correlation coefficient to +1 and see the results on
the portfolio possibilities curve.
Expected
return

Correlation Coefficient = +1
25,00%

20,00%

15,00%

10,00%

5,00%

0,00% Standard
10,00% 10,50% 11,00% 11,50% 12,00% 12,50% deviation

Partially positive correlation

There will also be a case, which is more real-world, where the correlation is moderately positive, for
example a correlation coefficient of 0.5. In this case, just like in the case of zero correlation, the assets
will have a non-linear portfolio possibility curve.
In such a case, the portfolio standard deviation calculation is reduced as follows:
1
σport = [w12 σ2i + w22 σ22 + w1 w2 σ1 σ2 ]2

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 9

Going back to our previous example, let us change the correlation coefficient to +0.5 and see the results
on the portfolio possibilities curve.

Expected
return

Correlation coefficient = +0.5

21,00%

20,00%

19,00%

18,00%

17,00%

16,00%

15,00%

14,00% Standard
8,50% 9,00% 9,50% 10,00% 10,50% 11,00% 11,50% 12,00% 12,50% deviation

Perfect negative correlation

The original portfolio standard deviation calculation is as follows:

1
σport = [w12 σ2i + w22 σ22 + 2w1 w2 r1,2 σ1 σ2 ]2

When there is perfect negative correlation between the two assets, in other words a correlation
coefficient of -1, then the portfolio standard deviation calculation is reduced as follows:

σport = w1 σi − w2 σ2 OR −w1 σi + w2 σ2

In this case, a portfolio with zero volatility can be constructed. The standard deviation will be set to zero
and the portfolio weight can be solved for as follows:

σ2
W1 =
σ1 +σ2

W2 = 1 – W1

In this case the standard deviation reduces to two linear equations, and the portfolio possibilities curve
will also be linear (two line segments).

Keep in mind that perfect diversification is achieved when the correlation coefficient is equal to -1.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 10

Going back to our previous example, let us change the correlation coefficient to -1 and see the results on
the portfolio possibilities curve.

Expected
return

Correlation coefficient = -1
22,00%
21,00%
20,00%
19,00%
18,00%
17,00%
16,00%
15,00%
14,00%
13,00% Standard
0,00% 2,00% 4,00% 6,00% 8,00% 10,00% 12,00% 14,00% deviation

As can be seen from the graph, in this case there are two lines for a perfectly negative correlation and in
the case of negative correlation we can construct a zero-volatility portfolio (in other words the portfolio
has a standard deviation of zero).

THE EFFICIENT FRONTIER

When looking at the shape of the portfolio possibilities curve (other than in cases of correlations of +1
or -1) the curve looks as follows:

Expected
return

21,00%
Efficient B
20,00% Frontier

19,00%

18,00%

17,00%
A Minimum Variance C
16,00% Portfolio

15,00%

14,00% Standard
deviation
8,50% 9,00% 9,50% 10,00% 10,50% 11,00% 11,50% 12,00% 12,50%

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 11

Points to consider:

There are two parts to the portfolio possibilities curve.

• The part above the minimum variance portfolio – that is concave.

• The part below the minimum variance portfolio – that is convex.

In the graph above every portfolio that lies on the efficient frontier (in our example portfolios A and B),
has either a higher rate of return for an equal level of risk or lower level of risk for an equal rate of
return.

From the graph we can deduce the following:

• Portfolio A dominates portfolio C because it has an equal rate of return but substantially less risk.
• Portfolio B dominates portfolio C because it has an equal risk but a higher expected rate of return.
• The curve shows that the slope of the efficient frontier curve decreases steadily as you move
upward. This is because, for any investor, a point will be reached where the investor will not be
willing to accept the increased risk, no matter what the expected return. In other words, there must
be some point where the expected return cannot be any greater, even though the risk has increased.

Portfolios on the efficient frontier cannot dominate one another as they all have different risk measures
and expected rates of return that become greater as risk increases.

The portfolio on the efficient frontier with the lowest risk is called the global minimum-variance
portfolio.

The efficient frontier and short sales

Up to this point we have been dealing with long-only transactions. We now have a look at the effect that
short sales have on the shape of efficient frontier.

Let us go back to our original example, and assume short sales of the asset with the lower expected
returns, in other words Asset 2.

We have worked on several different weightings in our example. However, keep in mind that in reality
there will be infinite possibilities for the weightings.

Weighting - Asset 1 0% 25% 50% 75% 100% 125% 150%

Weighting - Asset 2 100% 75% 50% 25% 0% -25% -50%

Expected return 15.00% 16.25% 17.50% 18.75% 20.00% 21.25% 22.50%

Standard deviation 10.00% 9.60% 9.85% 10.69% 12.00% 13.65% 15.52%

The expected return and standard deviation calculations for the portfolio have not been shown.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 12

The expected returns and standard deviations have been plotted in the following graph:

Expected
return

24,00%

22,00%

20,00%

18,00%

16,00%

14,00% Standard
8,00% 9,00% 10,00% 11,00% 12,00% 13,00% 14,00% 15,00% 16,00% deviation

Points to consider:

As a result of adding short sales to the efficient frontier, we observe the following:

• The frontier extends further to the right (in other words, higher return and risk portfolios).
• Shorting allows us to create portfolios that would not be possible otherwise.

We will now add in the risk-free asset to our discussion.

The efficient frontier and the risk-free rate

Once we introduce the risk-free asset into our portfolio, we will derive a linear formula for the expected
standard deviation of the portfolio. The shape of the efficient frontier changes from being curved to
being a straight line.

When a risk-free asset is combined with risky assets in a portfolio, the standard deviation of a portfolio
will be the linear proportion of the standard deviation of the risky asset portfolio. In other words, the
standard deviation of the whole portfolio will be determined by the standard deviation of the risky
assets only since the standard deviation of the risk-free asset is equal to zero.

The risk-return combination

Both the expected return and standard deviation of returns for a portfolio with a risk-free asset and
risky assets are linear combinations. Thus, a graph of risk and return would resemble a straight line.

An investor can attain any point along the straight line (risk free rate), by investing some portion of the
portfolio in the risk-free asset and the remainder in the risky asset portfolio on the efficient frontier.

Let us first have a look at the expected return on a portfolio comprising risky and risk-free assets.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 13

The equation for this can be expressed as follows:

E(Rp) = E(RR)WR + E(RF)WF

Where:

E(Rp) = Expected return on the portfolio

E(RR) = Expected return on Risky asset
WR = Weighting of Risky asset
E(RF) = Expected return on Risk-free asset
WF = Weighting of Risk-free asset

Now let us look at how the variance and standard deviation are affected when the risk-free asset is
introduced to the portfolio: The initial equation for the risky portfolio was expressed as follows:

σ2 port = w12 σ12 + w22 σ22 + 2w1 w2 r1,2 σ1 σ2

Keep in mind that the standard deviation of the risk-free asset is zero and so to the covariance between
the risky and risk-free asset is also zero. Based on this, the variance portfolio will reduce to:

σ2 port = wP2 σ2P

Where:

σ2 port = Variance on the total portfolio (Risky + risk-free assets)

σ2P = Variance of the risky asset
WP = Weighting of the risky asset

The standard deviation of the portfolio will equal:

σ port = √wP2 σ2P

Note that the efficient frontier has now become a straight line – we call this straight line the Capital
market line. This is indicated in the graph below.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 14

Looking at the graph we see that the CML has the following characteristics:

• The intercept is equal to the risk-free rate.

• The slope is equal to the risk-reward ratio for a risky portfolio.
• The capital market line lies tangent to the efficient frontier.
• Where the CML and the efficient frontier touch is called the market portfolio or M. This portfolio
contains all the risky assets in proportion to their total market values.

Now, depending on each investors risk appetite, they will choose where on the line they wish to invest.
However, all investors will use the same portfolio - called the market portfolio.

Now that we have worked on the fundamental concepts behind portfolio management let us begin with
the Capital Asset Pricing Model.

LO 5 b: Understand the derivation and components of the CAPM.

The total risk of an asset consists of two parts - the non-diversifiable risk and the diversifiable risk.

Systematic risk

This is the non-diversifiable risk. This is the variability of returns due to macroeconomic factors that
affect all risky assets. Since it affects all risky assets, it cannot be eliminated by diversification.

Unsystematic risk

This is the diversifiable risk that is unique to an asset, derived from its particular features.

Only systematic risk remains in the market portfolio. Systematic risk is measured by the standard
deviation of returns of the market portfolio and can transform over time with changes in the
macroeconomic variables that affect the appraisal of all risky assets. Such variables include interest rate
volatility, industrial production, corporate earnings and cash flow. Systematic risk is the only relevant
risk as unsystematic risk can be easily eliminated through diversification.

The concept of diversification

Diversification of investments, as a concept proposed by Markowitz, is used for the primary purpose of
reducing the total risk of a portfolio to enhance the potential of expected overall returns. Diversification
involves selecting investments that will work collectively to produce an expected ultimate positive
result for an investor.

As more securities are added to a portfolio, total risk is likely to fall at a decreasing rate. Some
researchers say that about 12-18 stocks means the portfolio is well diversified. Over 18 stocks, most of
the risk should be systematic, and there should be almost no specific risk left. Others say that 30 stocks
means the portfolio is well diversified.

Based on the definition and characteristics of the CML (which we dealt with in the introductory
reading), we saw that it can only be used to measure the expected return for a diversified portfolio. In
the event that a portfolio is an individual security or not diversified then the CAPM will need to be used.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 15

The whole concept behind the CAPM is that an investor is only compensated for taking systematic risk.
The risk-return trade-off is thus a graph of return versus beta, this line is called the Securities market
line (SML).
The Capital market line (CML) is a graph of return versus standard deviation (total risk). When a
portfolio is perfectly diversified, this means there is no unsystematic risk. In such a case, the standard
deviation or total risk comprises only systematic risk and thus gives the same measure as beta.

Now that we have understood the concepts behind diversification, the CAPM and the SML, let us look at
deriving the model.

The end result of the model can be expressed as follows:

E(r) = Rf + ß(Rm - Rf)

Where:

E(r) = The return required by equity holders

Rf = The risk-free rate (This is often displayed as the rate of return provided by a government
bond, both of which are considered to be almost risk free.)
ß = The beta of the share (The beta of a share is an indicator of risk. Thus, a share with a beta of 2
compared to a beta of 1 will on average be expected to generate twice the return on the market
compared to the risk-free rate. Thus, a beta of 2 shows more risk than a beta of 1)
Rm = Return on the market portfolio

Deriving the CAPM:

Step 1:

Beta identifies the level of risk that the investor needs to be compensated for. This risk is the
unsystematic or idiosyncratic risk. As the portfolio becomes more diversified this risk gets smaller and
we are eventually only left with systematic risk. Keep in mind that diversification does not cost anything
and as such the investor should only be compensated for systematic risks (in other words risks that
cannot be diversified away).

An equally weighted portfolio variance = Unsystematic risk (variance term) + Systematic risk
(Covariance term).

This can be expressed as follows:

1 n−1
𝜎P2 = ̅𝜎̅̅i2̅ + ̅̅̅̅̅
Cov
n n

What is interesting to note is that as the size of the portfolio increases, the variance gets close to the
average covariance. The reason for this is that the first term (variance term) gets close to zero as the
size of the ‘n’ increases, and the second term will approach the average covariance as ‘n’ increases,
n−1
because n will approach 1.

Step 2:

We need to remember that the expected return calculation (CAPM) function is a linear function of beta.
This can be proven mathematically. We need to remember that individual assets with the same beta
coefficients should offer the same expected returns. The reason for this is due to arbitrage.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 16

This linear function can be expressed as follows:

E(Rp) = a + ßp x m

Where:

Cov
ß = Beta = 𝜎2P,M
m
𝜎2m = Variance of returns on the market portfolio.

Since all assets with the same beta coefficients should offer the same expected returns. We arrive at the
following equation:

E(Ri) = a + ßi x m

This equation can be plotted onto a graph that looks as follows:

What we see from the graph is that the SML is merely a graphical representation of the CAPM.

Step 3:

Solve for the CAPM. To do this we use the two points on the line, the risk-free asset and the market
portfolio. The beta of the risk-free asset is zero (since it carries no risk) and the beta of the market
portfolio is equal to one. With this information we can solve for the slope or ‘m’. By solving for ‘m’
(slope) we arrive at ‘m’ = the ERP (the equity risk premium). Inserting this into our equation we arrive
at:

E(r) = Rf + ß(Rm - Rf)

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 17

LO 5 c: Describe the assumptions underlying the CAPM.

The Capital Asset Pricing Model (CAPM) is based on the following assumptions:

1. Investors hold diversified portfolios and that the Stock Exchange is an efficient market at the semi
strong level. When a stock exchange operates at semi strong efficiency, this implies that current
market prices reflect all publicly available information.

2. All investors have the same expectations regarding the expected returns, variances and covariances
of all shares – homogenous expectations.

3. Borrowing and lending opportunities are available to all investors at a publicly quoted risk free rate
of interest.

4. An unlimited amount of short selling is allowed.

5. There are no transaction costs to other impediments to trading – frictionless markets.

6. A perfect market exists with perfect information available to investors.

7. There is either no taxation or equal taxation for all participants.

8. All assets are infinitely divisible – divisible assets.

9. All assets are marketable.

10. An investor’s utility function is based solely expected return and risk.

11. The time period that investors are concerned about for expected return and risk is one year.

LO 5 d: Interpret and compare the capital market line and the security market line.

Once we introduce the risk-free asset into our portfolio, we will derive a linear formula for the expected
standard deviation of the portfolio. The shape of the efficient frontier changes from being curved to
being a straight line.

When a risk-free asset is combined with risky assets in a portfolio, the standard deviation of a portfolio
will be the linear proportion of the standard deviation of the risky asset portfolio. In other words, the
standard deviation of the whole portfolio will be determined by the standard deviation of the risky
assets only since the standard deviation of the risk-free asset is equal to zero.

The risk-return combination

Both the expected return and standard deviation of returns for a portfolio with a risk-free asset and
risky assets are linear combinations. Thus, a graph of risk and return would resemble a straight line.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 18

An investor can attain any point along the straight line (risk free rate), by investing some portion of the
portfolio in the risk-free asset and the remainder in the risky asset portfolio on the efficient frontier.

Note that the efficient frontier has now become a straight line – we call this straight line the Capital
market. This is indicated in the graph below.

Looking at the graph we see that the CML has the following characteristics:

• The intercept is equal to the risk-free rate.

• The slope is equal to the risk-reward ratio for a risky portfolio.
• The capital market line lies tangent to the efficient frontier.
• Where the CML and the efficient frontier touch is called the market portfolio or M. This portfolio
contains all the risky assets in proportion to their total market values.

Now, depending on each investors risk appetite, they will choose where on the line they wish to invest.
However, all investors will use the same portfolio - called the market portfolio.

The Market risk premium

Any investor that chooses to take on no risk will earn the risk-free rate.
The difference between the risk-free rate and the market rate is called the market risk premium.
This CML equation can be expressed as follows:
σP
E(Rp) = Rf + (E(Rm) − Rf)
σM

Based on this equation we can see that the investor can expect to earn:

1. The risk-free rate, +

2. The market risk premium X additional market risk that he is willing to accept.

The intercept of the CML is the risk-free rate.

The slope of the CML equates to the Sharpe ratio, also known as the market price of risk.
FRM Part I – Foundations of Risk Management
Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 19

(E(Rm) − Rf)
σM

Based on the definition and characteristics of the CML, it can only be used to measure the expected
return for a diversified portfolio. In the event that a portfolio is an individual security or not diversified
then the CAPM will used. For this we introduce the security market line.

LO 5 e: Apply the CAPM in calculating the expected return on an asset.

Evaluating a security

To determine whether an asset is an appropriate investment, we can compare the required rate of
return of the asset with the estimated rate of return over a specific investment horizon.

In order to make this comparison, an independent estimate is required of the return outlook of the asset
using either fundamental or technical analysis techniques.

The output of the analysis provides analysts with price and dividend projections that allow the analyst
to compute the estimated rates of return he/she would expect during the holding period.
The computed estimates are compared with the expected return computed below to determine whether
a security is under, over or properly valued.

The expected rate of return for a risky asset is determined by the risk-free rate (RFR) plus a risk
premium for the individual asset. The risk premium is determined by the systematic risk of the asset
(Beta), and the prevailing market risk premium.

Note:
The risk-free rate is sometimes referred to as the y-intercept of alpha.

Example: Evaluating a security

The following graph is provided:

Consider the security market line for stock C above with the following information provided:

The beta for stock C is 1.20.

How accurately is the stock priced?

FRM Part I – Foundations of Risk Management
Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 20

Step 1:

Calculate the expected rate of return using the SML equation:

E(r) = Rf + ß(Rm - Rf)

= 0.05 + 1.20 (0.13-0.05)

= 0.05 + 0.096

= 0.146

= 14.6%

Step 2:

Now compare the estimated return to the expected return. This is done by reading the value from the
graph. Let us assume that the corresponding return on the graph when the beta is 1.2 is 15%. We have
calculated that the expected return is 14.6%. Our calculation tells us that the expected return is 14.6%,
but we know from the graph that we should be achieving a 15% return from an investment that has this
level of risk. Therefore, if at this level of risk we should be achieving a 15% return, but we are only
managing a 14.6% return, then the asset is overvalued

• This illustrates that if the estimated return is more than the expected return, the stock is
undervalued.
• If the estimated return is the same as the expected return, the stock is properly valued.
• If the estimated return is less than the expected return, the stock is overvalued.

Example: Expected return

An analyst gathers the following data:

• Expected (estimated) rate of return on the market = 19%

• Risk free rate = 11%
• Expected (estimated) rate of return on Stock A = 22%
• Stock A’s beta = 1.30

Using the above data and the capital asset pricing model, what can you conclude about the value of Stock
A?

The capital asset pricing model assumes that all security returns fall on the SML. Stocks that plot above
the SML earn excess returns and stocks that plot below the SML are under performing.

Using the CAPM, the stock should get a return of:

E(r) = Rf + ß(Rm - Rf)

= 0.11 + (1.30)(0.19-0.11)

= .214

= 21.4%

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 21

The stock is expected to get a return of 22%, which is slightly more than its CAPM return above. Thus, it
is underpriced since it has an excess return.

The Capital Market Line (CML)

The capital market line (CML) is the line from the y-intercept that represents the risk-free rate tangent
to the original efficient frontier; it becomes the new efficient frontier since investments on this line
dominate all the portfolios on the original Markowitz efficient frontier. The existence of a risk-free asset
results in the derivation of a capital market line (CML).

The equation of the SML, together with estimates for the return on a risk-free asset and the market
portfolio, can be used to generate expected or required rates of return for any asset based upon its
systematic risk, beta.

In summary

The whole concept behind the CAPM is that an investor is only compensated for taking systematic risk.
The risk-return trade-off is thus a graph of return versus beta, i.e. the SML. The CML is a graph of return
versus standard deviation (total risk). When a portfolio is perfectly diversified, this means there is no
unsystematic risk. In such a case, the standard deviation or total risk comprises only systematic risk and
thus gives the same measure as beta.

TYPICAL EXAM QUESTION:

Suppose the S&P 500 Index has an expected annual return of 7.6% and volatility of 10.8%. Suppose
the Atlantis fund has an expected annual return of 7.2% and volatility of 8.8% and is benchmarked
against the S&P 500 index. If the risk-free rate is 2.0% per year, what is the beta of the Atlantis Fund
according to the CAPM?

A. 0.81
B. 0.93
C. 1.13
D. 1.23

Answer: B

Since the correlation or covariance between the Atlantis Fund and the S&P 500 Index is not known,
CAPM must be used to back out beta: Ṝι = RF + βί * (ṜM – RF).
(7.2%−2.0%)
Therefore: 7.2% = 2.0% + βί *(7.6% –2.0%); hence βί = (7.6%−2.0%) 𝑜𝑟 0.93

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 22

LO 5 f: Interpret beta and calculate the beta of a single asset or portfolio.

Beta is a standardized measure of systematic risk based upon an asset’s covariance with the market
portfolio.

We know that an asset that has a beta of 1 will mirror the performance of the market as a whole. For
such an asset, if the market as a whole shows an increase of 10% in a particular year, then we expect
that asset to also show an increase of 10% in the same year.

The market portfolio has a beta of 1. If the beta for an asset is more than 1.0, the asset has higher
normalized systematic risk than the market. This means that the asset is more volatile than the total
market portfolio.

To calculate beta in practice, we would need to regress the returns of the specific asset against the
returns of the market. When using the regression analysis, we would use the following inputs:

• Excess return on the specific asset = Dependent variable.

• Excess return on the market = Independent variable.
• The slope of the regression line = Estimate of Beta = Security line.

Calculation of Beta

The following formulas can be used to calculate Beta:

Covi,m
Beta =
𝜎2m
𝜎market 𝜎stock pmarket,stock
Beta =
𝜎 2 market
𝜎stock
Beta = ( ) (pmarket,stock )
𝜎market

Portfolio beta

The beta of a portfolio can be calculated by summing the individual company betas within the portfolio
multiplied by their weights in the portfolio. The formula would look as follows:

ßp = W1ß1 + W2ß2 + Wkßk

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 23

LO 5 g: Calculate, compare, and interpret the following performance measures: the

Sharpe performance index, the Treynor performance index, the Jensen performance
index, the tracking error, information ratio, and Sortino ratio.

The following three measures relate return to a certain level of risk.

1. Jensen’s Alpha (also known as ex post alpha or simply alpha)

The difference between the actual return and that required to compensate for the systematic risk is
called alpha. Alpha makes use of the ex post SML (security market line) as a benchmark for performance
appraisal.

αP = R Pt − R P

Where:

αP = Ex post alpha on the portfolio

R Pt = Actual return on the portfolio in period t
RP = R F + βP (R M − R F ) = The account return predicted by the SML and CAPM; where RF is the
risk-free rate of return, βP is the portfolio’s beta (systematic risk) and R M is the expected market
return.

2. The Treynor Measure

The Treynor measure is similar to alpha in that it uses beta, i.e., systematic risk. Graphically, a manager
with a positive alpha will plot above the SML. The slope of the line that connects the intercepts on the y-
axis, i.e., the risk-free rate and the portfolio is the Treynor measure. A portfolio with a positive alpha
will, of course, have a Treynor greater than the market’s Treynor, and vice versa.

E(Rp) − Rf
TP =
βA

Where:

E(R p ) = Portfolio return

Rf = Risk-free return
βA = Portfolio beta

3. The Sharpe Ratio

Unlike the previous two measures, the Sharpe ratio considers total, not only systematic risk. Standard
deviation is Sharpe’s measure of total risk, and, therefore, would be plotted against the CML, which also
formulates risk as standard deviation. The Sharpe ratio of the market is the slope of the CML. The line
between the risk-free rate and a portfolio’s return, standard deviation co-ordinate is the CAL of that
portfolio and its slope is its Sharpe ratio.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 24

A portfolio with superior performance will have a steep-sloping CAL and a high Sharpe ratio.

E(Rp) − Rf
SP =
σP

Where:

E(R p ) = Portfolio return

Rf = Risk-free return
σP = Portfolio standard deviation

Many consider the Sharpe ratio a more accurate historical performance measure and Treynor to be a
more forward-looking measure – as the beta measure needs to be estimated for.

There are definite relationships that exist between the three measures. Two common approximations
are:
𝜎P
1. For a well-diversified portfolio, BetaP ≈ 𝜎M
Treynor
2. Sharpe ratio ≈
𝜎M

Example: Performance measures

Data relating to four asset management firms is presented below for appraisal purposes:

Fund A Fund B Market Index

Return 8.88% 9.44% 7.40%
Beta 1.04 1.42 1.00
Std deviation 5.12% 3.69% 2.85%

The risk-free rate of return for the period was 3.5%. Calculate and rank the funds using Jensen’s alpha,
Treynor measure and the Sharpe ratio.

Fund A

αP = R Pt − R P
R P = R F + βP (R M − R F ) = 0.035 + 1.04(0.074 – 0.035) = 7.56%
Alpha = 8.88% - 7.56% = 1.32%

Treynor = (8.88 – 3.5)/1.04 = 5.17

Sharpe = (8.88 – 3.5)/5.12 = 1.05

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 25

Fund B

R P = 0.035 + 1.42(0.074 – 0.035) = 9.04%

Alpha = 9.44% - 9.04% = 0.4%

Treynor = (9.44 – 3.5)/1.42 = 4.18

Sharpe = (9.44 – 3.5)/3.69 = 1.61

Fund A ranks first according to Alpha and Treynor, but second according to Sharpe. This is a result of
Fund A having higher total risk but lower systematic risk than Fund B.

Summary

• Both Alpha and Treynor measure risk systematic risk (beta). A manager with a positive alpha will
have a Treynor measure above the market Treynor.

• Both Alpha and Treynor are criticized for making the assumptions of the CAPM and for using beta as
a risk measure. Aspects that have been criticized include: (1) the assumption of a single priced risk
without accounting for other sources of risk and (2) the use of a proxy (such as the ALSI Index) for
the market. Small changes in what is assumed to be the market can significantly alter the Alpha and
Treynor measures.

• The fact that these measures are all ex post means that they are historical and may not reflect the
future reality. A manger may perform better or worse in future periods. Therefore, these measures
are not always justifiably used in judging managers.

• Alpha, Treynor and Sharpe enjoy more widespread use than the other measures.

Jensen’s Alpha – extensions

In truth, Jensen’s measure is just a set return (portfolio return) in excess of a reference return.
Keep in mind that the reference return used in Jensen is the CAPM return. Based on this, the reference
return could be replaced by a more applicable return.
An example of such a return could the return that is based on CML or a return based on a multifactor
model.

This would change the Jensen formula to the following:

αP = R Pt − R REF

Where:

αP = Ex post alpha on the portfolio

R Pt = Actual return on the portfolio in period t
R REF = Return on the reference portfolio

Tracking error. Tracking error is the volatility (standard deviation) of the active return. If the
benchmark is suitable, the tracking error should be smaller than the volatility of the difference between
the portfolio return and that of the market. If this is not the case, the benchmark may not be capturing
the manager’s style to a sufficient degree.
Tracking error can be described as the standard deviation of active returns.
Another term that is used for the standard deviation is active risk.
FRM Part I – Foundations of Risk Management
Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 26

The equation used to express tracking error is:

TE = s(Rp – RB)

Where:

TE = Tracking error
s = Sample standard deviation
Rp = Portfolio return
RB = Benchmark return

An aggressive investment manager would seek to achieve a high tracking error.

A passive investment manager would seek to achieve a low tracking error.

We also use the information ratio (IR) when evaluating mean active returns per unit of active risk. The
IR is similar to the Sharpe measure, which looks at evaluating absolute returns.

The equation used to express the IR is:

Portfolio mean return−Benchmark mean return

IR =
Tracking error

The Sortino ratio

The Sortino ratio works with downside deviation. The numerator is calculated as the return on the
portfolio less a minimum acceptable return (MAR) for the portfolio. The denominator is the standard
deviation of returns that fall below the MAR. The reason that we only use standard deviation of returns
that fall below the MAR is that often the standard deviation of all returns unfairly inflates the risk
measure if very good returns are experienced.

R −MAR
Sortino ratio = Standard deviation pof returns below the MAR

Where:

MAR = Minimum acceptable return

(RPt −Rmin )2
Standard deviation of returns below the MAR = √
N

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 27

EDGE CHECKERS
Topic 5 – Modern Portfolio Theory and The Capital Asset Pricing
Model (CAPM)

1. 1
The following formula: σport = [w12 σ2i + w22 σ22 ]2 is the formula used to calculated portfolio standard
deviation, assuming a correlation coefficient of?

A. Zero
B. +1
C. -1
D. +0.5

2. When deriving the Capital Asset Pricing Model. Which of the following statements is correct?

I. The first term (variance term) gets close to zero as the size of the ‘n’ increases.
II. The second term will approach the average covariance as ‘n’ increases.

A. I only.
B. II only.
C. Both I and II.
D. Neither I or II.

3. Which of the following statements relating to the CAPM assumptions is not correct?

A. A limited amount of short selling is allowed.

B. There are no transaction costs to other impediments to trading – frictionless markets.
C. A perfect market exists with perfect information available to investors.
D. There is either no taxation or equal taxation for all participants.

4. Which of the following statements relating to the CAPM assumptions is not correct?

A. All assets are infinitely divisible – divisible assets.

B. All assets are marketable.
C. An investor’s utility function is based solely expected return.
D. The time period that investors are concerned about for expected return and risk is one year.

5. Which of the following statements relating to the Capital market line is not correct?

A. The intercept is equal to the risk-free rate.

B. The slope is equal to the risk-reward ratio for a risky portfolio.
C. The capital market line below the efficient frontier.
D. Where the CML and the efficient frontier touch is called the market portfolio or M.

6. Harry Markham, FRM, and his boss Jerry Jones, CFA, are discussing the capital market line (CML) and the
securities market line (SML).
Markham states that the CML can only be used in the case of a diversified portfolio.
Jones comments that in the event that a portfolio is not diversified then the CAPM will used.

Which of the above statements are correct?

A. Markham only.
B. Jones only.
C. Both Markham and Jones.
D. Neither Markham and Jones.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 28

7. An analyst gathers the following data:

• Expected rate of return on the market = 19%

• Risk free rate = 11%
• Expected rate of return on Stock A = 22%
• Stock A’s beta = 1.10

Using the above data and the capital asset pricing model, what can you conclude about the value of Stock A?

A. It is underpriced
B. It is overpriced
C. It is fairly priced
D. There is not enough information given to be able to calculate the price.

8. Which of the following statements relating to the Capital market line is are correct?

I. The capital market line (CML) is the line from the y-intercept that represents the risk-free rate tangent to
the original efficient frontier.
II. Investments on this line dominate all the portfolios on the original Markowitz efficient frontier

A. I only.
B. II only.
C. Both I and II.
D. Neither I or II.

9. An analyst developed the following data on Stock A and the market:

Return on the market = 0.16

Covariance between the return on Stock A and the return on the market = 0.038
Correlation coefficient between the return on Stock A and the return on the market = 0.70
Standard deviation of the returns on Stock A = 0.17
Standard deviation of the returns on the market = 0.23

What is the beta of Stock A?

A. 0.165
B. 0.718
C. 1.00
D. 1.315

10. Which of the following statements relating to beta are correct?

I. Beta is a standardized measure of systematic risk based upon an asset’s covariance with the market
portfolio.
II. To calculate beta in practice, we would need to regress the returns of the specific asset against the
returns of the market.

A. I only.
B. II only.
C. Both I and II.
D. Neither I or II.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 29

11. Data relating to Fund A is presented below for appraisal purposes:

Fund A Market Index

Return 9.00% 8.00%
Beta 1.04 1.00
Std deviation 5.00% 3.00%
Risk-free rate 4%

Jensen’s alpha is equal to:

A. 1.00%
B. -0.2%
C. 2.00%
D. 0.84%

12. Data relating to Fund A is presented below for appraisal purposes:

Fund A Market Index

Return 9.00% 8.00%
Beta 1.04 1.00
Std deviation 5.00% 3.00%
Risk-free rate 4%

The Fund’s Sharpe ratio is equal to:

A. 0.80
B. 1.00
C. 4.80
D. 3.85

13. Data relating to Fund A is presented below for appraisal purposes:

Fund A Market Index

Return 9.00% 8.00%
Beta 1.04 1.00
Std deviation 5.00% 3.00%
Risk-free rate 4%

The Fund’s Treynor ratio is equal to:

A. 5.00
B. 1.00
C. 4.80
D. 0.8

14. Which of the following statements are correct when dealing with the Sharpe ratio?

A. A portfolio with superior performance will have a steep-sloping CAL and a high Sharpe ratio.
B. The Sharpe ratio is considered to be a more accurate historical performance measure.

A. I only.
B. II only.
C. Both I and II.
D. Neither I or II.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 30

15. Which of the following statements are correct when dealing with Jensen’s alpha?

I. The difference between the actual return and that required to compensate for the unsystematic risk is
called alpha.
II. Alpha makes use of the e SML as a benchmark for performance appraisal.

A. I only.
B. II only.
C. Both I and II.
D. Neither I or II.

16. Which of the following statements are correct when dealing with Jensen’s alpha and the Treynor measure?

I. Both Alpha and Treynor measure risk systematic risk (beta).

II. A manager with a positive alpha will have a Treynor measure above the market Treynor.

A. I only.
B. II only.
C. Both I and II.
D. Neither I or II.

17. Data relating to Fund A is presented below for appraisal purposes:

Fund A Market Index

Return 9.00% 8.00%
Beta 1.10 1.00
Std deviation 5.00% 3.00%
Semi-standard deviation 6.00%
Minimum acceptable 3%
return
Active risk 3.00%
Risk-free rate 4%

The Fund’s Sortino ratio is equal to:

A. 0.83
B. 1.00
C. 5.45
D. 16.67

18. Data relating to Fund A is presented below for appraisal purposes:

Fund A Market Index

Return 9.00% 8.00%
Beta 1.10 1.00
Std deviation 5.00% 3.00%
Semi-standard deviation 6.00%
Minimum acceptable return 3%
Active risk 3.00%
Risk-free rate 4%

The Fund’s information ratio is equal to:

A. 0.33
B. 0.20
C. 0.50
D. 1.00

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 31

19. Which of the following statements are correct when dealing with the information ratio and the tracking
error?

I. An aggressive investment manager would seek to achieve a low tracking error.

II. A passive investment manager would seek to achieve a high tracking error.

A. I only.
B. II only.
C. Both I and II.
D. Neither I or II.

20. Which of the following statements are correct when dealing with the Sortino measure?

I. The numerator is calculated as the return on the portfolio less a minimum acceptable return (MAR) for
the portfolio.
II. The denominator is the standard deviation of returns that fall below the MAR.

A. I only.
B. II only.
C. Both I and II.
D. Neither I or II.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 32

EDGE CHECKERS - SOLUTIONS

Topic 5 – Modern Portfolio Theory and The Capital Asset Pricing
Model (CAPM)

1. A When there is no correlation between the two assets, then the third term in the equation
disappears (since the correlation is zero). Based on this, the portfolio standard deviation
calculation is as follows:
1
σport = [w12 σ2i + w22 σ22 ]2

2. B What is interesting to note is that as the size of the portfolio increases, the variance gets close
to the average covariance. The reason for this is that the first term (variance term) gets close to
zero as the size of the ‘n’ increases, and the second term will approach the average covariance
n−1
as ‘n’ increase, because will approach 1.
n

3. A • An unlimited amount of short selling is allowed.

• There are no transaction costs to other impediments to trading – frictionless markets.

• A perfect market exists with perfect information available to investors.

• There is either no taxation or equal taxation for all participants.

4. C • All assets are infinitely divisible – divisible assets.

• All assets are marketable.

• An investor’s utility function is based solely expected return and risk.

• The time period that investors are concerned about for expected return and risk is one year.

5. C • The intercept is equal to the risk-free rate.

• The slope is equal to the risk-reward ratio for a risky portfolio.
• The capital market line lies tangent to the efficient frontier.
Where the CML and the efficient frontier touch is called the market portfolio or M.
6. C Based on the definition and characteristics of the CML, it can only be used to measure the
expected return for a diversified portfolio. In the event that a portfolio is an individual security
or not diversified then the CAPM will used.

7. A Using the CAPM, the stock should get a return of:

E(r) = Rf + ß(Rm - Rf)

= 0.11 + (1.10)(0.19-0.11)

= .198
= 19.8%

The stock is expected to get a return of 22%, which is more than its CAPM return above. Thus it
is underpriced since it has an excess return.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations
 33

8. C The capital market line (CML) is the line from the y-intercept that represents the risk-free rate
tangent to the original efficient frontier; it becomes the new efficient frontier since investments
on this line dominate all the portfolios on the original Markowitz efficient frontier.

9. B Covi,m
Beta =
𝜎 2m
0.038
Beta = = 0.718
0.232

10. C Beta is a standardized measure of systematic risk based upon an asset’s covariance with the
market portfolio.

To calculate beta in practice, we would need to regress the returns of the specific asset against
the returns of the market.

11. D αP = R Pt − R P
R P = R F + βP (R M − R F ) = 0.04 + 1.04(0.08 – 0.04) = 8.16%

Alpha = 9.00% - 8.16% = 0.84%

12. B Sharpe = (9.00 – 4.00)/5.00 = 1.00

13. C Treynor = (9.00 – 4.00)/1.04 = 5.17

14. C A portfolio with superior performance will have a steep-sloping CAL and a high Sharpe ratio.
The Sharpe ratio is considered to be a more accurate historical performance measure.

15. B The difference between the actual return and that required to compensate for the systematic
risk is called alpha.
Alpha makes use of the ex post SML (security market line) as a benchmark for performance
appraisal.

16. C Both Alpha and Treynor measure risk systematic risk (beta).

A manager with a positive alpha will have a Treynor measure above the market Treynor.

17. B Sortino ratio =

Rp −MAR
Standard deviation of returns below the MAR

0.09−0.03
Sortino ratio = 0.06
= 1

18. A IR =
Portfolio mean return−Benchmark mean return
Tracking error

0.09−0.08
IR = = 0.33
0.03

19. D An aggressive investment manager would seek to achieve a high tracking error.
A passive investment manager would seek to achieve a low tracking error.

20. C The numerator is calculated as the return on the portfolio less a minimum acceptable return
(MAR) for the portfolio.
The denominator is the standard deviation of returns that fall below the MAR.

FRM Part I – Foundations of Risk Management

Topic 5: Modern Portfolio Theory and The Capital Asset Pricing Model [FRM-5] ©2023 Edge Designations