2. Learning Outcomes
•Measuring Risk and Return for Investors
•Portfolio Diversification
•Quantitative Techniques: Correlation & Regression
•Alternative Risk Measures
3. Fundamental Investor Preferences
1. Investment decisions are made on the basis of the
return and risk of an investment.
2. Investors like to maximise their utility.
3. Investors prefer less risk to more risk - risk averse.
4. Return is viewed as a set of probabilities that certain
outcomes will occur.
5. Risk is measured as the standard deviation of expected
returns.
What we know about investors (Markowitz Model):
4. •The purpose of location and dispersion measures is to indicate
the:
• Central/long run average value/return achieved – location
• The variability or spread of values around this point –
dispersion
•These measures provide us with an indication of:
• The expected future returns
• The risk we are facing
Location and Dispersion
5. Measurement of Risk & Return
The most commonly used measure of risk in investment
analysis and fund management:
Expected Return – Mean (A location measure)
Measure of Risk – Standard Deviation (A dispersion
measure)
6. Expected Return
The expected return for an asset can be determined
using the below formula:
E(R) = P1
R1
+ P2
R2
+...Pn
Rn
Pi = Probably of an outcome occurring
Ri = Return from the outcome
7. Sources of Risk
Uncertainty of
income
Interest rates
Inflation
Exchange Rates
Tax rates
Economic state
Default Risk
Liquidity Risk
Factors contributing to risk
in investment returns:
8. Example: Expected Return
A company’s stock is expected to generate the following
returns with attached likelihood (probability):
Expected Return Probability
-20% 30%
0% 10%
10% 40%
20% 20%
What is the stock’s
expected return?
(0.30x −0.20)+(0.10x0)+
(0.40x0.10)+(0.20x0.20)
=2.0%
9. •Standard deviation is a dispersion measure that is related to
the arithmetic mean.
•The idea is to establish how far each observed value falls from
the mean – standard deviation measures this divergence.
•The greater the divergence of the observed value from the
mean, the greater the standard deviation.
Dispersion Measures
10. Standard Deviation
Standard deviation and variance are key measures of an
asset’s risk.
It is a measure that is used to quantify the amount of
variation or dispersion of a set of values from its expected
(mean) return.
σ2
= Pi
(Ri
−E(R)#
$
%
&
2
∑
11. Steps to Calculate Standard Deviation
1. Calculate the mean (expected return) of a set of data.
2. Calculate the difference between observed values and the average.
3. Square the differences to remove negative signs and multiply them
by their probabilities.
4. Sum the figures and take the square root.
σ2
= Pi
(Ri
−E(R)#
$
%
&
2
∑
12. Example: Standard Deviation
Expected Return Probability
-20% 30%
0% 10%
10% 40%
20% 20%
What is the stock’s
variance and
standard deviation?
Using our previous example and its results:
13. Example: Standard Deviation
Expected Return Probability
-20% 30%
0% 10%
10% 40%
20% 20%
Variance = 2.36%
Standard Deviation = 15.36%
(0.30)(−0.20−0.02)2
+(0.10)(0−0.02)2
+(0.40)(0.10−0.02)2
+(0.20)(0.20−0.02)2
14. Portfolio Returns
If we are analysing a portfolio which holds two assets, the
expected return of the portfolio is calculated as follows:
E(Rp
) = w1
E(R1
)+ w2
E(R2
)
Asset value / Total portfolio value
Expected Return on an Asset
15. Example: Portfolio Returns
E(Rp
) = w1
E(R1
)+ w2
E(R2
)
An investor holds 40% of his portfolio in a stock with an
expected return of 10% and the balance in a stock with 20%
expected return.
What is the portfolio’s expected return?
16. Reducing Risk - Diversification
•By spreading a total investment over several securities
which offer the target level of return, we can achieve the
same return at lower risk.
•Therefore: Diversification reduces risk without necessarily
reducing returns.
17. Reducing Risk - Diversification
What is diversification?
The process of combining a variety of assets to a portfolio
to reduce risk.
18. Reducing Risk - Correlation & Regression
•To track how a stock moves with the broader market (eg.
S&P500, FTSE100):
•Correlation measures the strength of the relationship between
two variables.
•In the context of regression analysis, the correlation coefficient
gives an indication of how accurately the regression line
matches the observed values.
19. Regression and correlation analysis provides the means to
establish whether a relationship exists between two factors:
•Regression – the equation of the line of best fit
mathematically
•Correlation – an indication of the accuracy or strength of a
relationship between two factors.
Measuring Relationships
20. If we plot a straight line on a graph where y is the vertical
axis and x is the horizontal axis, the equation of the straight
line is:
Y = a + bx + e
X – Independent variableY – Dependent variable
Intercept – height where line cuts y axis
Slope – change in value of y for a unit change in x
Error term
Regression
21. •The method of estimating the
parameters a and b is the least
squares method ie. the line of
best fit.
•Minimising the sum of the
vertical distances of each value
from the straight line drawn.
Minimising the sum of the
squared errors.
Least Squares Regression
The objective is to minimise the
errors from the regression line
22. Reducing Risk - Correlation
•Values used to measure the relationship between two
variables are called correlation coefficients.
•Positive correlation implies returns move up or down
together.
•Negative correlation implies returns move in opposite
directions.
•The closer to value of 1, the stronger the relationship.
24. Interpreting Correlation
• What is the link between correlation and diversification?
• Returns move up or down
together in equal and
opposite direction.
• Perfectly offsets gains/
losses in one position.
Perfect Negative
Correlation
(-1)
Uncorrelated
(0)
Perfect Positive
Correlation
(+1)
• Returns move independently
of each other.
• Some diversification of
gains/losses by adding
another security.
• Returns move up or down
together in proportion.
• No diversification gains
by adding another
security.
25. Calculating Correlation Coefficients
•How do we calculate the correlation coefficient of
variables x and y?
Corx,y
=
Covariance(x,y)
σx
σy
Covariance is a measure of how much x and
y variables move together
Dividing by the standard deviation of both x
and y gives a normalised measure of their
relationship’s strength
26. Reducing Risk - Correlation
By increasing our level of diversification, we reduce risk.
Can we completely eradicate all investment risk?
27. Nature of Risk
Total Risk of Investment
σi
Systematic / Market Risk
σs
Idiosyncratic / Specific Risk
σu
Cannot be reduced
through diversification
Can be reduced through
diversification
28. Systematic Risk
What is Systematic Risk?
Market volatility caused by macroeconomic (broader market)
factors which affect all assets
29. Idiosyncratic Risk
What is Idiosyncratic (Specific) Risk?
Risk that is specific to an asset and is uncorrelated to
systematic risk.
30. Nature of Risk - Consequences
• An undiversified investor faces full risk, σi
• A diversified investor has only systematic risk - σs
σi
2 = σs
2 + σu
2
Total risk:
31. Measuring Portfolio Risk
•To measure portfolio risk requires:
• (i) weights of the assets in the portfolio
• (ii) the variance of each asset, and
• (iii) the correlation between the two assets
Portfolio Variance = w1
2
σ2
1
+ w2
2
σ2
2
+2w1
w2
Cov(R1
,R2
)
Portfolio Variance = w1
2
σ1
2
+ w2
2
σ2
2
+2w1
w2
σ1
σ2
Corr(1,2)
32. Measuring Portfolio Risk
•The formula indicates:
• By combining assets which are not perfectly correlated
(correlation =1), we can reduce portfolio risk
• Smaller the correlation coefficient, greater the risk
reduction
Portfolio Variance = w1
2
σ1
2
+ w2
2
σ2
2
+2w1
w2
σ1
σ2
Corr(1,2)
33. Question: Measuring Portfolio Risk
Stock 1 Stock 2
Amount 100,000 150,000
Expected
Return
10% 15%
Standard
Deviation
20% 25%
Correlation 0.3
What is the expected variance
for a 2-asset portfolio?
35. Application
• If a portfolio is fully diversified, an investor only faces
systematic risk.
• How do we compare investments to determine the
return an investor should expect based on a level of
systematic risk?
• The Capital Asset Pricing Model (CAPM) provides this
analytical tool. (See next course).
37. Value at Risk (VAR)
• Value at Risk (VAR) is the amount by which the value of
an investment or portfolio may fall over a given period of
time at a given probability level.
Example:
• VAR = £1m at 5% probability over 1 week
This indicates:
• 95% chance that losses will not be greater than £1m in a
week
• 5% chance of losses greater than £1m in a week
38. Value at Risk (VAR) - Pros & Cons
Benefits Limitations
Easy to understand Assumes normal distribution
Widely used by regulators Only focuses on downside
Used to allocate risk among divisions
Can be difficult to calculate for large
portfolios
39. Alternative Measures of Risk
Standard Deviation & Variance
• Dispersion measure
• Limitation – Assumes normal distribution
Tracking Error
• Used for tracking a specified index
• The greater the accuracy that a fund
tracks the index, lower the tracking error
Shortfall risk
• Probability of generating a return below
a target level
40. Recap
•The mean and standard deviation are standard measures
of return and risk in investment analysis.
•In portfolio management, diversification reduces risk
without necessarily reducing returns.
•Correlation and regression are quantitative methods to
measure the strength of a relationship between two
variables.
•There are alternative measures of risk (VAR, Shortfall risk,
tracking error.
