Investment Risk
Evaluation
Knut Larsen – The Brigus Group
CIFPs Annual Conference, Orlando FL, May 7, 2008
Focus on Investment Risk
¾ Investment Risk definition: the possibility of
losing money
¾ Investors lose money when their portfolios
have negative returns
¾ Returns fluctuate – sometimes they’re
positive, other times they’re negative
¾ More fluctuation in returns means more risk
May 7, 2008
Knut Larsen, Brigus Group
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Measuring Risk
¾ Ideal risk measure: the probability of
getting a negative return on investments
¾ Practical risk measure: the standard
deviation of annual returns
¾ Standard deviation reflects the typical
spread of returns around the mean return
May 7, 2008
Knut Larsen, Brigus Group
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Low Risk versus High Risk
Frequency distributions of annual returns
High Risk: st.dev. = 8
Low Risk: st.dev = 2
6%
5%
15%
Probability
Probability
20%
10%
5%
mean=5%
0%
4%
3%
2%
1%
mean=10%
0%
-10 -7 -4 -1
2
5
8 11 14 17 20
-10 -6 -2
Annual return (%)
May 7, 2008
2
6
10 14 18 22 26 30
Annual return (%)
Knut Larsen, Brigus Group
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Risk in Investment Portfolios
¾ Simple portfolio: a risk-free asset (t-bills)
and equities with non-zero risk
¾ Return on this portfolio: Rp = wtRt + wsRs
w = proportion in portfolio
R = rate of return
t = t-bill; s = stock (equities)
May 7, 2008
Knut Larsen, Brigus Group
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Returns of Investment Portfolios
Example:
30% invested in t-bills; 70% in equities
Return on t-bills: 5%
Return on equities: 17%
Return on this portfolio:
Rp = (0.3x5%) + (0.7x17%)
Rp = 1.5% + 11.9% = 13.4%
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Knut Larsen, Brigus Group
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Risk in Investment Portfolios
Example cont’d:
st.dev. of t-bills = 0 (risk-free)
st.dev. of equities = 18
st.dev. of portfolio = ws x st.dev. of equities
= 0.7 x 18
= 12.6
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Knut Larsen, Brigus Group
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Capital Market Line
¾ Shows the risk-return combinations for
varying proportions of the risk-free and the
risky assets
¾ Select the risky assets so that return is
maximized for a given level of risk, or risk
is minimized for any given level of return
(the portfolio is efficient)
¾ Point A: 100% t-bills; 0% equities
¾ Point M: 0% t-bills; 100% equities
May 7, 2008
Knut Larsen, Brigus Group
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Capital Market Line
20
Returns (%)
M
15
10
5
A
0
0
5
10
15
20
25
30
35
40
st.dev.
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Knut Larsen, Brigus Group
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Evaluation Using the CML
¾ The Capital Market Line (CML) represents
the best risk-return combinations available
¾ Any investment whose risk-return
combination is plotted below the CML has
underperformed
¾ Any investment whose risk-return plot lies
on the CML has performed as well as the
market
May 7, 2008
Knut Larsen, Brigus Group
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Capital Market Line
Returns (%)
20
M
15
10
P
Q
5
A
0
0
5
10
15
20
25
30
35
40
st.dev.
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Knut Larsen, Brigus Group
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A Practical Benchmark
¾
Use the Sharpe index (ratio):
Investment' s return - Riskfree return
Sharpe index =
Investment' s st.dev.
• Compare the Sharpe value of individual
investments with the Sharpe value of the market
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Knut Larsen, Brigus Group
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Calculating Sharpe values
17% − 5% 12% 2
Market’s Sharpe =
=
= ≅ 0.67
18%
18% 3
10% − 5% 5% 1
=
= ≅ 0.33
Investment P’s Sharpe =
15%
15% 3
10% − 5%
5%
2
Investment Q’s Sharpe =
=
= ≅ 0.67
7.5%
7.5% 3
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Knut Larsen, Brigus Group
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Evaluation of P and Q
¾ Investment P has underperformed
compared to the market
¾ Investment Q has performed just as well
as the market
¾ An investment’s Sharpe value has
meaning only in comparison with the
market’s Sharpe or with another
investment’s Sharpe value
May 7, 2008
Knut Larsen, Brigus Group
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Different Risks
¾ Total risk can be split into diversifiable and
non-diversifiable risk
¾ Diversifiable risk can be eliminated
through diversification; hence it’s assumed
to be zero
¾ Non-diversifiable risk depends on:
z
z
Market risk
The investment’s “sensitivity” to market risk
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Knut Larsen, Brigus Group
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Beta (β)
¾ Beta is a measure of an investment’s
sensitivity to market fluctuations
¾ Market’s beta = 1 (by definition)
¾ “Cyclical” investments have beta > 1
¾ “Defensive” investments have beta < 1
¾ An investment’s beta is estimated on the
basis of historical data
¾ Beta of a t-bill = 0
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Knut Larsen, Brigus Group
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Beta of a Portfolio
A portfolio’s beta is the weighted average of the
betas of the investments held in the portfolio:
β p = q1β1 + q2 β 2 + ... + qn β n
Here, the q’s are weights for the n investments in
the portfolio representing each investment’s
relative importance in the portfolio
May 7, 2008
Knut Larsen, Brigus Group
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Beta of Our Portfolio
Our simple portfolio contains one risk-free asset (tbills) and one non-zero-risk asset (“equities”)
β p = qt β t + qs β s
Since t-bills’ beta is zero, we have:
β p = qs β s
The portfolio’s beta equals the equities’ beta
multiplied by the portfolio’s proportion of equities
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Knut Larsen, Brigus Group
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Calculating Our Example Portfolio’s Beta
- assuming we still have 70% equities (and 30%
t-bills) and that equities have a beta of 1.5
β p = qs β s = 0.7 ×1.5 = 1.05
Let’s now vary the proportions held in t-bills and
equities in the portfolio the way we did to create
the Capital Market Line (CML)
May 7, 2008
Knut Larsen, Brigus Group
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Security Market Line (SML)
Return (%)
20
15
10
5
0
0
0.5
1
1.5
Beta
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Knut Larsen, Brigus Group
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Evaluation Using the SML
¾ The Security Market Line (SML) shows the
equilibrium trade-offs between returns and
betas of securities
¾ Investments with a return-beta plot that
lies below the SML have underperformed
¾ Investments with a return-beta plot that
lies on the SML have performed just as
well as the market
May 7, 2008
Knut Larsen, Brigus Group
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Security Market Line (SML)
Return (%)
20
15
10
C
D
5
0
0
0.5
0.75
1
1.2
1.5
Beta
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Knut Larsen, Brigus Group
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Treynor Index
Instead of plotting return-beta values, we use the
Treynor Index, defined as:
Investment' s Return - Riskfree Return
Treynor =
Investment' s Beta
Compare the Treynor value of individual
investments with the Treynor value of the market
May 7, 2008
Knut Larsen, Brigus Group
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Calculating Treynor values
17 % − 5 %
12 %
=
= 12 %
Market’s Treynor =
1
1
Investment C’s Treynor =
10% − 5% 5%
=
≅ 4.17%
1.2
1.2
Investment D’s Treynor =
6.5% − 5% 1.5%
=
= 2%
0.75
0.75
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Knut Larsen, Brigus Group
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Evaluation of C and D
¾ Both investment C and D have
underperformed compared to the market
¾ Investment D has underperformed the
most and ranks below Investment C
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Knut Larsen, Brigus Group
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CML versus SML
– or Sharpe versus Treynor
¾ Which one of the two approaches should
be used?
¾ Both!
¾ Treynor (SML) is used to evaluate an
investment’s returns relative to nondiversifiable risk
¾ Sharpe (CML) is used to evaluate an
investment’s returns relative to total risk –
including its diversifiable risk
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Knut Larsen, Brigus Group
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Possible Outcomes
SML (Treynor)
On or
above
CML
CML
(Sharpe)
Below
CML
May 7, 2008
On or above
SML
Below SML
Good
Investment
Not Possible
Watch it –
not fully
diversified!
Bad
investment
Knut Larsen, Brigus Group
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Jensen’s Alpha (α)
¾ Alpha measures the amount by which an
investment (over- or) underperformed the
SML
¾ It is measured in percentage points
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Knut Larsen, Brigus Group
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Alpha Calculations
Alpha = Investment’s actual return minus the
Investment’s expected return, where
expected return = Rt + β ( RM − Rt )
For example, Investment C - from earlier - has a
beta of 1.2, so its alpha is -9.4, calculated as:
α = 10% − [5% + 1.2(17% − 5%)] = 10% − 19.4%
Investment C has not performed well at all – there is
lots of room for improvement through diversification!
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Knut Larsen, Brigus Group
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Conclusions
¾ Use the Sharpe index to evaluate large,
well diversified portfolios, e.g. large mutual
funds, which are likely to be efficient
¾ Use the Treynor index to evaluate any
investment; avoid the investment if its
Treynor value is below the market’s value
¾ Use Jensen’s Alpha to measure an
investment’s potential performance
improvement
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Knut Larsen, Brigus Group
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Thank you very much
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Knut Larsen, Brigus Group
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