Portfolio
Performance
18 Evaluation
1
T PERFORMS BETTER?
Fund T had a return of 20% last year
Fund S had a return of 15% last year
Why would many investors go for a fund with
lower risk-adjusted return?
What is risk-adjusted return?
2
OUTCOMES
Apply different performance measures to
evaluate portfolio return
Calculate performance attribution
Note: How would you invest if half of the fund
managers underperform the S&P500 index?
3
18.1 RISK-ADJUSTED RETURNS
In order to tell how well your investments perform,
you have to know the strategy employed in your
investment
• Passive Management
• Diversified portfolio without searching for security mispricing
• You may not be looking for very high return
• Active Management
• Forecasting broad markets and/or identifying mispriced
securities to achieve higher returns
• Market Timing
• Switching between risky portfolio and cash
• Switching from low to high beta stocks 4
• Switching among different sectors/industries.
18.1 RISK-ADJUSTED RETURNS
Comparison Universe
Knowing your investment strategy/style, we can then
determine how to compare your investment performance
Set of portfolio managers with similar investment styles
used to assess relative performance
E.g. a collection of funds to which performance is
compared
E.g. portfolio of high-yield bonds if you are investing in
risky bonds.
You are not going to compare the performance of a risky
fund with an index fund!
5
FIGURE 18.1 UNIVERSE COMPARISON
Short term performance fluctuates, which one is better?
95th and
75th
percentile
managers
5th and
25th 6
percentile
managers
18.1 RISK-ADJUSTED RETURNS
Or CAPM
Excess return RPt during
period t
With 18% return, you outperform if the
E(Rp) = 16%, given RM=10%
𝐸 𝑅𝑃 =β𝑃𝐸 𝑅𝑀 + α𝑃 Beta measures the risk you are taking
7
CAPM assumes zero alpha, so +ve alpha is required to outperform the market
REMEMBER THE STEPS (WILL NOT BE
TESTED)?
Obtain time series of RPt and RMt
Run regression of RPt on RMt to get alpha, beta and residual
SD
Spreadsheet like Excel (using data analysis) will output the
alpha, beta and other information like residual SD, etc. Like
Table 18.1 (next 3 slides)
8
18.1 RISK-ADJUSTED RETURNS
You have to decide the risk measure.
Should it be the Total risk, i.e. sigma or
the Systematic risk, i.e. beta?
Covered in previous chapter
To calculate the standard deviation of portfolio 9
18.1 RISK-ADJUSTED RETURNS
You have to decide the risk measure.
Should it be the Total risk => Sharpe ratio or
• the Systematic risk beta => CAPM
10
TABLE 18.1 PERFORMANCE OF TWO
MANAGED PORTFOLIOS
P better than Q? 0.398 vs 0.344 11
How to interpret?
Not easy unless they have the
same standard deviation sigma
18.1 RISK-ADJUSTED RETURNS
• Return difference with same volatility
=18.5(0.394-0.344)=1%
σ𝑀
𝑅ത 𝑝 ∗= 𝑅ത 𝑝
σ𝑃
18.5
𝑅ത 𝑝 ∗= 𝑥9.6% = 0.7676𝑥9.6% = 7.37%
24.1 12
M2= 7.37%-6.37% = 1%
M-SQUARE EXPLAINED
Form portfolio p*of same risk as the market by investing
0.7676 in P and (1-0.7676) in Rf
0.7676x13.6%+(1-0.7676)x4% = 11.37%, given Rf=4%
Volatility would be 0.7676x24.1% +(1-0.7676)x0 = 18.5%
same as market standard deviation
M2 = 11.37%-10.4%(should be 10.37%) =1%
Excess return =11.37%-4% =7.37%
Or using excess return M2= 7.37%-6.37% = 1%
13
FIGURE 18.2 M2 OF PORTFOLIO
14
18.1 RISK-ADJUSTED RETURNS
15
FUND ON FUND EXAMPLE, WHY USE BETA?
2 equally weighted similar Q with uncorrelated residuals
Table 18.1:
Average Excess Return still 5.5%,
Same Beta of value 0.5,
Residual standard deviation of Q: 15.44%,
residual SD = sqrt (0.52x0.15442 + 0.52x0.15442) =10.92%,
total SD = sqrt[(βσM)2 + Var(e)] = sqrt[(0.5x0.185)2+0.10922]
= 14.31%,
Sharpe ratio = 5.5%/14.31% = 0.384 (vs original 0.306)
M2= 0.185(0.384-0.344) =0.74%, positive (vs
previously -0.72%),
Treynor measure not Sharpe ratio to be used 16
Residual risk reduced due to diversification, so systematic
risk not total risk should be considered
ACTIVE PORTFOLIO ADDED TO THE PASSIVE
PORTFOLIO
Information Ratio
Ratio of alpha to residual standard deviation
𝛼𝐻
𝑖𝑠 𝑡ℎ𝑒 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜
𝜎(𝑒𝐻)
𝛼
𝑆𝑝2 = 𝑆𝑀2 + (𝜎(𝑒𝐻)
𝐻
)2
improvement of adding hedge fund H to the index portfolio M
𝛼𝐻 2
𝑆𝑜 = 𝑆𝑞𝑟𝑡[𝑆𝑀2 + ( )]
𝜎(𝑒𝐻)
Should P or Q be considered as both have positive alphas when
adding to a passive portfolio?
Consider information ratio in equation above,
choose the one has higher Sharpe ratio S0. 17
18.2 STYLE ANALYSIS
Performance evaluation introduced by William
Sharpe
Regress fund returns on indexes representing a
range of asset class (style)
Recent studies of mutual fund performance show
> 90% of return variation can be explained by
funds’ allocations to T-bills, stocks, and bonds
18
TABLE 18.3 SHARPE’S STYLE PORTFOLIOS
FOR MAGELLAN FUND
12 asset
classes
*Regressions are constrained to have nonnegative coefficients and to have
19
coefficients that sum to 100%. Don’t worry if you have never done regression
with constrained variables.
18.3 MORNINGSTAR’S RISK-ADJUSTED
RATING (RAR)
Company peer groups established based on
Morningstar style definitions, compare each fund to a
peer group
Risk-adjusted performance ranked; then stars assigned
according to table Percentile Stars
0-10 1
10-32.5 2
32.5-67.5 3
67.5-90 4
90-100 5
5 stars for ARKK ETF. you may need to start the trial before you can see it
https://www.morningstar.com/etfs/arcx/arkk/performance 20
http://www.morningstar.com/Cover/Funds.aspx
https://www.morningstar.com/best-investments/medalist-funds
18.4 RISK ADJUSTMENTS WITH CHANGING
PORTFOLIO COMPOSITION
Problems with Performance Measures
Assume fund maintains constant level of risk
Particularly problematic for funds engaging in
active asset allocation
In large universe of funds, some will have abnormal
performance each period by chance
Survivorship bias
Upward bias in average fund performance due to
failure to account for failed funds over sample
period
21
PORTFOLIO RETURNS
1st and 2nd year, the
manager is like
maintaining a better than
passive Sharpe ratio
But he is taking higher
risk in the 2nd year
1st Year 2nd Year market Over the 2 year
excess return 1% 9% 5%
standard deviation 2% 18% 13.42%
Sharpe ratio 0.5 0.5 0.4 0.37
FIGURE 18.7 PORTFOLIO RETURNS (SIMILAR
EXAMPLE)
Read the text and go through the example yourselves
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18.5 PERFORMANCE ATTRIBUTION
PROCEDURES
Decomposing overall performance into
components
Determined by specific portfolio choices
Broad asset allocation
Industry weighting in equity portfolio
Security choice
Timing
You may ask why an ETF is performing so well.
Is it because how the manager’s ability in choosing the right stocks,
the right industry or her choice between stocks/bonds/commodity, etc.
Is it because of good timing or the country she chooses (currency appreciations
help)
24
TABLE 18.4 PERFORMANCE OF MANAGED
PORTFOLIO
• Bogey Where does it come from ?
• Benchmark portfolio comprised of three indexes with given
weights
• Bogey return represents return on unmanaged portfolio
• Weights represent standard portfolio for typical risk
tolerance of given type of client or typical fund in category 25
TABLE 18.5 PERFORMANCE ATTRIBUTION
Excess weights
26
Excess return on asset
TABLE 18.6 SECTOR ALLOCATION WITHIN
EQUITY MARKET (NOT TESTED)
27
TABLE 18.7 PORTFOLIO ATTRIBUTION:
SUMMARY (NOT TESTED)
From previous slide
Excess return col.3
Table 18.5
28
Using similar approach Using similar approach
18.6 MARKET TIMING
Adjust asset allocation for movements in market
Shift between stocks and money market instruments
or bonds, etc.
Little evidence of market-timing ability
29
FIGURE 18.9A CHARACTERISTIC LINES
Why zero intercept? What if non-zero intercept?
+ve implies stock picking ability. 30
FIGURE 18.9B CHARACTERISTIC LINES
If investors can time the market correctly and shift funds 31
into the market when the market is performing well
