Portfolio Management Performance Evaluation
Jan 12, 2016
Portfolio Management
Performance Evaluation
One period returns
• Gross return
• Net return
0
1Payoff1
P
Pr
00
01
0
1 gain CapitalPayoffPayoff1
Payoff
PP
PP
P
Pr
Average Returns
• Arithmetic Mean
• Geometric Mean
n
t
tAM
n
rr
1
n
tt
nGM rr
1
)1(1
1)1(/1
1
nn
ttGM rr
Example
Period Price Dividend
0 501 53 22 54 2
%66.553
25354
%1050
25053
2
1
r
r
Example
• Arithmetic Mean
• Geometric Mean
%83.72
66.510
AMr
0781.110566.110.1 2/1 GMr
0566.0110.0.112
GMr
Arithmetic vs Geometric
• Past Performance - generally the geometric mean is preferable to arithmetic
• Predicting Future Returns- generally the arithmetic average is preferable to geometric
Example
• A stock price doubles or halves
• Same probability
• We observePeriod Price
0 10
1 20
2 10
Example
• (True) Average mean
• (Observed) Geometric mean
%252
50100
AMr
%015.02 2/1 GMr
ExamplePeriod Price Return
0 10 0
1 20 100%
2 10 -50%
Time weighted return = arithmetic average return
(100-50) = 25%
Example
Period PriceNumber of shares bought
0 10 100
1 20 100
2 10 -200
Dollar weighted return = Internal Rate of Return
21
2000
1
20001000
rr
% 8.26r
Measuring Returns
Dollar-weighted returns
• Internal rate of return considering the cash flow from or to investment
• Returns are weighted by the amount invested in each stock
Time-weighted returns
• Not weighted by investment amount
• Equal weighting
Adjusting for risk
• Mean returns are not enough and one must also adjust for risk
• Find the appropriate comparison universe
• Mean-variance risk adjustments
The Sharpe Ratio
• Sharpe’s measure: expected excess return per unit of risk (measured as total volatility)
• Apropriate scenario: Evaluate a portfolio which represents the entire investor’s initial wealth
• Slope of the CAL
P
fPP
rrES
The Sharpe Ratio and M2
• Equates the volatility of the managed portfolio with the market by creating a hypothetical portfolio made up of T-bills and the managed portfolio
• If the risk is lower than the market, leverage is used and
• The hypothetical portfolio is compared to the market
The Sharpe Ratio and M2
• Find the value
where a is the value for which
• Then:
M
PMPMP r
rarrarr
22222*
PfPraErarE 1*
MPrErEM *
2
The Sharpe Ratio and M2E(r)
P*
rf
M
P
σMσP
M2
Jensen’s alpha & AP
• Jensen’s measure: the expected return of the portfolio above its CAPM counterpart
• The appraisal ratio: alpha divided by the portfolio’s nonsystematic risk
fMPfPP rrErrE
PP
P eAR
Jensen’s alpha & AP
• The AP is used in situations where the portfolio to be evaluated will be mixed with the market
• Why? For the optimal mix, the complete portfolio’s sharpe ratio is
• It measures improvement in the Sharpe ratio
2
22
P
PMC eSS
Treynor’s measure
• Treynor’s measure: excess expected return per unit of systematic risk (measured as beta)
• Appropriate when the portfolio is part of a large investment portfolio
• The slope of the T-line
P
fPP
rrET
Treynor’s measure
E(r)
E(rM)
rf
SML
= 1.0
Slope(SML)=TM=E(rM)- rf
Q
P
TQ
TP
P
P
PMPP TTT
2
Some Issues
• Assumptions underlying measures limit their usefulness– Constant distributions– Preferences
• When the portfolio is being actively managed, basic stability requirements are not met– An example: market timing
Market Timing
• Adjusting portfolio for up and down movements in the market– Low Market Return - low ßeta
– High Market Return - high ßeta
• Regression:
pfMPfMPPfP errcrrbirr 2
An Example of Market Timing
******
**
**
**
**
**
**
****
****
******
******
****
****
rp - rf
rm - rf
Steadily Increasing the Beta
Market Timing
• A simple alternative:– Beta is large if the market does well– Beta is small otherwise
• Regression
ppfMPfMPPfP eDrrcrrbirr
Market timing
the Beta takes only two values
rp - rf
**
****
**
******
**
**
**
**
******
****
******
********
rm - rf
ppfMPfMPPfP eDrrcrrbirr
Performance Attribution• Decomposing overall performance into
components
• Components are related to specific elements of performance
• Example components– Broad Allocation– Industry– Security Choice
Performance Attribution
• Set up a ‘Benchmark’ or ‘Bogey’ portfolio– Use indexes for each component: depends on
the asset class– Use target weight structure: neutral, depend
on preferences of the client
• BKM give the example:– 10% cash, 15% bonds and 75% equity for
risk-tolerant client. – 45% cash, 20% bonds and 35% equity for
risk-averse.
A question
• The bond-to-equity ratio is
– 15/75 = 0.2 (low risk aversion)
– 20/35 = 0.57 (high risk aversion)
• If cash is riskless, does it make sense according to standard assumptions?
Asset allocation puzzle
• Canner, Mankiew and Weil:
”Popular financial advisors appear not to follow the mutual-fund separation theorem. When these advisors are asked to allocate portfolios among stocks, bonds, and cash,
they recommend more complicated strategies than indicated by the theorem”
• And so do BKM!!!!
Performance Attribution
• Calculate the return on the ‘Bogey’ and on the managed portfolio
• Explain the difference in return based on component weights or selection
• Summarize the performance differences into appropriate categories
Performance Attribution
PiBi
n
iPi
n
iBiBiPiBiBi
n
iPiPi
n
iBiBi
n
iPiPiBP
n
iPiPiP
n
iBiBiB
wrrrwwrwrw
rwrwrr
rwrrwr
)()(
(managed) (bogey)
111
11
11
Asset Allocation Security Selection
Performance Attribution
Contribution for asset allocation (wpi - wBi) rBi
+ Contribution for security selection wpi (rpi - rBi)
= Total Contribution from asset class wpirpi -wBirBi
Style analysis
• Regress the returns under evaluation on a sufficiently representative set of asset classes
• This allows identification of the capital allocation decision
• The proportion not explained: security selection
Style analysis
• Magellan Fund– Growth stocks 47%– Medium cap 31%– Small stocks 18%– European stocks 4%
Some Complications
• Two major problems– Need many observations even when portfolio
mean and variance are constant– Active management leads to shifts in
parameters making measurement more difficult
• To measure well– You need a lot of short intervals– For each period you need to specify the
makeup of the portfolio