This document discusses measuring risks and returns of portfolio managers. It covers three key performance measures: the Treynor measure, Sharpe measure, and Jensen's Alpha. The Sharpe measure evaluates reward per unit of total risk, the Treynor measure evaluates reward per unit of beta risk, and Jensen's Alpha measures actual returns against the CAPM model. The document also discusses how investors apply modern portfolio theory, with some believing markets are strongly efficient so only a naïve diversified portfolio is needed, while others believe in semi-strong efficiency and analyze stocks for a diversified growth portfolio. Diversification, long-term performance measurement, and balancing growth versus diversification are important implications for investors.

1. CHAPTER 22
MEASURING RISKS AND
RETURNS OF PORTFOLIO
MANAGERS
FIN 330
Principles of Investing

2. STUDENT LEARNING OBJECTIVES
A. Learning from Historical Trends
1. Measuring Holding Period Returns: Geometric vs. Arithmetic
2. Fund Objectives and Risk Attributes
B. Three measures of investment performance based on modern
portfolio theory
C. Past performance as a predictor of future performance
D. Applying modern portfolio theory to investment decisions

3. Learning from Historical Trends
A. Measuring Holding Period Returns
1. Arithmetic: simple averages of daily weekly, monthly, quarterly, or
annual stock or index returns. (dollar weighted)
E(r) = 𝑛=1
𝑁
𝑟𝑡 / N
2. Geometric: the n-root of the product of n-period returns (time
weighted)
E(𝑟
𝑔 ) = 1 + 𝑟1 ∗ 1 + 𝑟2 ∗ ⋯ 1 + 𝑟𝑇
1/T - 1
3. Arithmetic mean returns are upwardly biased v-v Geometric mean
returns.

4. Learning from Historical Trends
A. Example of upward bias in arithmetic returns
1. Three daily closing prices:
a. P1 = 1.00
b. P2 = 1.10
c. P3 = 1.00
2. 2 daily returns
a. R1,2 = (.10 / 1.00) = 0.10 or 10% gain
b. R2,3 = (-.10 / 1.10) = -0.0909 or 9.09% loss
c. Mean r = (0.10 + -0.09090) / 2 = 0.0046 or 0.46% gain
d. Geometric r = [(1.10) * (0.9091)]1/2 – 1 = 0.0000 or 0% gain
Note that 1 + -.0909 = .9091

5. Three performance measures
1. Treynor measure is reward per unit of beta risk
 (Actual Rp – Rf) / Beta
2. Sharpe measure is reward per unit of total risk (total risk =
std. dev.)
 (Actual Rp – Rf) / sp
3. Jensens’s Alpha measures the actual mean excess return
minus the CAPM return
 a = (Actual Rp – Rf) – b (Rm – Rf)

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Performance Measures
A. Sharpe Performance Index (1966)
1. Reward to Variability (risk) (CML construct)
2. S = (Rp – Rf) / sp
3. S is the slope of a line whose intercept is the risk free rate (Rf)
4. the STEEPER the line, the better the performance.
5. Best used to [performance] rank portfolios

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Performance Measures
A. Treynor Performance Index
1. Reward per unit of Beta Risk (SML construct)
2. T = (Rp – Rf) / bp
3. Beta computed using historical rates of return
4. How well did the investment portfolio do in terms of percentage
return on a risk-adjusted basis.

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Performance Measures
A. Jensen’s Alpha:
1. Mean Excess Return minus the CAPM return
2. Excess return = Rp – Rf
3. CAPM Return = b (Rm – Rf)
4. a = (Rp – Rf) – b (Rm – Rf)
5. One problem with Jensen’s measure is that we do not know the
magnitude of non-systematic risk incurred in order to achieve the
excess.

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Supplemental Material: EMT
A. Gauging impact of efficient markets theory on investor
behavior
1. How do investors implement efficient market theories?
a. True Believers
b. Doubtful
c. Percentage players

10. Applying EMT/MPT to investor decisions
A. Different groups of investors apply EMT-MPT differently
depending on how strongly they believe in market efficiency
1. Group 1 MPT investors believe the market is strong-form efficient and
will invest in any naïve diversified portfolio
2. Passive or naïve strategy invests in a well-diversified portfolio because
one cannot “beat the market” – index portfolio

11. Applying EMT/MPT to investor decisions
A. Group 2 MPT investors believe in Semi-strong market
efficiency and invest in a well-diversified portfolio of growth
stocks to gain both benefits
1. Group 2 investors will analyze securities to determine which stock to
include in a well-diversified portfolio
2. Group 2 investors will also analyze optimal allocation of the portfolio

12. Copyright © 1998 by Harcourt Brace & Company
Applying EMT/MPT to investor decisions
A. Third group is somewhere between group 1 and group 2
B. They believe the market offers undervalued and overvalued
stocks, but that finding them is nearly impossible, so they may
act as group 1 investors
C. Other investors scorn EMT/MPT
D. Technical Analysts may fall in this group

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Implications for investors
A. Diversify by investing in several securities or in mutual funds
B. Measure performance using reward per risk to determine fund
performance
C. Measure performance over a long period of time, perhaps five
years or more
D. Understand the tradeoffs between picking high growth stocks
over a well-diversified portfolio

14. Homework
A. Discussion Questions: 1, 3, 4, 5, 6, 7
B. Problems: 1, 2 (parts a & c)