This document discusses various methods for evaluating portfolio performance, including the Sharpe ratio, Treynor ratio, and Jensen performance index. The Sharpe ratio measures risk-adjusted return using standard deviation, while the Treynor ratio uses beta to measure performance relative to the overall market. The Jensen performance index calculates excess return above what is expected based on market risk to evaluate portfolio manager performance. Taken together, these metrics provide different ways to analyze both the risk and return of portfolios and their managers.

1. PORTFOLIO
SELECTION USING
SHARPE ,TREYNOR
& JENSEN
PERFORMANCE
INDEX

2. The process by which one
chooses the securities,
derivatives, and other assets
to include in a portfolio. In
making securities selections,
one considers the risk, the
return, the ethical
implications, and other
factors affecting both
individual securities and the
portfolio as a whole.
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PORTFOLIO SELECTION

3. SOME BASICTERMINOLOGIES
• Beta- Beta measures a security’s or
portfolio’s performance (asset’s
risk and return) in relation to the
movements in the market. Beta is
a relative measure used for
comparison and does not show a
security’s individual behavior. A
beta value of 1 show that the
security is performing in line with
the market’s performance and a
beta of less than 1 show that
security’s performance is less
volatile than the market. A beta of
more than 1 show that a security’s
performance more volatile than
the benchmark.
3

4. • Standard deviation-standard
deviation shows the volatility of stocks,
bonds, and other financial instruments
that are based on the returns spread over
a period of time. As the standard
deviation of an investment measures the
volatility of returns, the higher the
standard deviation, the higher volatility
and risk involved in the investment.
Measures both systematic and
unsystematic risks
• Risk free return-Risk-free return is
the theoretical return attributed to an
investment that provides a guaranteed
return with zero risk. The risk-free rate
represents the interest on an investor's
money that he or she would expect from
an absolutely risk-free investment over a
specified period of time.
4

5. • Risk adjusted return-Risk-adjusted
return refines an investment's return by
measuring how much risk is involved in
producing that return, which is generally
expressed as a number or rating. Risk-
adjusted returns are applied to
individual securities, and portfolios
• Expected Return- is calculated as the
weighted average of the likely profits of
the assets in the portfolio, weighted by
the likely profits of each asset class.
Expected return is calculated by using
the following formula:
E(R) = w1R1 + w2Rq + ...+ wnRn
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6. DIFFERENCE BETWEEN BETA AND STANDARD DEVIATION
• Beta measures a security’s or
portfolio’s performance (asset’s risk
and return) in relation to the
movements in the market
• Beta measures only systematic risk
(market risk).
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• standard deviation shows the
volatility of stocks, bonds, and other
financial instruments that are based
on the returns spread over a period of
time
• Standard deviation on the other hand
measures the total risk, which is both
systematic and unsystematic risk
BETA STANDARD DEVIATION

7. SHARPE RATIO
7
• The Sharpe Ratio was developed by Nobel laureate William F. Sharpe
• The Sharpe ratio is a measure of an investment's excess return
• The Sharpe ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk
• It is calculated by taking the return of the investment, subtracting the risk-free rate, and dividing
this result by the investment's standard deviation
• A higher Sharpe ratio is better
• Sharpe ratio = (Mean portfolio return − Risk-free rate)/Standard deviation of portfolio return

8. CONTD.
8
• Mutual Fund A returns 12% over the past year and had a standard deviation of 10%. Mutual Fund
B returns 10% and had a standard deviation of 7%. The risk-free rate over the time period was
3%. The Sharpe ratios would be calculated as follows:
Mutual Fund A: (12% - 3%) / 10% = 0.9
Mutual Fund B: (10% - 3%) / 7% = 1
• Even though Mutual Fund A had a higher return, Mutual Fund B had a higher risk-adjusted
return, meaning that it gained more per unit of total risk than Mutual Fund A.
• Fund B is more efficiently earning returns per unit of unsystematic risk

9. TREYNOR RATIO
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• The Treynor ratio is calculated the same way as the Sharpe ratio, but it uses the investment's
beta in the denominator
• A higher Treynor ratio is better
• Mutual fund A returns 12% over the past year and had a beta of 0.75. Mutual fund B returns 10%
and a beta of again 0.75. The risk free rate of return over the time period was 3%. The
calculations are
Mutual Fund A: (12% - 3%) / 0.75 = 0.12
Mutual Fund B: (10% - 3%) / 0.75 = 0.09
• Here, Mutual Fund A has a higher Treynor ratio, meaning that the fund is earning more return
per unit of systematic risk than Fund B.

10. SHARPE RATIOVSTREYNOR RATIO
• Sharpe ratio aims to reveal how well an
equity investment portfolio performs as
compared to risk free investment.
• Common benchmark is Treasury bills
• Uses standard deviation to measure excess
return per unit of risk
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• Rather than measuring a portfolios return for
risk free investment, treynor ratio examines
how well a portfolio outperforms the equity
market as a whole
• Benchmark is market index
• Uses beta to evaluate performance of equity
against market performance
TREYNOR RATIO
SHARPE RATIO

11. JENSON PERFORMANCE INDEX
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• Named after its creator, Michael C. Jensen, the Jensen measure calculates the excess return that a portfolio
generates over its expected return
• This measure of return is also known as alpha
• The Jensen ratio measures how much of the portfolio's rate of return is attributable to the manager's ability
to deliver above-average returns, adjusted for market risk
• The higher the ratio, the better the risk-adjusted returns
• A portfolio with a consistently positive excess return will have a positive alpha, while a portfolio with a
consistently negative excess return will have a negative alpha
• The formula is broken down as follows
Jensen's Alpha = Portfolio Return – Benchmark Portfolio Return
Where: Benchmark Return (CAPM) = Risk-Free Rate of Return + Beta
(Return of Market – Risk-Free Rate of Return)
:

12. CONTD.
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• if we once again assume a risk-free rate of 5% and a market return of 10%, what is the alpha for the
following funds?
•
• First, we calculate the portfolio's expected return:
ER(D)= .05 + 0.90 (.10-.05) = .0950 or 9.5% return
ER(E)= .05 + 1.10 (.10-.05) = .1050 or 10.50% return
ER(F)= .05 + 1.20 (.10-.05) = .1100 or 11% return
• Then, we calculate the portfolio's alpha by subtracting the expected return of the portfolio from the actual
return:
Alpha D = 11%- 9.5% = 1.5%
Alpha E = 15%- 10.5% = 4.5%
Alpha F = 15%- 11% = 4.0%
Manager Average Annual Return Beta
Manager D 11% 0.90
Manager E 15% 1.10
Manager F 15% 1.20

13. CONTD.
13
• Which manager did best? Manager E did best because, although manager F had the same annual
return, it was expected that manager E would yield a lower return because the portfolio's beta
was significantly lower than that of portfolio F.

14. CONCLUSION
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• By studying all the different performance indexes which gives us the risk adjusted returns
we can conclude that each model uses different variables to compute returns
• Best would be to use the combination of all so as to analyze both systematic, unsystematic
as well as the performance of portfolio managers in providing the excess returns
• Sharpe uses standard deviation to calculate the excess profit gained as compared to the risk
free return
• On the other hand tryenor ratio uses beta to determine how well a portfolio or equity
performs against the market index
• Jenson index measures how much excess returns a portfolio manages to get against it’s
expected return. This denotes the performance of portfolio mangers
• Thus we need to consider all these factors while selecting our investments for the portfolio

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