Topic 2 - Term 2: Portfolio Performance Evaluation

Question 1

Two requirements of an active portfolio manager:
1. The ability to derive above-average ? for a given risk class.
Superior risk-adjusted returns can be derived from:
- superior ?
- superior ??

2. The ability to ? the portfolio to eliminate ? risk relative to the portfolio’s benchmark.

Answer 1

Two requirements of an active portfolio manager:
1. The ability to derive above-average returns for a given risk class.
Superior risk-adjusted returns can be derived from:
- superior timing
- superior security selection

2. The ability to diversify the portfolio to eliminate unsystematic risk relative to the portfolio’s benchmark.

Question 2

I. Treynor Ratio (aka ?-to-? ratio):
Treynor ratio = T = (Average Return of a ? –Average Returnof the ?? Rate)/ ? of the Portfolio
The risk variable, beta, measures ? (?) risk => implicitly assumes that the portfolio is ??.

Answer 2

I. Treynor Ratio (aka reward-to-volatility ratio):
Treynor ratio = T = (Average Return of a Portfolio –Average Returnof the Risk-Free Rate)/Betaof the Portfolio
The risk variable, beta, measures systematic (market) risk => implicitly assumes that the portfolio is completely diversified.

Question 3

I. Treynor Ratio (aka reward-to-volatility ratio):
If the portfolio’s T value ? the market (T_m)
=> the portfolio would plot ? the SML (i.e. beat the market).

Answer 3

I. Treynor Ratio:
If the portfolio’s T value > the market (T_m)
=> the portfolio would plot above the SML (i.e. beat the market).

Question 4

I. Treynor Ratio (T):
Negative T values can occur if:
- avg portfolio return ? market return => portfolio underperformed (? performance) => plot ? SML.
- portfolio beta < 0 => ? correlation bet portfolio and market => ? performance => plot ? SML.

Thus, T ratio can be confusing => we can plot the portfolio on an ? graph or to calculate the expected return for the portfolio using the SML equation ( E(R_i) = ….? ) and then compare this expected return to the ? return.

Answer 4

I. Treynor Ratio (T):
Negative T values can occur if:
- avg portfolio return < market return => portfolio underperformed (bad performance) => plot below SML.
- portfolio beta < 0 => negative correlation bet portfolio and market => good performance => plot above SML.

Thus, T ratio can be confusing => we can plot the portfolio on an SML graph or to calculate the expected return for the portfolio using the SML equation ( E(R_i) = R_f + Beta_i * (R_m - R_f) ) and then compare this expected return to the actual return.

Question 5

I. Treynor ratio - Limitations:
It assumes a portfolio manager has diversified away all of the ? (company-specific) risk, and that only systematic (market) risk is left.
This limits the use of the Treynor ratio to comparisons involving extremely ?? portfolios.
The Treynor ratio is based on ? beta, so it shares beta’s limitations.

Answer 5

I. Treynor ratio - Limitations:
It assumes a portfolio manager has diversified away all of the unsystematic (company-specific) risk, and that only systematic (market) risk is left.
This limits the use of the Treynor ratio to comparisons involving extremely well-diversified portfolios.
The Treynor ratio is based on CAPMs beta, so it shares beta’s limitations.

Question 6

II. Sharpe ratio:

• The Sharpe Ratio (S) is ??? earned per unit of ??.
• Has been one of the most referenced risk/return measures used in finance
• Similar to Treynor Ratio but S use ?? instead of Systematic Risk.
• Sharpe compares portfolios to the ? while Treynor to the ?.

Answer 6

II. Sharpe ratio:

• The Sharpe Ratio (S) is risk premium return earned per unit of total risk.
• Has been one of the most referenced risk/return measures used in finance
• Similar to Treynor Ratio but S use Total Risk instead of Systematic Risk.
• Sharpe compares portfolios to the CML while Treynor to the SML.

Question 7

II. Sharpe ratio - Limitations:
- The Sharpe ratio uses the??of returns as its proxy of ? portfolio ?, which assumes that returns are ??.
- The Sharpe ratio can also be manipulated by hedge funds or portfolio managers seeking to boost their apparent risk-adjusted returns history. This can be done by:
> Lengthening the measurement interval
> Compounding the monthly returns but calculating the standard deviation from thenotcompounded monthly returns.
> Eliminating extreme returns

Answer 7

II. Sharpe ratio - Limitations:
- The Sharpe ratio uses thestandard deviationof returns as its proxy of total portfolio risk, which assumes that returns are normally distributed.
- The Sharpe ratio can also be manipulated by hedge funds or portfolio managers seeking to boost their apparent risk-adjusted returns history. This can be done by:
> Lengthening the measurement interval
> Compounding the monthly returns but calculating the standard deviation from thenotcompounded monthly returns.
> Eliminating extreme returns

Question 8

III. Jensen’s alpha:
Jensen’s index, or alpha, is a measure of how much a portfolio’s ? return differs from the return it should have achieved, according to ?.

Answer 8

III. Jensen’s alpha:
Jensen’s index, or alpha, is a measure of how much a portfolio’s actual return differs from the return it should have achieved, according to CAPM.

Question 9

III. Jensen’s alpha:
CAPM: E(R_i) = R_f+β_i(R_m−R_f)
=> Actual return = R_i= R_f + β_i(R_m−R_f)

=> R_i−R_f = β_i*(R_m−R_f)

To detect and measure superior performance we need to add an intercept term.

This allows us to measure any positive or negative difference from the model.
=> R_i−R_f = α_i + β_i * (R_m−R_f)

Answer 9

III. Jensen’s alpha:
CAPM: E(R_i)=R_f+β_i(R_m−R_f)
=> Actual return = R_i= R_f + β_i(R_m−R_f)

=> R_i−R_f = β_i*(R_m−R_f)

To detect and measure superior performance we need to add an intercept term.
This allows us to measure any positive or negative difference from the model.
=> R_i−R_f = α_i + β_i * (R_m−R_f)

Question 10

III. Jensen’s alpha:
R_i−R_f = α_i + β_i * (R_m−R_f)

A superior manager has ?? α and an inferior manager has ?? α.
α is therefore a measure of how good the manager is at obtaining above-average returns, adjusted for risk.
The higher a fund’s alpha, the ? its returns have been relative to the amount of investment risk it has taken.

Answer 10

III. Jensen’s alpha:
R_i−R_f = α_i + β_i * (R_m−R_f)

A superior manager has significant positive α and an inferior manager has significant negative α.
α is therefore a measure of how good the manager is at obtaining above-average returns, adjusted for risk.
The higher a fund’s alpha, the better its returns have been relative to the amount of investment risk it has taken.

Question 11

III. Jensen’s alpha - Limitations:
The Jensen index measuresrisk premiumsin terms ofbeta(β); therefore, it is assumed that the portfolio being evaluated is??.

The Jensen index requires using a differentrisk-free ratefor each time interval measured during the specified period.
This calculation method contrasts with both the Treynor and Sharpe measures in that both examine the average returns for the total period for all variables, which include the portfolio, market and risk-free asset.

Answer 11

III. Jensen’s alpha - Limitations:
The Jensen index measuresrisk premiumsin terms ofbeta(β); therefore, it is assumed that the portfolio being evaluated iswell-diversified.

The Jensen index requires using a differentrisk-free ratefor each time interval measured during the specified period.
This calculation method contrasts with both the Treynor and Sharpe measures in that both examine the average returns for the total period for all variables, which include the portfolio, market and risk-free asset.

Question 12

III. Jensen’s alpha - Pros:
Easy to interpret - e.g. α = 0.02 shows a manager generated 2% per period more than implied by the level of risk.

As it is a regression model, we can make a judgement about the ?? of manager’s skill level.

Flexible enough to be applied to ? models.
For example, we can extend the model to encompass the Carhart 4-factor model.

We would include SMB, HML and, momentum factors.
Carhart (1997) demonstrated that once we adjust for risk more fully i.e. change from basic CAPM, the apparent performance of active mutual fund managers is much reduced.
α is reduced and after the change very few funds have significant α.
Unlike Sharpe, we could include a factor for ?.

Answer 12

III. Jensen’s alpha - Pros:
Easy to interpret - e.g. α = 0.02 shows a manager generated 2% per period more than implied by the level of risk.

As it is a regression model, we can make a judgement about the statistical significance of manager’s skill level.

Flexible enough to be applied to multifactor models.
For example, we can extend the model to encompass the Carhart 4-factor model.

We would include SMB, HML and, momentum factors.
Carhart (1997) demonstrated that once we adjust for risk more fully i.e. change from basic CAPM, the apparent performance of active mutual fund managers is much reduced.
α is reduced and after the change very few funds have significant α.
Unlike Sharpe, we could include a factor for ?.

Question 13

The Jensen index allows the comparison of portfolio managers’ performance relative to one another, or relative to the market itself.
When applying alpha, it’s important to compare funds within the sameasset ?.
Comparing funds from one asset class against a fund from another asset class is meaningless, because you are essentially comparing apples and oranges.
The right ? should also be considered.
For example, the benchmark more frequently used to measure the market for US funds is the S&P 500 Index, which serves as a proxy for “the market”.
However, some portfolios and funds include asset classes with characteristics that do not accurately compare against the S&P 500 such as bond funds, sector funds, real estate etc.
Therefore, the S&P 500 may not be the appropriate benchmark to use in these cases (we will consider this issue a bit more in seminar)

Answer 13

The Jensen index allows the comparison of portfolio managers’ performance relative to one another, or relative to the market itself.
When applying alpha, it’s important to compare funds within the sameasset class.
Comparing funds from one asset class against a fund from another asset class is meaningless, because you are essentially comparing apples and oranges.
The right benchmark should also be considered.
For example, the benchmark more frequently used to measure the market for US funds is the S&P 500 Index, which serves as a proxy for “the market”.
However, some portfolios and funds include asset classes with characteristics that do not accurately compare against the S&P 500 such as bond funds, sector funds, real estate etc.
Therefore, the S&P 500 may not be the appropriate benchmark to use in these cases (we will consider this issue a bit more in seminar)

Question 14

Select the right measures/ ratios:
Much of this decision owes to the different ways that methods have of dealing with risk. If it is important to capture how well a manager diversifies then ? is appropriate.
If diversification can be safely assumed, only non-diversifiable risk matters and we can use a β based measure such as ?.
Treynor and Jensen measures are concerned with ? risk.
Sharpe and M2 are concerned with ? risk.
- But we can calculate different ratios, each tells us a different story

Answer 14

Select the right measures/ ratios:
Much of this decision owes to the different ways that methods have of dealing with risk. If it is important to capture how well a manager diversifies then Sharpe is appropriate.
If diversification can be safely assumed, only non-diversifiable risk matters and we can use a β based measure such as Jensen.
Treynor and Jensen measures are concerned with systematic risk.
Sharpe and M2 are concerned with total risk.
- But we can calculate different ratios, each tells us a different story

Question 15

IV. M2 ratio:
Sharpe Ratio Formula (SR) = ?? /?
portfolio return = SR * ? + ?
M2 value = portfolio return - benchmark return

Answer 15

IV. M2 ratio:
Sharpe Ratio Formula (SR) = (r_p – r_f) / σ_p
The portfolio return = SR * σ_benchmark + (r_f)

Question 16

IV. M2 ratio - Limitations:
The data used for the calculation of the M2 measures incorporate only historical risk and return (the same as Sharpe).
It therefore has the same limitations as Sharpe i.e. can be manipulated by hedge fund or portfolio managers seeking to boost their apparent risk-adjusted returns history.

Answer 16

IV. M2 ratio - Limitations:
The data used for the calculation of the M2 measures incorporate only historical risk and return (the same as Sharpe).
It therefore has the same limitations as Sharpe i.e. can be manipulated by hedge fund or portfolio managers seeking to boost their apparent risk-adjusted returns history.

Question 17

V. Information ratio (aka Appraisal ratio):

• a ratio ofportfolio returnsabove the returns of abenchmark(index) to thevolatilityof those returns.
• The information ratio (IR) measures a portfolio manager’s ability to generateexcess returnsrelative to a benchmark and their ?.

Answer 17

V. Information ratio (aka Appraisal ratio):

• a ratio ofportfolio returnsabove the returns of abenchmark(index) to thevolatilityof those returns.
• The information ratio (IR) measures a portfolio manager’s ability to generateexcess returnsrelative to a benchmark and their CONSISTENCY.

Question 18

V. Information ratio:
IR_p= ???

(R_p) ̅ = the average return for the portfolio during a specified time period

(R_i) ̅ 	= the average return on index/benchmark during the same time period

σ_(p−i) 	= Tracking error(standard deviation of the difference between returnsof the portfolio and the returns of the index – excess returns)

Answer 18

V. Information ratio:
IR_p= ( (R_p) ̅−(R_i) ̅) / σ_(p−i)

(R_p) ̅ = the average return for the portfolio during a specified time period

(R_i) ̅ 	= the average return on index/benchmark during the same time period

σ_(p−i) 	= Tracking error(standard deviation of the difference between returnsof the portfolio and the returns of the index – excess returns)

Question 19

V. Information ratio:
To interpret IR , the mean return differential in the numerator represents the manager’s ability to their skill and information to generate a portfolio return that differs from that of the benchmark against which their performance is being measured.
In fact, it can be considered the manager’s average ? if the average return to the benchmark is taken to be the expected return for the actively managed portfolio.

Answer 19

V. Information ratio:
To interpret IR , the mean return differential in the numerator represents the manager’s ability to their skill and information to generate a portfolio return that differs from that of the benchmark against which their performance is being measured.
In fact, it can be considered the manager’s average ALPHA if the average return to the benchmark is taken to be the expected return for the actively managed portfolio.

Question 20

V. Information ratio:
Conversely, the denominator of the IR statistic measures the amount of residual (?) risk that the manager incurred in pursuit of those incremental returns.
The coefficient σ_p−i is called the ?? of the portfolio, and it is a cost of active management in that fluctuations in the periodic σ_p−i values represent random noise, beyond a manager’s control, that could hurt performance.

Answer 20

V. Information ratio:
Conversely, the denominator of the IR statistic measures the amount of residual (unsystematic) risk that the manager incurred in pursuit of those incremental returns.
The coefficient σ_p−i is called the tracking error of the portfolio, and it is a cost of active management in that fluctuations in the periodic σ_p−i values represent random noise, beyond a manager’s control, that could hurt performance.

Question 21

V. Information Ratio:
The higher the IR, the more ? a manager, with consistency being an ideal trait.
Conversely, the lower the IR, the ? consistency.
The IR serves as a tool that investors often use when selecting exchange-traded funds (ETFs) or mutual funds.

Answer 21

V. Information Ratio:
The higher the IR, the more consistent a manager, with consistency being an ideal trait.
Conversely, the lower the IR, the less consistency.
The IR serves as a tool that investors often use when selecting exchange-traded funds (ETFs) or mutual funds.

Question 22

The Information Ratio vs Sharpe:

The Sharpe ratio is similar to the IR and is used to measure risk-adjusted returns.
However, the Sharpe ratio measures the difference between an asset’s return and the risk-free rate of return divided by the standard deviation of the asset’s returns.
The difference between the Sharpe ratio and the IR is that the IR aims to measure the risk-adjusted return in relation to a benchmark/index.
Also, the IR measures the consistency of an investment’s performance, while the Sharpe ratio measures how much an investment portfolio outperformed the risk-free rate of return on a risk-adjusted basis.

Answer 22

The Information Ratio vs Sharpe:

The Sharpe ratio is similar to the IR and is used to measure risk-adjusted returns.
However, the Sharpe ratio measures the difference between an asset’s return and the risk-free rate of return divided by the standard deviation of the asset’s returns.
The difference between the Sharpe ratio and the IR is that the IR aims to measure the risk-adjusted return in relation to a benchmark/index.
Also, the IR measures the consistency of an investment’s performance, while the Sharpe ratio measures how much an investment portfolio outperformed the risk-free rate of return on a risk-adjusted basis.