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F303 – Intermediate Investments

Class 9

Portfolio Performance Evaluation

Andrey UkhovKelley School of Business

Indiana University

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Outline of This Subject

Why do we need performance analysis?

The risks associated with performance measurement.

The importance of benchmarking.

Risk-adjusted returns.

Stock-picking and market timing.

Different tools to analyze portfolio performance

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Motivation for Performance Analysis

Index funds simply try to replicate performance of an index (S&P 500) . An S&P 500 fund effectively holds a Market Portfolio.

But there are actively managed funds.

Investors who pay a fund manager to manage their portfolio must get timely information about the investment’s performance.

The analysis will identify sources of weaknesses and strengths.

The big question: Has any good performance resulted from good luck or was it the result of skill?

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The Basic Problems

There are two basic problems with performance analysis:

Even if the mean and variance of returns are assumed to be constant, there is the need to obtain many observations to achieve significant results (skills vs good luck);

In the case of active management (as opposed to passively managed funds) shifting parameters and styles could make evaluation quite difficult.

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Measuring Return: Time Interval

Performance must be investigated over a long period of time: minimum of five years, using monthly returns.

Without such a sample size, determining portfolio performance become a hazardous task…difficult to really assess whether performance was due to good (bad) luck or skill (or lack of it).

Question: What to do when different fund managers have managed the fund over some period?

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Measuring Returns: A Simple Case

Consider the simple case when an investor neither deposit nor withdraws money from the fund during a certain time period. How are we going to calculate the rate of return?

This is the simple case where all we need to know is the fund’s market value at the beginning of the period and end of the period.

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Measuring Returns: A Simple Case

In the simple case, the fund’s return is found as:

Example: Consider a portfolio with a market value of $50 million at the beginning and $56 at the end of the period.

The return is 12% [=($56m - $50m)/$50m]

BEGINNINGV

BEGINNINGV

ENDV

turn

Re

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Measuring Returns: A Complication

In reality, performance measurement is not that simple!

The main problem is that the investor may deposit or withdraw cash from the fund.

In this case, the market values will be influenced by these cash flows and just using the previous formula will give you misleading results.

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An Example With Deposits

Consider a case of a fund whose market value at the beginning of the period was $200 million.

Towards the end of the quarter an investor deposits $10 million in the fund. At the end of the quarter the fund’s market value is $206 million. When you use the previous formula, disregarding the cash inflow, you find that the return was 3%.

But this is incorrect! The increase in market value was not the result of the fund manager’s skills.

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An Example With Deposits

The accurate measure of the fund’s returns over the quarter must take into consideration the $10 million cash inflows…

Considering this fact, results in a return of -2% [={($206m -$10m) - $200m}/$200m].

The identification of the timing of deposits and withdrawals is crucial in determining the correct return of a fund. Take into considerations these cash flows to adjust the rate of return.

Ignoring these adjustments will lead to misleading conclusions!

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Dollar-Weighted Returns

One method that can be used in order to calculate returns when deposits or withdrawals occur is the Dollar-Weighted Returns Method.

Similar to the Internal Rate of Return calculation where the beginning-of-period value is set equal to the discounted values of the cash flows and the end-of-period value:

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206$

1

10$200$

r

m

r

mm

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Time-Weighted ReturnsOne alternative method is the Time-Weighted Returns Method.

This methodology uses the fund’s market value before each cash flow takes place.

Continuing from our previous example, assume that the deposit was made in the middle of the quarter. The fund’s value was $195m before the deposit, so that soon after the $10m deposit its market value went to $205m.

Hence the return for the first half would be -2.5% [=($195m-$200m)/$200m] while the return for the second half is 0.5% [=($206m-$205m)/$205m].

These two semi-quarter returns can give us the total quarter returns as follows: [={1+(-0.025)} x {1+(0.005)} - 1]= -2%

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Dollar-weighted versus Time-weighted Returns

The dollar-weighted returns are influenced by both the size and the timing of the cash flows.

The time-weighted returns do not present this problem.

For this reason, the time-weighted return is generally preferred to dollar-weighted in evaluating portfolio performance.

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Benchmark ReturnsIn performance analysis you need to make relevant comparisons.

The investor has to compare the returns of his/her manager with the returns that would have been obtained had he/she invested in an alternative portfolio with identical risk.

Performance must be evaluated on a relative basis; not on absolute basis!

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Benchmark ReturnsThe investor must make use of benchmark portfolios to assess the fund manager’s performance. These benchmark portfolios must be relevant (similar risk), feasible and known in advance.

For example, let us say that you decide to invest in a diversified equity portfolio with average risk. You see that your return was 20%.

Is this good or bad?

Now let us say that you find out that the Wilshire 5000 Index has gone up, for the same period, 14%. Then you can say that your fund, for this period in particular, had a superior return.

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A Fund’s Performance Against Its Peers

-20

-10

0

10

20

30

40

50

60

Year1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7

5th Percentile Median 95th Percentile Fund

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Risk-Adjusted Returns

One simple way to investigate the fund’s performance is to consider risk-adjusted returns…remember that the CAPM tells us that the more market risk you take on, the higher should be the return.

A widespread measure is the Sharpe ratio:

The Sharpe measure divides average portfolio excess returns by the portfolio’s risk. This measures the reward to total variability trade-off.

P

fR

PR

Sharpe

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Risk-Adjusted Returns: Treynor’s Measure

Treynor’s measure:

The Treynor’s measure gives you the excess return per unit of risk.

However, unlike the Sharpe Ratio, it uses the systematic risk instead of total risk.

P

fR

PR

Treynor

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Risk-Adjusted Returns: Jensen’s Measure

Jensen’s measure:

Like Jack Treynor and William Sharpe, Michael Jensen recognized the CAPM’s implications for performance measurement.

The Jensen’s measure is the average fund’s return above the predicted return from the CAPM, given the portfolio’s Beta and average market return.

fR

MR

PfR

PR

PJensen

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Risk-Adjusted Returns: Appraisal ratio

Appraisal ratio:

The appraisal ratio is simply the portfolio’s alpha divided by the non-systematic risk embedded in the portfolio.

In some sense, this gives us the abnormal return per unit of the non-systematic risk: i.e. the risk that can be diversified away by holding a completely diversified portfolio.

P

PAppraisal

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Which Measure Should We Use?

To decide on compensation: Use the Jensen measure, which should provide

you the amount you are willing to pay the manager.

To decide on optimal portfolio choices: Use Sharpe ratio if portfolio represents entire

investment;

Use Treynor’s measure for a fund that is just one sub-portfolio out of a large set of passively-managed portfolios;

Use Appraisal ratio for an actively-managed portfolio that is mixed with a passively-managed portfolio.

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Criticisms of Risk-adjusted Returns

The risk-adjusted returns that we have seen have come under some attack for the following reasons:

They use a market surrogate instead of the “true” market portfolio;

Unable to statistically distinguish luck from skill;

Using an inappropriate risk-free rate;

They rely on the validity of CAPM.

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Risk-adjusted Returns: An Example

Calculate the Sharpe ratio, the Jensen measure, Treynor measure and appraisal ratio for Portfolios P and M.Assume that the risk-free rate was 6%.

Portfolio P Portfolio M

Average Return 35% 28%

Beta 1.20 1.00

Standard Deviation

42% 30%

Non-systematic risk

18% 0%

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Risk-Adjusted ReturnsContinuing from the previous example…you should find that the Sharpe Ratio for Portfolio P is 0.69 and 0.73 for Portfolio M. This means that Portfolio P underperformed the market index…

But is the difference (0.04) significant? Is it economically meaningful?

The investor must pay a lot of attention how to use and interpret these ratios.

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The M-M ( ) Performance Measure

The Modigliani-Modigliani measure, , is a variant of the Sharpe ratio.

It focuses on total volatility but the risk-adjusted return is, essentially, a differential return relative to the benchmark returns.

The measure takes into account the fund’s average return and determines what it would have been if the fund (under investigation) had the same degree of total risk as the market portfolio (for example, Wilshire 5000 or S&P 500 indices).

2M

2M

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The M-M ( ) Performance Measure

The first step is to generate the so-called “adjusted portfolio”, P*.

This portfolio is made up of two parts: The managed portfolio, P; and A position in T-Bills in a way that the total

volatility of the adjusted portfolio matches the market index’s volatility.

2M

MR

PRM *

2

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The M-M ( ) Performance Measure

Let us take the example mentioned before to build the adjusted portfolio.

Portfolio P has a standard deviation of 42% while the market portfolio has a standard deviation of 30%. Hence, the adjusted portfolio, P*, will be formed by mixing the managed portfolio, P, and T-bills in the following way:

Weight of 30/42=71.4% in portfolio P; and 100% - 71.4% = 28.6% in T-bills.

The expected return of P* is 26.7% (why?) This means that it has underperformed…by how

much?

2M

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The Performance Measurement

2M

2M

P

P*P

M

M

E(R)

CAL(P)

CML

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Market Timing

A good market timer structures a portfolio to have a relatively high Beta when the market is expected to rise and low Beta when the market is expected to drop.

In other words, the market timer wants to do the following strategy:

Hold a high-Beta portfolio when Hold a low-Beta portfolio when

fR

MR

fR

MR

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Market TimingIf the fund manager really has good timing abilities (good and accurate forecasts of market movements), then the portfolio will do better than a benchmark portfolio that has a constant Beta (that is equal to the average Beta of the timer’s portfolio).

To really “time the market”, the fund manager must change:

the average Beta of the risky securities held in the portfolio; or

the relative amounts invested in the risk-free asset and the risky assets.

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Superior Stock Selection

fR

PR

Alpha

fR

MR

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Stock SelectionThe previous exhibit indicates that the relationship between the portfolio’s excess returns and the market’s excess return was linear.

This result suggests that the portfolio’s Beta was, roughly speaking, the same during the entire period under consideration.

In this case, it appears that the investment manager successfully identified and invested in some underpriced securities (alpha is positive).

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Superior Market Timing

fR

PR

Alpha

fR

MR

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Market TimingThe previous exhibit indicates that the relationship between the portfolio’s excess returns and the market’s excess return was not linear.

The exhibit suggests that the portfolio consisted of high-Beta securities during periods when the market return was high and low-Beta securities when the market dropped.

In this case, it appears that the investment manager successfully identified market timing (alpha is positive).

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Key Points to Remember

Basic problems with performance evaluation;

The issue of skill against good luck; Dollar-weighted returns and time-weighted

returns; Benchmark returns from benchmark

portfolios; Risk-adjusted returns (Sharpe ratio,

Treynor’s measure, Jensen measure, appraisal ratio, );

Why risk-adjusted returns have been criticized;

Market timing versus stock selection.

2M