Portfolio Performance Evaluation

Measuring Portfolio Return

Time-Weighted Returns

The geometric average is a time-weighted average.

Each period return has equal weight.

Dollar-Weighted Returns

Internal rate of return considering the cash flow to and from investment

Returns are weighed by the amount invested in each period.

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G rrrr 1...111 21

nn

r

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r

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Example

Dollar Weighted Return

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Example

Time Weighted Return

rG = [ (1.1) (1.0566) ]1/2 1 = 7.81%

The dollar-weighted average is less than the time-weighted average in this example because more money is invested in year two, when the return was lower.

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Problem with TWR and DWR

Do not control for risk!! Frequently used in investment reports, media and news.

Easiest way to adjust for risk is to compare investments with similar risk characteristics. For example a fund investing in technology stocks is

compared to another fund investing in technology stocks. Then time-weighted returns of each fund are ordered

(among comparison universe) and receive a percentile ranking.

The problem with this approach is to hard to determine a good comparable fund. For example, one fund investing in technology stocks could be investing in internet start-ups, another could be investing in telephone companies.

Risk Adjusted Performance

Sharpe Measure:

rp = Average return on the portfolio rf = Average risk free rate = Standard deviation of portfolio return Assumptions needed (same as during portfolio

optimization) No changes in portfolio composition Securities have constant means, variances and covariances over

evaluation period. Held for significant time

If investors can be summarized by mean-variance preferences, investors want to maximize their Sharpe ratio!

P

FP rr

)(

P

M2 Measure

It can be hard to interpret Sharpe Measure. If one portfolio has a Sharpe ratio of 0.69 and another has a Sharpe ratio of 0.73, what is the economic difference?

M2 helps make the Sharpe measure economically intuitive: Match volatility of the managed portfolio to that of the

index/portfolio which we are comparing to.

This can be done by creating a new imaginary portfolio, which includes a positive/negative proportion of a risk free investment.

If the managed portfolio has higher volatility than index, a positive proportion invested in the risk-free rate will reduce volatility.

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2

When to use Sharpe ratio/M2

If the portfolio represents the entire investment.

Assume that past security performance is representative of expected performance

Determine the benchmark portfolio in the case of a passive strategy

Compare Sharpe/M2 of benchmark to investment portfolio

Is P better or Q?

How would you choose between P & Q when one of them is to be added to a portfolio which includes numerous other investment funds?

Is P better or Q

When your portfolio consists of many funds, non-systematic risk gets diversified away.

Beta is the appropriate risk measure!

Using Treynors Measure

TP =Portfolio with P and T-bills (risk free investment). TQ =Portfolio with Q and T-bills (risk free investment).

Risk Adjusted Performance

Treynor Measure:

rp = Average return on the portfolio

rf = Average risk free rate

= Beta of portfolio return

Normally used when the fund is a small part of the total investor portfolio.

Non-systematic risk is diversified away in a large portfolio, therefore this measure only uses systematic risk

P

FP rr

)(

P

Risk Adjusted Performance

Jensens Alpha:

rp = Average return on the portfolio

rf = Average risk free rate

rM = Average return on the market portfolio

= Beta of portfolio return

Used in both Treynor and Sharpe Measure, but ranking of portfolios will be different.

( )P P f P M fr r r r

P

Mispriced assets-review

When choosing the weight of a portfolio of mispriced assets, the following equation is used:

The improvement in Sharpe ratio of the total portfolio is:

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ew

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HMP

eSS

Risk Adjusted Performance

Information Ratio:

The information ratio divides the alpha of the portfolio by the nonsystematic risk.

Useful for actively managed funds, e.g. hedge funds.

These funds are usually additions to core positions in more traditional portfolios.

eP is the residual from single index model regressions.

)( P

P

e

Risks in Hedge Fund Investing

Risk Profile may change rapidly

Hedge funds tend to invest in illiquid assets.

Apparent profits over the long term, but possibility of infrequent, but large losses. Also called as tail risk.

Survivorship bias in a major consideration because turnover is far higher than that of investment companies.

Is P better than Q?

If P and Q are each going to be used as the ONLY investment? If P and Q are competing for a role as one of a number of subportfolios? If we seek an active portfolio to mix with an index portfolio?

Market Timing

Imagine the fund manager is fully invested in a risk-free asset and a market-index portfolio.

A sign of manager skill would be the ability to time bull and bear markets accurately and allocate funds appropriately between the two assets .

Here timing ability is defined as the ability to recognize when the stock market is going to have positive returns and increasing the proportion of funds invested in the market-index portfolio.

Characteristic Line

Characteristic Line is the equation of the line for the regression single index model.

In this example, no market timing. Beta is constant

fP rr

fM rr

Market Timing(Henrikkson and Merton)

Add a dummy variable to the single index model:

D is a dummy variable that equals 1 when rM >rF .

Beta of the portfolio is b in bear markets and b+c in bull markets.

c needs to be positive for an indication of superior timing ability.

fP rr

fM rr

( ) ( )P f M f M f Pr r a b r r c r r D e

Market Timing(Treynor and Mazuy)

Add a square term to the single index model :

The investor adds funds to the stock market while it is going up and withdraws when it is going down.

This would give the investor a higher beta when rM is high, giving the line a steeper slope. c needs to be positive for an indication of superior timing ability.

fP rr

fM rr

2( ) ( )P f M f M f Pr r a b r r c r r e

Asset allocation and security selection

Market timing was an example of shifting proportions in the market and risk-free assets. This is an asset allocation decision.

If the risky asset is not the market-index portfolio, but a portfolio of selected stocks, then the decision to pick certain stocks is called security selection.

Now we will look at two methods to distinguish between asset allocation and security selection ability.

Performance attribution

In this case we are trying to attribute performance to asset allocation or security selection.

Here we will examine the difference in returns between a managed portfolio, P, and a selected benchmark portfolio, B, called the bogey.

For each asset class, a benchmark and weight in the asset class is chosen. The return of the bogey portfolio can be broken down into:

n

i

BiBiB rwr1

Performance attribution

The fund manager will choose weights in each asset class and securities within each asset class.

The difference between the two rates of return is:

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BiBi

n

i

pipi

n

i

BiBi

n

i

pipiBp rwrwrwrwrr

Asset Allocation vs. Security Seleciton

The last part of the previous equation:

This can be simplified into two parts:

Contribution from asset allocation:

Contribution from security selection:

BiBiPi rww )(

)( BiPiPi rrw

BiBiPiPi rwrw

Example: Performance Attribution

Consider a managed fund which has invested 70%, 7% and 23% in equity, fixed income and money markets respectively.

The bogey portfolios has weights of 60/30/10.

The managed fund returned 5.34%. The bogey portfolio returned 3.97%. What explains the difference of 1.37%?

The equity, fixed income and cash benchmarks returned 5.81%,1.45% and 0.48% respectively

The managed fund returned 7.28% and 1.89% for the equity and fixed income portfolios respectively.

Example: Performance Attribution

Problems with portfolio evaluation

We need a very long observation period to measure performance with any precision, even if the return distribution is stable with a constant mean and variance.

Shifting parameters when portfolios are actively managed makes accurate performance evaluation all the more elusive.