PPERFORMANCEERFORMANCE

EEVALUATION VALUATION

Portfolio Management

Prof. Ali Nejadmalayeri

Why Evaluate Performance?

• Fund Sponsor’s Perspective– “Quality Control” for managers

• A feedback from managers and their funds to determine what happened, why it happened and what should be done

– IPS enhancement• Enhances the direction and guidance provided by the

investment policy statements

– Improved Reaction & Response• Allows for timely response to an ever changing complex

investment world at large

Why Evaluate Performance?

• Fund Manager’s Perspective– “Pay Day” guidelines

• The performance ultimately determines the compensation through performance based pay structures

– Internal “Quality Control”• Models and ideas are as good as the outcome they

produce; without a evaluation managers won’t know if their models and ideas are working at all

– “Luck” effect• Separating luck from skill; without evaluation managers

won’t know if they are just darn lucky or mighty skillful

Components of Performance

• Performance Measurement– What was the account performance?

• Need to define how to measure performance

• Performance Attribution– Why did the account produce the performance?

• Need to define what factors affecting the results; stock movements, interest rates, commodity prices, etc.

• Performance Appraisal– Is the performance due to luck or skill?

• Need to find out how much of the performance would have occurred no matter what the managers did

Performance Measurement

• PM without intra-period external cash flows– The prototypical holding period return

• Beginning and end cash infusions can occur

• Total Return– Old fashion measurement with intra-period cash

• Fixed income, limited computing power, less competition

• Time-weighted ReturnTime-weighted Return– Compound growth for an initial $1 of investment

• Money-weighted ReturnMoney-weighted Return– Compound growth for all funds invested

Performance w/o intra CFs• Without initial or ending cash infusions:

• With initial cash flows

• With ending cash flows

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Time-weighted Return• First, separate time-line into subperiods where cash

flows occurred– Let’s say there are T such periods

• Second, compute return per subperiod– This return is the holding period return with end cash flow

• Then the time-weighted returntime-weighted return is defined as:

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Example: Time-Weighted Return• A $1,000,000 account recorded a month-end value of

$1,080,000. The account has received two cash flows during a month: a $30,000 contribution on day 5 and a $20,000 contribution on day 16. Using a daily pricing system, we know that the account value was $1,045,000 and $1,060,000 on days 5 and 16, respectively. What’s TWR?

– Three subperiods: days 1 – 5, days 6 – 16, days 17 – 30

– For each subperiod the holding period return is:• Days 1 – 5: r1 = [(1,045,000 – 30,000) – 1,000,000 ]/1,000,000

• Days 6 – 16: r2 = [(1,060,000 – 20,000) – 1,045,000 ]/1,045,000

• Days 17 – 30: r3 = [(1,080,000 – 1,060,000 ]/1,060,000

– Then, TWR for the month is:• rtwr = ( 1 + r1 ) ( 1 + r1 ) ( 1 + r1 ) – 1 =

= (1 + 0.0150) (1 + -0.0048) (1 + 0.0189) – 1 = 0.0292 = 2.92%

Money-weighted Return• The money-weighted returnmoney-weighted return is the defined as the rate

that solves the following equation:

• where– m = number of time units in the evaluation period

– CFi = the ith cash flow

– L(i) = number of time units by which the ith cash flow is separated from the beginning of the evaluation period

)()1(101 111 nLm

nLmm RCFRCFRMVMV

Example: Money-Weighted Return• A $1,000,000 account recorded a month-end value of

$1,080,000. The account has received two cash flows during a month: a $30,000 contribution on day 5 and a $20,000 contribution on day 16. Using a daily pricing system, we know that the account value was $1,045,000 and $1,060,000 on days 5 and 16, respectively. What’s MWR?

– The MWR would solve:

– By trial-and-error we have: R = 2.90%

– We can set this up in Excel: • First have three columns, one for dollar values, one for time passed,

and one for present value which incorporates dollar values and time.

• Then the sum of all present values should be zero. Use goal seek!

163053030 1000,201000,301000,000,1000,080,1 RRR

TWR vs. MWR

• With small cash flows, the two are very close

• With large cash flows or when subperiod return is very volatile, then there can large differences– If funds are contributed prior to a period of strong

performance, then MWR will be greater than TRW– If funds are withdrawn after a period of strong

performance, then TWR will be greater than MRW

• Bank Administration Institute as well as Global Investment Performance Standards (GIPS®) recommend TWR.

Linked Internal Rate of Return

• TWR has a major disadvantage: accounts must be re-evaluated at every data that external cash flow take place!

• BAI study recommends:– TWR should approximated with MWR over

reasonably frequent time intervals and then those returns should be chain-linked. This is called the Linked Internal Rate of Return (Dietz, 1966)

– In our example, the let’s choose one week as a reasonably short interval, then:

• rtwr = (1 + 0.021) (1 + 0.0016) (1 + -0.014) (1 + 0.018) – 1 = = 0.0265 = 2.65%

Performance Attribution

• Fama (1972) shows that selection and diversification can be the source of returns

• State-of-Art considers two main sources:– Macro Attribution

• Inputs: Policy allocations, Benchmark portfolio returns, fund returns, valuations, and external cash flows

• Then various factors affect the macro attribution:– Net contributions, risk-free asset, asset categories, benchmarks,

investment managers, allocation effects

– Micro Attribution • Finds out how manager performs relative to designated

benchmarks

Macro AttributionThe general idea is to find out what give rise to the return:

1. Net contribution

2. Risk-free Asset

3. Asset Category

4. Benchmark

5. Investment Managers

6. Allocation Effects

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Asset Category’s Attribution• The IPS states objectives in terms of strategic

allocation. Return generated then is greatly affected by the corresponding indexes which represent the IPS allocation.

IPS weight for the ith asset category

Benchmark return for the jth manager

in the ith asset category

Benchmark return for the ith asset

category

Benchmarks’ Attribution• Since managers hired to achieve the IPS’s objectives

have varying styles, their corresponding benchmarks may not maps onto the IPS’s benchmarks. Thus we need to determine to what role these benchmarks play.

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IPS weight for the ith asset category

IPS weight for the jth manager in ith

asset category

Benchmark return for the jth manager

in the ith asset category

Benchmark return for the ith asset

category

Managers’ Attribution• Managers can choose to tactically deviate from their

own benchmarks to create value. Returns then can be attributed to their intentions, or investment managers’ attribution to the performance.

IPS weight for the ith asset category

IPS weight for the jth manager in ith

asset category

Benchmark return for the jth manager

in the ith asset category

Benchmark return for the ith asset

category

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Others’ Attributions• Net contribution:

– Cash in- and out-flows can distort the returns. Depending on whether cash flows made at the beginning or at the end of the period, returns may or not may not affected. End-of-period cash contributions do not change the performance.

• Risk-free Asset:– The most basic return is the return on a risk-free investments.

This part of the performance is a function of the economy at large and not anything else. IPS, allocation, managers are not responsible for this part of the return.

• Allocation Effects:– Any return that cannot attributed to contributions, risk-free

rate, asset categories, benchmarks, and managers.

Performance Attribution: ExampleA pension has the following policy allocation:

• This means that to obtained the desired exposure to the large equity index, the pension hires two managers whose style (and relevant benchmarks) may be different from the desired exposure. The same is also true for the fixed incomes portion of the pension. Two manager with potentially different style (and benchmarks) may be used to attain the exposure to the large fixed income index.

Asset CategoryAsset Category Policy WeightPolicy Weight

Large Equity Index 75%

Equity Manager 1 65% of equity portion

Equity Manager 2 35% of equity portion

Large Fixed Income Index 25%

Fixed Income (FI) Manager 1 55% of FI portion

Fixed Income (FI) Manager 2 45% of FI portion

Performance Attribution: ExampleFor the past year, the pension performance has been:

• During the same period, the risk-free rate was 0.31%. IFIF the contributions are made at the end of each period (year), then for our computation of return, we use the ratio of end value minus contribution to beginning value minus one. So for total value, the return is (194,816,599 – 950,000)/186,419,405 – 1 which is equal to 3.9949%. Now if not all cash flows made at the beginning or end of the year, then 3.99% reported would be correct!

Beginning ValueBeginning Value End ValueEnd Value ContributionsContributions Actual Actual ReturnReturn

Benchmark Benchmark ReturnReturn

Large Equity Category $143,295,254 $148,747,228 ($1,050,000) 4.5300%

4.04%

Eq. Manager 1 $93,045,008 $99,512,122 $1,950,000 4.7600%

4.61%

Eq. Manager 2 $50,250,246 $49,235,106 ($3,000,000) 4.1300%

4.31%

Fixed Income Category $43,124,151 $46,069,371 $2,000,000 2.1600%

2.56%

FI Manager 1 $24,900,250 $25,298,754 $0 1.6000%

1.99%

FI Manager 2 $18,223,900 $20,770,617 $2,000,000 2.9100%

2.55%

Total $186,419,405 $194,816,599 $950,000 3.9900%

3.94%

Performance Attribution: ExampleThe performance attribution of the pension would be:

• For asset category, the return contribution is 0.754.04% + 0.252.56% which is 3.67% or after risk-free a return of 3.36%. For benchmarks’ return contribution, we have 0.750.65(4.61%–4.04%) + 0.750.65(4.31%–4.04%) + 0.250.55(1.99%–2.56%) + 0.250.45(2.55%–2.56%) or 0.27%. The managers’ contribution is 0.750.65(4.76%–4.61%) + 0.75 0.65(4.13%–4.31%) + 0.250.55(1.60%–1.99%) + 0.250.45(2.91% –2.55%) or 0.012%.

Decision-Making LevelDecision-Making Level Fund ValueFund Value Incremental ReturnIncremental Return Incremental ValueIncremental Value

Beginning value $186,419,405 —— ——

Net contribution $187,369,405 0.0000% $950,000

Risk-free asset $187,944,879 0.3100% $575,474

Asset category $194,217,537 3.3600% $6,272,658

Benchmarks $194,720,526 0.2693% $502,989

Investment Managers $194,746,106 0.0128% $25,580

Allocation effects $194,816,600 0.0409% $70,494

Total Fund $194,816,600 3.9900% $8,397,195

Micro Attribution• A manager’s value added can be written as:

where, • wpj = Portfolio weight of sector j

• wBj = Benchmark weight of sector j

• rpj = Portfolio return of sector j

• rBj = Benchmark return of sector j

• S = Number of sectors

S

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Micro Attribution Components• A manager’s value can be decomposed into three parts:

– Pure Sector Allocation

– Allocation/Selection Interaction

– Within Sector Selection

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Performance Appraisal

• Whether skill or luck generated results

• Investment Skill:– Manager’s ability to outperform a benchmark

consistently over time – Risk Adjusted Appraisal Measures

• Ex-post alpha, Treynor ratio, Sharpe ratio, M2 ratio, information ratio

– Quality Control Charts • Cumulative value-added and its confidence bounds over

time (years of experience)

Ex Post Alpha• Also known as Jensen’s alpha, uses the empirical SML:

– Run a regression of excess portfolio return on market’s excess return, where,

• RA,t = portfolio A’s return for period t

• rf,t = risk-free rate of return for period t

• RM,t = Market portfolio’s return for period t

• εt = residual return for period t

• βA = beta for portfolio A

• αA = Jensen’s alpha

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Treynor Ratio• Also known as reward-to-volatility ratio (or, reward to

undiversifiable risk):

– where, • RA = average return for portfolio A

• rf, = average risk-free rate of return

• βA = beta for portfolio A

• TA = Treynor ratio

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Sharpe Ratio• Also known as reward-to-variability ratio:

– where, • RA = average return for portfolio A

• rf, = average risk-free rate of return

• σA = standard deviation of portfolio A returns

• SA = Sharpe ratio

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M2 Ratio• Also known as Modigliani and Miller ratio:

– where, • RA = average return for portfolio A

• rf, = average risk-free rate of return

• σA = standard deviation of portfolio A returns

• σM = standard deviation of market portfolio returns

• M2A = M2 ratio

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Information Ratio• Also known as active return-to-risk ratio:

– Numerator is Active Return and denominator is Active Risk, where,

• RA = average return for portfolio A (e.g., account)

• RB = average return for benchmark B (e.g., market)

• σA-B = standard deviation of the difference between portfolio A returns and benchmark B returns

• IRA = information ratio

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