PERFORMANCE EVALUATION 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 Return Compound growth for an initial $1 of investment Money-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

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 return is defined as:

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 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

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!

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 Attribution The general idea is to find out what give rise to the return: Net contribution Risk-free Asset Asset Category Benchmark Investment Managers Allocation Effects

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. Benchmark return for the ith asset category Benchmark return for the jth manager in the ith asset category IPS weight 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. IPS weight for the jth manager in ith asset category 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

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 jth manager in ith asset category 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

Others’ Attributions Net contribution: Risk-free Asset: 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: Example A 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 Category Policy 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: Example For the past year, the pension performance has been: During the same period, the risk-free rate was 0.31%. IF 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 Value End Value Contributions Actual Return Benchmark Return 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 2.9100% 2.55% Total $186,419,405 $194,816,599 $950,000 3.9900% 3.94%

Performance Attribution: Example The performance attribution of the pension would be: For asset category, the return contribution is 0.754.04% + 0.252.56% which is 3.67% or after risk-free a return of 3.36%. For benchmarks’ return contribution, we have 0.750.65(4.61%–4.04%) + 0.750.65(4.31%–4.04%) + 0.250.55(1.99%–2.56%) + 0.250.45(2.55%–2.56%) or 0.27%. The managers’ contribution is 0.750.65(4.76%–4.61%) + 0.75 0.65(4.13%–4.31%) + 0.250.55(1.60%–1.99%) + 0.250.45(2.91% –2.55%) or 0.012%. Decision-Making Level Fund Value Incremental Return Incremental 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 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

Micro Attribution Components A manager’s value can be decomposed into three parts: Pure Sector Allocation Allocation/Selection Interaction Within Sector Selection

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

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

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

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

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