1. PERFORMANCE EVALUATION
Portfolio Management
Prof. Ali Nejadmalayeri

2. 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

3. 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

4. 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

5. 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

6. Performance w/o intra CFs
Without initial or ending cash infusions:
With initial cash flows
With ending cash flows

7. 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:

8. 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 = = 2.92%

9. 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

10. 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!

11. 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.

12. 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 = = = 2.65%

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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.

19. 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

20. 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 %. 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%

21. Performance Attribution: Example
The performance attribution of the pension would be:
For asset category, the return contribution is 0.754.04% 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%) 0.65(4.31%–4.04%) 0.55(1.99%–2.56%) 0.45(2.55%–2.56%) or 0.27%. The managers’ contribution is 0.750.65(4.76%–4.61%)  0.65(4.13%–4.31%) 0.55(1.60%–1.99%) 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

22. 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

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

24. 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)

25. 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

26. 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

27. 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

28. 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

29. 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