1. Performance Calculations 101
Monday, October 19, 2009
Public Pension Financial Forum
John D. Simpson, CIPM
The Spaulding Group, Inc.

2. What we’ll do today
We’ll cover a few basic formulas that are used to calculate rates of return and risk
“Nature is pleased with simplicity”
Issac Newton, Principia
We will try to make this easy to comprehend
But, we have a fair amount to cover and limited time
Feel free to ask questions

3. Rates of Return: Time-weighting vs. Money-weighting
Time-weighted returns measure the performance of the portfolio manager
Money-weighted returns measure the performance of the fund or portfolio

4. Time-weighting
Time-weighting eliminates or reduces the impact of cash flows
Because managers don’t control the flows
Two general approaches:
Approximations, which approximate the exact, true, time-weighted rate of return
Exact, true, time-weighted rate of return

5. Approximation methods we’ll discuss
Original Dietz
Modified Dietz
Modified BAI
(a.k.a. Modified IRR and Linked IRR)

6. What Are Cash Flows? Two types: Specifics:
External: impact the portfolio
Internal: impact securities, sectors
Specifics:
External: contributions/withdrawals of cash and/or securities
Internal: buys/sells, interest/dividends, corporate actions

7. Visualizing Flows

8. The scenario we will use to demonstrate the various formulas:

9. Original Dietz
Assumes constant rate of return on the portfolio during the period
Very easy method to calculate
Provides approximation to the true rate of return
Returns can be distorted when large flows occur
Also, return doesn’t take into account market volatility, which further affects the accuracy
Weights each cash flow as if it occurred at the middle of the time period

10. Original Dietz

11. Modified Dietz Method
Assumes constant rate of return on the portfolio during the period
Provides an improvement in the approximation of true time-weighted rate of return, versus the Original Dietz formula
Disadvantage greatest when: (a) 1 or more large external cash flows; (b) cash flows occur during periods of high market volatility
Weights each external cash flow by the amount of time it is held in the portfolio

12. Modified Dietz Method

13. Modified Dietz Method

14. Modified BAI (Modified IRR, Linked IRR)
Determines internal rate of return for the period
Takes into account the exact timing of each external cash flow
Market value at beginning of period is treated as cash flow
Disadvantage: Requires iterative process solution – difficult to calculate manually

15. Modified BAI Method

16. Modified BAI Method

17. True, exact TWRR Value portfolio every time external flows occur
Advantage: calculates true time-weighted rate of return
Disadvantage: requires precise valuation of the portfolio on each day of external cash flow

18. True, exact TWRR

19. Money-weighted returns Internal Rate of Return (IRR)
Takes cash flows into consideration
Cash flows will impact the return
Only uses cash flows and the closing market value in calculation (don’t revalue during period)
Produces the return that equates the present value of all invested capital

20. Solving for the IRR It’s an iterative process
We solve for r, by trial-and error
The general rule is to use the Modified Dietz return as the “first order approximation” to the IRR

21. IRR Method

22. IRR Method

23. Calculation Question
Why did the Modified BAI and IRR yield the same returns (2.63%)?
ANSWER: Because both the Modified BAI and IRR use the same formula: the IRR. The difference is that with Modified BAI, we calculate the return for subperiods (e.g., months) and then geometrically link them; however, for the IRR, we do not link subperiod returns … we calculate the IRR across the entire period (e.g., if we were calculating a return for a year, we’d geometrically link the 12 monthly Modified BAI returns but we’d only calculate a single IRR, valuing the portfolio only at the start and the end of the year!

24. Contrasting IRR with time-weighting
IRR values portfolio at the beginning and end of the period
TWRR values at various times throughout the period

25. We’ll use an example to compare TWRR and MWRR
Our investment is a mutual fund
Where two investors begin with 100 shares
And both make two additional purchases during the year of 100 shares each
But at different times
And at different prices

26. Our fund’s end-of-month NAVs

27. Our investors’ purchases
Believes
Buy high/
Sell low
Believes
Buy low/
Sell high

28. The investments’ unrealized gains/losses
Paper gain of
$600!
Paper loss of
$600!

29. What’s our return?
The fund’s return (using an exact TWRR method):

30. How about our investors?
But this investor lost $600
And this investor made $600
Because time weighting eliminates the effect of cash flows!

31. How about money-weighting?
Investor #1’s IRR = %
Investor #2’s IRR = %

32. As a Plan Sponsor … Which returns make more sense to you?
Which are more meaningful?
TWRR judges portfolio manager
MWRR judges the portfolio

33. Multi-period rates of return
We don’t just want to report returns for a month
We want to link our returns to form quarterly, annual, since inception, etc. returns
How do we do this?

34. Geometric linking
The process used to link sub-period returns to create returns for extended periods:
e.g., We want to take January, February, and March returns to create a return for 1Q
We geometrically link in order to compound our returns

35. Geometric linking Step-by-step process:
Convert the returns to a decimal
Add 1
Multiply these numbers
Subtract 1
Convert the number to a percent

36. Geometric linking

37. Before we move to risk, are there any questions?

38. Risk measures Two categories Formulas that measure risk
We’ll look at standard deviation and tracking error
Formulas that adjust the return per unit of risk
We’ll look at Sharpe Ratio and Information Ratio

39. Standard Deviation Measures volatility of returns over time
The most common and most criticized measure to describe the risk of a security or portfolio.
Used not only in finance, but also statistics, sciences, and social sciences.
Provides a precise measure of the amount of variation in any group of numbers.

40. Standard Deviation; based on the Bell-shaped (normal) curve

41. Standard Deviation Formulas
Note: This is represented in Excel as the STDEVP Function
Note: This is represented in Excel as the STDEV Function

42. An example of standard deviation

43. Tracking Error
The difference between the performance of the benchmark and the replicating portfolio
Measures active risk; the risk the manager took relative to the benchmark
Measured as annualized standard deviation
Standard deviation of excess returns
Standard deviation of the difference in historical returns of a portfolio and its benchmark

44. Tracking Formula: Volatility of Past Returns vs. Benchmark
Tracking error measures how closely the portfolio follows the index and is measured as the standard deviation of the difference between the portfolio and index returns.

45. An example of Tracking Error
To annualize, multiply by square root of 12

46. The Sharpe Ratio Also known as Reward-to-Variability Ratio
Developed by Bill Sharpe –
Nobel Prize Winner
Equity Risk Premium (Return) / Standard Deviation (Risk)

47. Sharpe Ratio Formula Equity Risk Premium divided by standard deviation of portfolio returns

48. An example of Sharpe Ratio
To annualize, multiply by square root of 12

49. Information Ratio
The Information Ratio measures the excess return of an investment manager divided by the amount of risk the manager takes relative to the benchmark
It’s the Excess Return (Active Return) divided by the Tracking Error (Active Risk)
IR is a variation of the Sharpe Ratio, where the Return is the Excess Return and the Risk is the Excess or Active Risk

50. Information Ratio
IR serves as a measure of the “special information” an active portfolio manager has
Value Added (excess return) / Tracking Error
Typically annualize

51. Information Ratio
Active Return
on the account
Account’s
Active Risk

52. An example of Information Ratio

53. What have we covered today
Hopefully you’ll agree a lot in a short time
Return measures
TWRR approximation measues
Original Dietz
Modified Dietz
Modified BAI
TWRR exact measure
True daily
Geometric Linking

54. What have we covered today
Risk measures
Measurements of risk
Standard deviation
Tracking error
Measurements of risk-adjusted returns
Sharpe ratio
Information ratio

55. Questions? John D. Simpson jsimpson@spauldinggrp.com 1.310.500.9640