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**Performance Calculations 101**

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

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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

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**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

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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

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**Approximation methods we’ll discuss**

Original Dietz Modified Dietz Modified BAI (a.k.a. Modified IRR and Linked IRR)

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**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

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Visualizing Flows

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**The scenario we will use to demonstrate the various formulas:**

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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

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Original Dietz

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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

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Modified Dietz Method

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Modified Dietz Method

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**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

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Modified BAI Method

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Modified BAI Method

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**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

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True, exact TWRR

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**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

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**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

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IRR Method

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IRR Method

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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!

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**Contrasting IRR with time-weighting**

IRR values portfolio at the beginning and end of the period TWRR values at various times throughout the period

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**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

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**Our fund’s end-of-month NAVs**

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**Our investors’ purchases**

Believes Buy high/ Sell low Believes Buy low/ Sell high

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**The investments’ unrealized gains/losses**

Paper gain of $600! Paper loss of $600!

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

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**How about our investors?**

But this investor lost $600 And this investor made $600 Because time weighting eliminates the effect of cash flows!

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**How about money-weighting?**

Investor #1’s IRR = % Investor #2’s IRR = %

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**As a Plan Sponsor … Which returns make more sense to you?**

Which are more meaningful? TWRR judges portfolio manager MWRR judges the portfolio

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**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?

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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

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**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

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Geometric linking

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**Before we move to risk, are there any questions?**

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**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

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**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.

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

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**Standard Deviation Formulas**

Note: This is represented in Excel as the STDEVP Function Note: This is represented in Excel as the STDEV Function

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**An example of standard deviation**

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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

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**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.

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**An example of Tracking Error**

To annualize, multiply by square root of 12

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**The Sharpe Ratio Also known as Reward-to-Variability Ratio**

Developed by Bill Sharpe – Nobel Prize Winner Equity Risk Premium (Return) / Standard Deviation (Risk)

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

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**An example of Sharpe Ratio**

To annualize, multiply by square root of 12

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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

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

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

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**An example of Information Ratio**

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**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

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**What have we covered today**

Risk measures Measurements of risk Standard deviation Tracking error Measurements of risk-adjusted returns Sharpe ratio Information ratio

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

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