Equity Portfolio Tracking Error

Raman Vardharaj

Quantitative Portfolio Manager

Guardian Life Insurance

This presentation is based on the following unpublished paper: “Determinants of tracking error for equity portfolios” by Raman Vardharaj, Frank Jones and Frank Fabozzi.

How do you measure the risk of a stock portfolio?

• Absolute risk: Volatility of returns

• Relative risk: Volatility relative to a benchmark• Example: Tracking error or Active risk

Definition of Tracking Error

• Active return = Portfolio return - Benchmark return

• Tracking error = standard deviation of active returns

Application: Information ratio = alpha / tracking error

(Alpha = average active return)

• Estimation of tracking error:

- Trailing active returns (backward looking estimate)

- Risk model e.g., Barra (forward looking estimate)

A Measure of Risk

• An important risk metric for portfolios that are managed versus a benchmark

• Typical values:

- 0% for index funds

- less than 2% for enhanced index funds

- 5% for active large cap stock funds

Today

One yearfrom now

Time

CumulativeReturn

Tracking Error = 4%

BenchmarkCumulativeReturn

Rtn(Benchmark)

Rtn(BM) +4%

Rtn(BM) -4%

1 Std Devor66% confident

Tracking Error: An Example

Tracking Error = 1%

Today Time

CumulativeReturn

BenchmarkCumulativeReturn

Rtn(Benchmark) Rtn(BM) +1%

Rtn(BM) -1%

1 Std Dev or66% confident

One yearfrom now

Tracking Error: An Example

Determinants of tracking error

• Number of stocks held - those in the benchmark and those not in the benchmark

• Size or Style or Sector bets

• Beta

• Benchmark volatility

Effect of number of stocks

•Tracking error falls as the portfolio includes more and more of the stocks in the benchmark

• An optimally constructed portfolio of just 50 stocks can track the S&P 500 within 2%

• Tracking error rises as the portfolio starts to include stocks that are not in the benchmark

Tracking Error vs. the Number of Benchmark Stocks in the Portfolio

0%

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0 50 100 150 200 250 300 350 400 450 500

number of benchmark (S&P 500) stocks in the portfolio

Tra

ck

ing

Err

or Benchmark: S&P 500

Large Cap

Tracking Error is Reduced as More Benchmark Stocks are Included

Tracking Error vs. the Number of Benchmark Stocks in the Portfolio

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1%

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6%

0 50 100 150 200 250 300 350 400

number of benchmark (S&P 400) stocks in the portfolio

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Mid Cap

Benchmark: S&P 400

Tracking Error Reduction Requires More Benchmark Stocks for Mid Cap than for Large Cap

Tracking Error vs. the Number of Benchmark Stocks in the Portfolio

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0 50 100 150 200 250 300 350 400 450 500 550 600

number of benchmark (S&P 600) stocks in the portfolio

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Small Cap

Benchmark: S&P 600

Tracking Error Reduction Requires More Benchmark Stocks for Small Cap than for Mid Cap

Tracking Error vs. the Number of Non-Benchmark Stocks in the Portfolio

Note: All of the S&P 100 stocks are present in the S&P 500. We start w ith a portfolio that has all 100 of the stocks in the S&P 100 index and

progressively add to it stocks that are not in the S&P 100 index but are in the S&P 500 index. The tracking error for such a portfolio versus the

S&P 100 index is show n above. So, for example, w hen the portfolio has 200 of the S&P 500 stocks in addition to the S&P 100 stocks (i.e., 300 in all)

then its tracking error, upon optimal choice, is 6% as show n above.

5%

6%

7%

8%

9%

10%

100 150 200 250 300 350 400

number of non-benchmark (S&P 100) stocks in the portfolio

Tra

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ing

err

or

Benchmark: S&P 100

Portfolio Universe: S&P 500

Tracking Error Rises With the Increase in Non-Benchmark Stocks

Effect of size and style

• Tracking error rises as the portfolio deviates from its benchmark in terms of average market cap (size) or investment valuation (style)

• Different portfolios can have the same tracking error

In v e s tm e n t v a lu a t io n i s a lo n g th e h o r iz o n ta l a x is a n d m a rk e t c a p ( s iz e ) is a lo n g t h e v e r t ic a l a x i s .

L a r g e C a pP o r tfo l io 1 h a s a t r a c k in g e r r o r o f 0 % . P o r tfo l io s 2 , 3 , 4 h a v e n e a r ly s i m ila r t r a c k in g e r ro r s o f a r o u n d 2 .1 % .P o r tfo l io s 5 , 6 , 7 h a v e n e a r l y s i m i la r t r a c k in g e r ro r s o f a ro u n d 4 .2 % . P o r tfo l io 8 h a s a t r a c k in g e r ro r o f8 .5 % . A ll o f th e a b o v e t r a c k in g e r ro r s a r e v e r s u s t h e S & P 5 0 0 , th e la rg e c a p in d e x .

S m a ll C a pP o r tfo l io 1 1 h a s a t r a c k in g e r ro r o f 0 % . P o r tfo l io s 2 2 , 3 3 , 4 4 h a v e n e a r ly s im ila r t r a c k in g e r ro r s o f a ro u n d1 .7 % . P o r tfo l io s 5 5 , 6 6 , 7 7 h a v e n e a r ly s im ila r t r a c k in g e r r o r s o f a ro u n d 3 .4 % . P o r tfo l io 8 8 h a s a t r a c k in ge r ro r o f 4 .9 % . A ll o f th e s e t r a c k in g e r ro r s a r e v e r s u s th e S & P 6 0 0 , th e s m a ll c a p in d e x .

425

31

17

6

8 , 8 8

V a lu e B le n d G ro w th

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M id

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3 3

6

1 1

5 5 7 74 42 2

6 6

Effect of Size and Style Deviations on Tracking Error

Effect of sector bets and beta

•Tracking error rises as the portfolio’s sector allocations begin to differ from those of the benchmark

• Tracking error rises as the portfolio’s beta with respect to the benchmark begins to differ from 1

• Holding cash decreases a portfolio’s overall volatility but increases its tracking error

Using data pertaining to an actual mutual fund, this figure illustrates that the fund's tracking error with respect to the S&P 500 increased during

the calendar year 2000 as the fund placed increasingly larger sector bets. The tracking error values are predictive estimates from Barra.

We define sector deviation as the fund portfolio weight in a sector in excess of the benchmark weight in that sector. By definition, the sum of

all sector deviations as well as their average would be zero. We define the "Abs S ector Bet" as the average of the absolute values of the sector

deviations. The "RMS S ector Bet" is the root mean square sector deviation. That is, we first square the sector deviations, and calculate

their average. Then, we find the square root of this average. This is the RM S sector bet. The Abs Sector Bet and the RM S sector bet are two

indicators of the overall level of sector bets in the portfolio. Notice that the RM S sector bet measure appears to move more closely in line with

the Tracking Error than the Abs sector bet measure. Irrespective of which measure is used, tracking error increases as sector bets increase.

Tracking Error Increases as Sector Bets Increase

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Abs Sector bet RMS Sector bet Tkg Err

The Effects of Beta on Tracking Error

0.00 0.50 1.00 1.50 2.00

Beta

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rro

r

Why is it important to monitor the portfolio tracking error?

• Probability of dramatic shortfall (active return < -10%) rises with tracking error

• Probability of dramatic outperformance also rises with tracking error

• Management consequences of the two are asymmetric

Note: 1) Dramatic shortfall is assumed to be a shortfall of 10% or more. 2) Portfolio excess returns

w ere assumed to be normally distributed. 3) Portfolio alpha w as set at -0.93%. During the 15 years

ended Sept 2002, the median active domestic large cap stock fund had an annalized return that w as

0.93% low er than that of the S&P 500 over that period, according to Morningstar.

Probability of a dramatic shortfall rises with tracking error

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0% 5% 10% 15% 20% 25%

Tracking Error

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Appendix

Tracking Error of Enhanced Index Fund

Suppose, enhanced index fund = 10% active + 90% indexed Let tracking error of active = 5% Then, tracking error of enhanced index fund = 5%*10% = 0.5%

Subscripts: p = enhanced index portfolio; i = indexed portfolio; b = benchmark; a = activeNotation: r = return; w = weight; Var = variance; std = standard deviation; Corr = correlation

rp = wi*ri + wa*ra

rp – rb = wi*(ri - rb) + wa*(ra - rb), since wi + wa = 1.

Var ( rp - rb ) = Var {wi*(ri - rb)} + Var {wa*(ra - rb)} + 2*wi*wa*Corr(ri - rb, ra - rb)*std(ri - rb)* std(ra - rb)

Var ( rp - rb ) = Var {wa*(ra - rb )} since ri = rb

std( rp - rb ) = wa*std(ra - rb)

Tracking error rises with benchmark volatility

Subscript: p = portfolio; b = benchmarkNotation: r= return in excess of cash; e = error term; Var = variance; = beta in a single index market model

rp = *rb + e

rp - rb = (-1)*rb + e

Var(rp - rb) = (-1)2 * Var(rb) + Var(e)

There would be no correlation between rb and the error term due to the regression.

Tracking error and beta

Consider a combination of the market portfolio and cash.

Subscript: m = market in the context of a single index market model; p = portfolioNotation: r = return in excess of cash; w = weight; = beta; Var = variance;Cov = covariance; = absolute value

rp = w*rm + (1-w)*0 = w*rm , since the excess return of cash is zero.

= Cov(rp , rm) / Var(rm) = w * Var(rm) / Var(rm) = w

rp - rm = (w-1)*rm = (-1)*rm

Var (rp - rm) = (w-1)2 * Var(rm) = (-1)2 * Var(rm)

Tracking error = w-1 * std(rm) = -1 * std(rm)