Collection Of Portfolio Tracking Error Calculation 536s4n

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Raman Vardharaj Quantitative Portfolio Manager Guardian Life Insurance This presentation is based on the following unpublished paper: “Determinants of tracking error for equity portfolios” by Raman Vardharaj, Frank Jones and Frank Fabozzi.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

Tracking Error: An Example Tracking Error = 4%

Cumulative Return

Rtn(BM) +4%

Benchmark Cumulative Return

Rtn(Benchmark) Rtn(BM) -4%

Today

Time One year from now

1 Std Dev or 66% confident

Tracking Error: An Example Cumulative Return

Tracking Error = 1%

Benchmark Cumulative Return

Today

Rtn(BM) +1% Rtn(Benchmark) Rtn(BM) -1%

Time One year from now

1 Std Dev or 66% confident

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 is Reduced as More Benchmark Stocks are Included

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

14% 12%

Large Cap

Tracking Error

10% Benchm ark: S&P 500

8% 6% 4% 2% 0% 0

50

100

150

200

250

300

350

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

400

450

500

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

6% Mid Cap

5% Tracking Error

Benchm ark: S&P 400

4% 3% 2% 1% 0% 0

50

100

150

200

250

300

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

350

400

Tracking Error Reduction Requires More Benchmark Stocks for Small Cap than for Mid Cap Tracking Error vs. the Number of Benchmark Stocks in the Portfolio

7%

Tracking Error

6% Small Cap

5% 4%

Benchmark: S&P 600

3% 2% 1% 0% 0

50

100

150

200

250

300

350

400

450

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

500

550

600

Tracking Error Rises With the Increase in Non-Benchmark Stocks

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

10%

Tracking error

Benchmark: S&P 100

9% 8%

Portfolio Universe: S&P 500

7% 6% 5% 100

150

200

250

300

350

number of non-benchmark (S&P 100) 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.

400

Effect of size and style • Tracking error rises as the portfolio deviates from its benchmark in of average market cap (size) or investment valuation (style) • Different portfolios can have the same tracking error

Effect of Size and Style Deviations on Tracking Error Value

Blend

1

2

Growth

4

5

7

Large 3 1

6

8, 88

Mid 66 6

33

11

55

22

Small 44

77

Investment valuation is along the horizontal axis and market cap (size) is along the vertical axis. Large Cap Portfolio 1 has a tracking error of 0%. Portfolios 2, 3, 4 have nearly similar tracking errors of around 2.1%. Portfolios 5, 6, 7 have nearly similar tracking errors of around 4.2%. Portfolio 8 has a tracking error of 8.5%. All of the above tracking errors are versus the S&P 500, the large cap index. Small Cap Portfolio 11 has a tracking error of 0%. Portfolios 22, 33, 44 have nearly similar tracking errors of around 1.7%. Portfolios 55, 66, 77 have nearly similar tracking errors of around 3.4%. Portfolio 88 has a tracking error of 4.9%. All of these tracking errors are versus the S&P 600, the small cap index.

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

Tracking Error Increases as Sector Bets Increase 12% 10% 8% 6% 4% 2%

Abs Sector bet

RMS Sector bet

Tkg Err

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.

3/31/01

12/31/00

9/30/00

6/30/00

3/31/00

12/31/99

9/30/99

6/30/99

3/31/99

0%

Tracking Error

The Effects of Beta on Tracking Error

0.00

0.50

1.00 Beta

1.50

2.00

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

Probability of Shortfall

Probability of a dramatic shortfall rises with tracking error 40% 35% 30% 25% 20% 15% 10% 5% 0% 0%

5%

10%

15%

20%

Tracking Error

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.

25%

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 = active Notation: 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 )} std( rp - rb ) = wa*std(ra - rb)

since ri = rb

Tracking error rises with benchmark volatility Subscript: p = portfolio; b = benchmark Notation: 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 = portfolio Notation: 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)