equity portfolio tracking error raman vardharaj quantitative portfolio manager guardian life...
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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This presentation is based on the following unpublished paper: “Determinants of tracking error for equity portfolios” by Raman Vardharaj, Frank Jones and Frank Fabozzi.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%
2%
4%
6%
8%
10%
12%
14%
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
0%
1%
2%
3%
4%
5%
6%
0 50 100 150 200 250 300 350 400
number of benchmark (S&P 400) stocks in the portfolio
Tra
ck
ing
Err
or
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
0%
1%
2%
3%
4%
5%
6%
7%
0 50 100 150 200 250 300 350 400 450 500 550 600
number of benchmark (S&P 600) stocks in the portfolio
Tra
ck
ing
Err
or
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
ck
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
L a rg e
M id
S m a l l
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
0%
2%
4%
6%
8%
10%
12%
3/3
1/9
9
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0/9
9
9/3
0/9
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/31
/99
3/3
1/0
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6/3
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9/3
0/0
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/31
/00
3/3
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1
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
Tra
ckin
g E
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
0%
5%
10%
15%
20%
25%
30%
35%
40%
0% 5% 10% 15% 20% 25%
Tracking Error
Pro
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f S
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rtfa
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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)