Investment Course III – November 2007
Topic Four:
Portfolio Risk Analysis
Notion of Tracking Error
When managing an active investment portfolio against a well-defined benchmark (such as the
Standard & Poor’s 500 or the IPSA index), the goal of the manager should be to generate a return
that exceeds that of the benchmark while minimizing the portfolio’s return volatility relative to the
benchmark. Said differently, the manager should try to maximize alpha while minimizing
tracking error.
Tracking error can be defined as the extent to which return fluctuations in the managed portfolio
are not correlated with return fluctuations in the benchmark. The concept is analogous to the
statistic (1 – R2) in a regression context.
A flexible and straightforward way of measuring tracking error can be developed as follows:
Let:
wi = investment weight of asset i in the managed portfolio
Rit = return to asset i in period t
Rbt = return to the benchmark portfolio in period t.
With these definitions, we can define the period t return to managed portfolio as:
R pt 
N
w
i 1
i
Rit
where:
N = number of assets in the managed portfolio
and:
N
w
i 1
i
 1 (i.e., the managed portfolio is fully invested).
4-1
Notion of Tracking Error (cont.)
We can then specify the period t return differential between the managed portfolio and the benchmark as:
t 
N
w R
i
i 1
it
- R bt  R pt - R bt .
Notice two things about the return differential . First, given the returns to the N assets in the managed portfolio and the
benchmark, it is a function of the investment weights that the manager selects (i.e.,  = f({wi}/{Ri}, Rb)). Second,  can be
interpreted as the return to a hedge portfolio where wb = -1.
With these definitions and a sample of T return observations, calculate the variance of  as follows:
T
 ( t -  ) 2
 2  t 1
(T - 1)
.
Then, the standard deviation of the return differential is:
 
 2 = periodic tracking error,
so that annualized tracking error (TE) can be calculated as:
TE =   P
where P is the number of return periods in a year (e.g., P = 12 for monthly returns, P = 252 for daily returns).
4-2
Notion of Tracking Error (cont.)

Generally speaking, portfolios can be separated into the following
categories by the level of their annualized tracking errors:

Passive (i.e., Indexed): TE < 1.0% (Note: TE < 0.5% is normal)

Structured: 1.0% < TE < 3%

Active: TE > 3% (Note: TE > 5% is normal for active managers)
4-3
Index Fund Example: VFINX
4-4
ETF Example: SPY
4-5
“Large Blend” Active Manager: DGAGX
4-6
Tracking Errors for VFINX, SPY, DGAGX
4-7
Chile AFP Tracking Errors: Fondo A
(Rolling 12-month historical returns relative to Sistema)
4-8
Chile AFP Tracking Errors: Fondo E
(Rolling 12-month historical returns relative to Sistema)
4-9
Risk and Expected Return Within a Portfolio

Portfolio Theory begins with the recognition that the total risk and
expected return of a portfolio are simple extensions of a few basic
statistical concepts.

The important insight that emerges is that the risk characteristics of a
portfolio become distinct from those of the portfolio’s underlying assets
because of diversification. Consequently, investors can only expect
compensation for risk that they cannot diversify away by holding a
broad-based portfolio of securities (i.e., the systematic risk)

Expected Return of a Portfolio:
n
E(R p ) =
 w i * E(R i )
i = 1
where wi is the percentage investment in the i-th asset

Risk of a Portfolio:
 p2  [w1212  ...  w 2n n2 ]  [2w1w 21 2 1,2  ...  2w n-1w n n1 n n 1,n ]
Total Risk = (Unsystematic Risk) + (Systematic Risk)
4 - 10
Example of Portfolio Diversification:
Two-Asset Portfolio


Consider the risk and return characteristics of two stock positions:
E(R1) = 5%
1 = 8%
E(R2) = 6%
2 = 10%
1,2 = 0.4
Risk and Return of a 50%-50% Portfolio:
E(Rp) = (0.5)(5) + (0.5)(6) = 5.50%
and:
p = [(.25)(64) + (.25)(100) + 2(.5)(.5)(8)(10)(.4)]1/2 = 7.55%
Note that the risk of the portfolio is lower than that of either of the
individual securities
4 - 11
Another Two-Asset Class Example:
Suppose that a portfolio is divided into two different subportfolios consisting of stocks and bonds,
respectively. Further assume that the subportfolios have the following risk and expected return
characteristics:
E(Rstock) = 12.0%
E(Rbond) = 5.1%
stock = 21.2%
bond = 8.3%
 = 0.18
Then, an overall portfolio consisting of a 60%-40% mix of stocks and bonds would have the
following characteristics:
E(Rp) = (0.6)(0.120) + (0.4)(0.051) = 0.0924 or 9.24%
and
p = [(0.6)2(0.212)2 + (0.4)2(0.083)2] + [2(0.6)(0.4)(0.212)(0.083)(0.18)] = 0.0188
or
p = (0.0188)1/2 = 0.1371 or 13.71%
For different asset mixes and different levels of correlation between stocks and bonds, the portfolio
variance is given as:
( = 0.18)
( = 1)
( = -1)
Portfolio
wstock
wbond
E(Rp)
p
p
p
1
2
3
4
5
6
7
0.00
0.25
0.28
0.40
0.50
0.75
1.00
1.00
0.75
0.72
0.60
0.50
0.25
0.00
5.10%
6.83
7.04
7.86
8.55
10.28
12.00
8.30%
8.87
9.16
10.58
12.06
16.40
21.20
8.30%
11.53
11.93
13.46
14.75
17.98
21.20
8.30%
0.93
0.00
3.50
6.45
13.83
21.20
4 - 12
Example of a Three-Asset Portfolio:
Suppose that a portfolio is divided into three different subportfolios consisting of stocks, bonds, and
cash equivalents, respectively. Further assume that the subportfolios have the following risk and
expected return characteristics:
E(Rstock) = 12.0%
E(Rbond) = 5.1%
E(Rcash) = 3.6%
stock = 21.2%
bond = 8.3%
cash = 3.3%
stock,bond = 0.18
cash,stock = -0.07
cash,bond = 0.22
Then, an overall portfolio consisting of a 60%-30%-10% mix of stocks, bonds, and cash equivalents
would have the following characteristics:
E(Rp) = (0.6)(0.120) + (0.3)(0.051) + (0.1)(0.036) = 0.0909 or 9.09%
and:

p = [(0.6)2(0.212)2 + (0.3)2(0.083)2 + (0.1)2(0.033)2] +
{[2(0.6)(0.3)(0.212)(0.083)(0.18)] + [2(0.6)(0.1)(0.212)(0.033)(-0.07)]
+ [2(0.3)(0.1)(0.083)(0.033)(0.22)]} = 0.01793
or
p = (0.01793)1/2 = 0.1339 or 13.39%
4 - 13
Diversification and Portfolio Size: Graphical Interpretation
Total Risk
0.40
0.20
Systematic Risk
Portfolio Size
1
20
40
4 - 14
Advanced Portfolio Risk Calculations
Total Portfolio Risk
Suppose you have formed a portfolio consisting of N asset classes. Suppose also the portfolio
weight in the j-th asset class is denoted as wj while jk represents the covariance between assets j
and k (where jk equals the variance, j2, when j = k). With this notation, the return variance, p2,
of the portfolio is given by:
σ 2p 
N
N
 w w
j
j 1 k 1
k
σ jk
(1)
or, equivalently:
σ
2
p

N
w
j1
2
j
σ
2
j

N
N
 w
j1 k 1
j k
j
w k σ jk
(2)
The standard deviation of the portfolio is:
σp
N
  w 2j σ 2j 
 j1


w j w k σ jk 


j1 k 1

j k
N
1/ 2
N
(3)
4 - 15
Advanced Portfolio Risk Calculations (cont.)
Marginal Asset Risk
In order to compute the contribution of asset k’s risk to the overall risk of the portfolio, we can take
the derivative of equation (3) with respect to asset k’s weight in the portfolio:
σ p
w k

1N

w 2j σ 2j 


2 j1


w j w k σ jk 


j1 k 1
j k

1 / 2


N
2 w σ 2  2 w σ 

k k
j jk


j1
j k


N
 w 2σ 2 
j
j

j1


w j w k σ jk 


j1 k 1
j k

1 / 2

w σ 2 
 k k

=
 
= σ
2 -1/2
p
N
N

w σ 2 
 k k

N
N

w j σ jk 


j1
j k

N

w j σ jk 


j1
j k

N
which can be simplified to:
σ p
w k
 1  N
 1  N


    w j σ jk      w j σ j σ k ρ jk 
 σ p   j1
 σ p   j1


(4)
where j and k are the standard deviations of asset classes j and k, respectively, and jk is the
correlation coefficient between them.
4 - 16
Advanced Portfolio Risk Calculations (cont.)
Equation (4) shows that the marginal volatility of asset k in a portfolio is the weighted sum of the kth row (or, equivalently, the k-th column) of the return covariance matrix divided by the standard
deviation of the portfolio. Notice that the magnitude of this marginal risk contribution is determined
by three factors: (i) the volatility of the asset itself, (ii) the asset’s weight in the portfolio, and (iii) the
asset’s covariance with all of the other portfolio holdings and their investment weights.
A convenient property of marginal volatilities is that the weighted sum over all assets is, in fact, the
overall volatility of the portfolio. By contrast, recall that the standard deviation of a portfolio is not
simply a weighted average of the standard deviations of the underlying assets whenever the
correlations between the asset classes are less that +1.0. However, by redefining the risk of asset k
within the portfolio taking those correlations into account—which is what equation (4) does—it is
possible to view overall portfolio risk as an additive statistic. To see this notice that:
σ p
N
w
k 1
k
w k
 1  N
 
  w k   w j σ jk  
 σ p  j1
k 1
 
N
1
=
σp
N
N
 w w
j1 k 1
j
k
σ jk 
σ 2p
σp
 σp
(5)
4 - 17
Advanced Portfolio Risk Calculations (cont.)
This feature provides a mechanism for the portfolio manager to “roll-up” marginal volatilities to a
higher level (e.g., the sector or country level) without having to recompute the derivatives. In other
words, assume that the marginal volatility of each of the N assets has been calculated. Assume also
that we are interested in knowing the aggregate marginal volatility of a collection of M assets where k
= 1 to M < N (i.e., the M assets comprise a subset of the total portfolio). The marginal volatility of
this sub-portfolio is given by:
σ p
M
σ p
w M

w
k 1
k
w k
(6)
M
w
k 1
k
This additive property also allows the portfolio manager to interpret the weighted marginal volatilities
directly as the asset’s contribution to overall portfolio risk or as the contribution to tracking error if
asset class returns are defined in excess of the returns to a benchmark. That is:
σ p
Asset k’s Marginal Risk:
Asset k’s Total Contribution to Risk:
w k
wk
σ p
w k
(7)
(8)
Once again, equations (7) and (8) highlight two facts: (i) Asset k’s marginal volatility within the
portfolio depends not only on its own inherent riskiness (i.e. k) but also how it interacts with every
other asset held in the portfolio (i.e., jk), and (ii) Asset k’s total contribution to the risk of the overall
portfolio also depends on how much the manager invests in that asset class (wk).
4 - 18
Example of Marginal Risk Contribution Calculations
4 - 19
Dollar Allocation vs. Marginal Risk Contribution:
UTIMCO - March 2007
4 - 20
Fidelity Investment’s PRISM Risk-Tracking System:
Chilean Pension System – March 2004
PRISM (CR) / Return, Volatility and Tracking Error for 200401
Obs
1
2
3
4
5
6
7
8
AFP
LLLL
MMMM
NNNN
PPPP
QQQQ
RRRR
SSSS
SISTEMA
AFP
Id
Assets
Volatility
1
2
3
4
5
6
7
8
733
738
635
257
469
54
31
2918
0.081731
0.081423
0.080780
0.077193
0.079808
0.073797
0.061490
0.080525
Obs
1
2
3
4
5
6
7
8
AFP
LLLL
MMMM
NNNN
PPPP
QQQQ
RRRR
SSSS
SISTEMA
Assets
733
738
635
257
469
54
31
2918
0.3646
0.3514
0.4187
0.4713
0.3791
0.8308
2.0927
0.0000
Mean
Portfolio
Return
Tracking
Error
0.003646
0.003514
0.004187
0.004713
0.003791
0.008308
0.020927
0.000000
PRISM (CY) / AFP Value at Risk for 200401
Tracking
Error
(%)
March 17, 2004
0.23543
0.23476
0.22947
0.21804
0.22660
0.22981
0.20014
0.23089
Mean
Excess
Return
0.004531
0.003862
-0.001425
-0.012857
-0.004296
-0.001082
-0.030757
0.000000
March 17, 2004
50bp
Shortfall
Probability
(%)
8.5130
7.7410
11.6213
14.4383
9.3576
27.3640
40.5582
.
200bp
Shortfall
Probability
(%)
0.0000
0.0000
0.0001
0.0011
0.0000
0.8034
16.9613
.
4 - 21
Chilean Sistema Risk Tracking Example (cont.)
PRISM (CX) / Risk Diagnostics
Sistema-Relative Tracking Error for 200401
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
AFP
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
Asset Class
PRD_BCD
PDBC_PRBC
BCP
RECOGN_BOND
CERO
PRC_BCU
BCE
PCD_PTF
ZERO
DEPOSITS_OFFNO
DEPOSITS_OFFUF
LHF
BEF
BSF
CC2
CFI
CORPORATE_BOND
CTC_A
ENDESA
COPEC
ENTEL
CMPC
SQM-B
CERVEZAS
COLBUN
D&S
ENERSIS
Chile_SMALL_CAP
US
EUROPE
UK
JAPAN
GLOBAL
ASIA_EX_JAPAN
LATIN_AMERICA
EASTERN_EUROPE
EMERGING_MARKETS
G7_BOND
HY
EMB
MM
March 17, 2004
Active
Weight
(%)
Worst Case
Contribution
(%)
Contribution
to
Tracking Error
(%)
-0.2144
-1.1800
0.1830
-1.2731
-0.2823
-1.1605
0.0385
-0.0002
-0.0184
0.4018
3.7074
-0.9636
-0.0080
-0.0395
-0.2398
-0.3106
-0.1932
-0.3847
-0.0505
0.1789
-0.0672
0.2043
-0.1966
0.0069
0.0184
-0.4237
0.3731
0.6830
0.8959
0.0364
-0.0954
0.0710
0.8999
-2.1238
-0.5469
-0.1620
1.8244
0.1123
0.0364
0.2535
0.0094
==========
0.0000
-0.0194
-0.0331
0.0111
-0.1965
-0.0026
-0.0822
0.0011
-0.0000
-0.0017
0.0041
0.1039
-0.0363
-0.0011
-0.0029
0.0000
-0.0212
-0.0084
-0.0826
-0.0095
0.0424
-0.0179
0.0436
-0.0356
0.0018
0.0032
-0.1334
0.1107
0.0677
0.1332
0.0063
-0.0158
0.0149
0.1219
-0.3475
-0.0898
-0.0299
0.2729
0.0062
0.0041
0.0304
0.0008
============
-0.1870
0.0023
-0.0097
-0.0003
0.1040
0.0002
0.0193
0.0003
-0.0000
0.0002
0.0001
0.0304
0.0074
0.0002
0.0005
0.0000
0.0004
0.0006
0.0148
0.0001
0.0031
0.0015
0.0104
-0.0010
-0.0001
-0.0001
0.0127
0.0286
0.0039
0.0607
0.0022
-0.0064
0.0029
0.0518
-0.0165
-0.0272
-0.0061
0.0638
0.0008
0.0011
0.0069
0.0002
==============
0.3640
1
Implied
View
(%)
-1.0527
0.8206
-0.1709
-8.1706
-0.0617
-1.6626
0.8206
0.8206
-1.0363
0.0347
0.8206
-0.7657
-2.4299
-1.3547
0.0000
-0.1428
-0.3317
-3.8358
-0.2352
1.7448
-2.1698
5.1077
0.5129
-1.2372
-0.4791
-2.9882
7.6550
0.5649
6.7785
6.0088
6.7521
4.0322
5.7571
0.7754
4.9767
3.7790
3.4975
0.7247
3.0627
2.7101
2.5109
4 - 22
Chilean Sistema Risk Tracking Example (cont.)
PRISM (CX) / Risk Diagnostics
Sistema-Relative Tracking Error for 200401
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
AFP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
Asset Class
PRD_BCD
PDBC_PRBC
BCP
RECOGN_BOND
CERO
PRC_BCU
BCE
PCD_PTF
ZERO
DEPOSITS_OFFNO
DEPOSITS_OFFUF
LHF
BEF
BSF
CC2
CFI
CORPORATE_BOND
CTC_A
ENDESA
COPEC
ENTEL
CMPC
SQM-B
CERVEZAS
COLBUN
D&S
ENERSIS
Chile_SMALL_CAP
US
EUROPE
UK
JAPAN
GLOBAL
ASIA_EX_JAPAN
LATIN_AMERICA
EASTERN_EUROPE
EMERGING_MARKETS
G7_BOND
HY
EMB
MM
March 17, 2004
Active
Weight
(%)
0.5674
-1.1800
-0.5650
-0.2379
-0.2119
0.8570
-0.0128
-0.0002
-0.0184
6.9872
-3.3930
1.2495
-0.0080
-0.0790
-0.2401
-1.2552
-0.6841
-0.4095
-0.1138
-0.0081
0.0744
-0.0372
0.0650
0.0145
0.1316
-0.4237
0.2388
0.8843
-0.1981
-0.2826
-0.0419
0.0953
0.1448
0.0479
-0.5673
-0.3052
-1.7698
-0.1264
0.3431
-0.0563
0.5247
==========
-0.0000
Worst Case
Contribution
(%)
Contribution
to
Tracking Error
(%)
0.0515
-0.0331
-0.0343
-0.0367
-0.0020
0.0607
-0.0004
-0.0000
-0.0017
0.0717
-0.0951
0.0471
-0.0011
-0.0057
0.0000
-0.0858
-0.0298
-0.0880
-0.0213
-0.0019
0.0198
-0.0079
0.0118
0.0038
0.0232
-0.1334
0.0708
0.0876
-0.0295
-0.0492
-0.0069
0.0200
0.0196
0.0078
-0.0931
-0.0563
-0.2647
-0.0070
0.0386
-0.0068
0.0460
============
-0.5113
0.0224
0.0101
0.0010
0.0009
0.0001
0.0138
0.0001
0.0000
-0.0006
0.0131
0.0290
0.0084
-0.0002
-0.0010
0.0000
0.0207
-0.0005
0.0246
0.0045
0.0001
-0.0017
0.0007
-0.0013
0.0001
-0.0009
0.0434
-0.0028
-0.0165
0.0147
0.0297
0.0038
-0.0090
-0.0122
-0.0051
0.0635
0.0318
0.2148
0.0005
-0.0167
0.0023
-0.0150
==============
0.4708
4
Implied
View
(%)
3.9475
-0.8560
-0.1737
-0.3819
-0.0523
1.6098
-0.8560
-0.8560
3.3578
0.1876
-0.856
0.6692
1.9925
1.2991
0.0000
-1.6488
0.0663
-6.0021
-3.9277
-0.8357
-2.2350
-2.0070
-2.0004
0.4622
-0.6567
-10.2519
-1.1930
-1.8696
-7.4319
-10.5277
-8.9775
-9.4234
-8.3883
-10.5855
-11.1915
-10.4358
-12.135
-0.3814
-4.8702
-4.1709
-2.863
4 - 23
Notion of Downside Risk Measures:

As we have seen, the variance statistic is a symmetric measure of
risk in that it treats a given deviation from the expected outcome the
same regardless of whether that deviation is positive of negative.

We know, however, that risk-averse investors have asymmetric
profiles; they consider only the possibility of achieving outcomes that
deliver less than was originally expected as being truly risky. Thus,
using variance (or, equivalently, standard deviation) to portray
investor risk attitudes may lead to incorrect portfolio analysis
whenever the underlying return distribution is not symmetric.

Asymmetric return distributions commonly occur when portfolios
contain either explicit or implicit derivative positions (e.g., using a put
option to provide portfolio insurance).

Consequently, a more appropriate way of capturing statistically the
subtleties of this dimension must look beyond the variance measure.
4 - 24
Notion of Downside Risk Measures (cont.):

We will consider two alternative risk measures: (i) Semi-Variance, and (ii) Lower Partial Moments

Semi-Variance: The semi-variance is calculated in the same manner as the variance statistic,
but only the potential returns falling below the expected return are used:
E(R)
Semi - Variance
=

R p = -

p p (R p - E(R)) 2
Lower Partial Moment: The lower partial moment is the sum of the weighted deviations of each
potential outcome from a pre-specified threshold level (t), where each deviation is then raised to
some exponential power (n). Like the semi-variance, lower partial moments are asymmetric risk
measures in that they consider information for only a portion of the return distribution. The
formula for this calculation is given by:
t
LPM n =

R p = -
p p (t - R p ) n
4 - 25
Example of Downside Risk Measures:
To see how these alternative risk statistics compare to the variance consider the following
probability distributions for two investment portfolios:
Potential
Return
-15%
-10
-5
0
5
10
15
20
25
30
35
Prob. of Return for
Portfolio #1
Prob. of Return for
Portfolio #2
5%
8
12
16
18
16
12
8
5
0
0
0%
0
25
35
10
7
9
5
3
3
3
Notice that the expected return for both of these portfolios is 5%:
E(R)1 = (.05)(-0.15) + (.08)(-0.10) + ...+ (.05)(0.25) = 0.05
and
E(R)2 = (.25)(-0.05) + (.35)(0.00) + ...+ (.03)(0.35) = 0.05
4 - 26
Example of Downside Risk Measures (cont.):
Clearly, however, these portfolios would be viewed differently by different investors.
nuances are best captured by measures of return dispersion (i.e., risk).
These
1. Variance
As seen earlier, this is the traditional measure of risk, calculated the sum of the weighted squared
differences of the potential returns from the expected outcome of a probability distribution. For
these two portfolios the calculations are:
(Var)1 = (.05)[-0.15 - 0.05]2 + (.08)[-0.10 - 0.05]2 + ... + (.05)[0.25 - 0.05]2 = 0.0108
and
(Var)2 = (.25)[-0.05 - 0.05]2 + (.35)[0.00 - 0.05]2 + ... + (.03)[0.35 - 0.05]2 = 0.0114
Taking the square roots of these values leaves:
SD1 = 10.39%
and
SD2 = 10.65%
4 - 27
Example of Downside Risk Measures (cont.):
2. Semi-Variance
The semi-variance adjusts the variance by considering only those potential outcomes that fall
below the expected returns. For our two portfolios we have:
(SemiVar)1 = (.05)[-0.15 - 0.05]2 + (.08)[-0.10 - 0.05]2 + (.12)[-0.05 - 0.05]2 +
(.16)[0.00 - 0.05]2 = 0.0054
and
(SemiVar)2 = (.25)[-0.05 - 0.05]2 + (.35)[0.00 - 0.05]2 = 0.0034
Also, the semi-standard deviations can be derived as the square roots of these values:
(SemiSD)1 = 7.35%
and
(SemiSD)2 = 5.81%
Notice here that although Portfolio #2 has a higher standard deviation than Portfolio #1, it's semistandard deviation is smaller.
4 - 28
Example of Downside Risk Measures (cont.):
3. Lower Partial Moments
For these two portfolios, we will consider two cases (n = 1 and n = 2), both having a threshold level of 0% (i.e., t = 0):
(i) LPM1
(LPM1)1 = (.05)[0.00 - (-0.15)] + (.08)[0.00 - (-0.10)] + (.12)[0.00 - (-0.05)] = 0.0215
and
(LPM1)2 = (.25)[0.00 - (-0.05)] = 0.0125
(ii) LPM2
(LPM2)1 = (.05)[0.00 - (-0.15)]2 + (.08)[0.00 - (-0.10)]2 + (.12)[0.00 - (-0.05)]2
= 0.0022
and
(LPM2)2 = (.25)[0.00 - (-0.05)]2 = 0.0006
For comparative purposes, it is also useful to take the square root of the LPM2 values. These are:
(SqRt LPM2)1 = 4.72%
and
(SqRt LPM2)1 = 2.50%
Notice again that Portfolio #2 is seen as being less risky when the lower partial moment risk measures are used.
4 - 29