Investment Course III – November 2007

Topic Four:

Portfolio Risk Analysis

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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:

N

iitR

1ipt w R

where: N = number of assets in the managed portfolio and:

1 w1

i

N

i

(i.e., the managed portfolio is fully invested).

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Notion of Tracking Error (cont.)

We can then specify the period t return differential between the managed portfolio and the benchmark as:

.R - R R - w btptbt1

it

N

iitR

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:

.1) - (T

) - (

T

1t

2t

2

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).

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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)

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Index Fund Example: VFINX

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ETF Example: SPY

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“Large Blend” Active Manager: DGAGX

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Tracking Errors for VFINX, SPY, DGAGX

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Chile AFP Tracking Errors: Fondo A(Rolling 12-month historical returns relative to Sistema)

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Chile AFP Tracking Errors: Fondo E(Rolling 12-month historical returns relative to Sistema)

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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:

where wi is the percentage investment in the i-th asset

Risk of a Portfolio:

Total Risk = (Unsystematic Risk) + (Systematic Risk)

E(R ) = w E(R )p ii = 1

n

i *

]w2w ... w[2w ] w ... [w n,1nn1nn1-n2,121212n

2n

21

21

2p

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Example of Portfolio Diversification: Two-Asset Portfolio

Consider the risk and return characteristics of two stock positions:

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

E(R1) = 5% 1 = 8% 1,2 = 0.4

E(R2) = 6% 2 = 10%

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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% stock = 21.2% = 0.18 E(Rbond) = 5.1% bond = 8.3% 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 0.00 1.00 5.10% 8.30% 8.30% 8.30% 2 0.25 0.75 6.83 8.87 11.53 0.93 3 0.28 0.72 7.04 9.16 11.93 0.00 4 0.40 0.60 7.86 10.58 13.46 3.50 5 0.50 0.50 8.55 12.06 14.75 6.45 6 0.75 0.25 10.28 16.40 17.98 13.83 7 1.00 0.00 12.00 21.20 21.20 21.20

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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% stock = 21.2% stock,bond = 0.18 E(Rbond) = 5.1% bond = 8.3% cash,stock = -0.07 E(Rcash) = 3.6% cash = 3.3% 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%

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Diversification and Portfolio Size: Graphical Interpretation

Total Risk

Portfolio Size1 20 40

0.20

0.40

Systematic Risk

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Advanced Portfolio Risk CalculationsTotal 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, p

2,

of the portfolio is given by:

N

1j

N

1kjkkj

2p σww σ (1)

or, equivalently:

N

1j

N

kj1k

jkkj

N

1j

2j

2j

2p σww σw σ (2)

The standard deviation of the portfolio is:

2/1

N

1j

N

kj1k

jkkj

N

1j

2j

2jp σww σw σ

(3)

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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:

N

kj1j

jkj2kk

2/1

N

1j

N

kj1k

jkkj

N

1j

2j

2j

k

p σw2 σw2 σww σw 2

1

w

σ

=

N

kj1j

jkj2kk

2/1

N

1j

N

kj1k

jkkj

N

1j

2j

2j σw σw σww σw

=

N

kj1j

jkj2kk

1/2-2p σw σw σ

which can be simplified to:

N

1jjkkjj

p

N

1jjkj

pk

p ρσσw σ

1 σw

σ

1

w

σ (4)

where j and k are the standard deviations of asset classes j and k, respectively, and jk is the

correlation coefficient between them.

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Advanced Portfolio Risk Calculations (cont.)

Equation (4) shows that the marginal volatility of asset k in a portfolio is the weighted sum of the k-

th 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:

N

1jjkj

p

N

1kk

k

pN

1kk σw

σ

1 w

w

σw

= pp

2p

N

1j

N

1kjkkj

p

σ σ

σ σww

σ

1

(5)

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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:

M

1kk

M

1k k

pk

M

p

w

w

σw

w

σ (6)

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:

Asset k’s Marginal Risk: k

p

w

σ

(7)

Asset k’s Total Contribution to Risk: k

pk w

σw

(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).

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Example of Marginal Risk Contribution Calculations

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Dollar Allocation vs. Marginal Risk Contribution:UTIMCO - March 2007

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Fidelity Investment’s PRISM Risk-Tracking System: Chilean Pension System – March 2004

PRISM (CR) / Return, Volatility and Tracking Error for 200401 March 17, 2004 Mean Mean AFP Tracking Portfolio Excess Obs AFP Id Assets Volatility Error Return Return 1 LLLL 1 733 0.081731 0.003646 0.23543 0.004531 2 MMMM 2 738 0.081423 0.003514 0.23476 0.003862 3 NNNN 3 635 0.080780 0.004187 0.22947 -0.001425 4 PPPP 4 257 0.077193 0.004713 0.21804 -0.012857 5 QQQQ 5 469 0.079808 0.003791 0.22660 -0.004296 6 RRRR 6 54 0.073797 0.008308 0.22981 -0.001082 7 SSSS 7 31 0.061490 0.020927 0.20014 -0.030757 8 SISTEMA 8 2918 0.080525 0.000000 0.23089 0.000000

PRISM (CY) / AFP Value at Risk for 200401 March 17, 2004 50bp 200bp Tracking Shortfall Shortfall Error Probability Probability Obs AFP Assets (%) (%) (%) 1 LLLL 733 0.3646 8.5130 0.0000 2 MMMM 738 0.3514 7.7410 0.0000 3 NNNN 635 0.4187 11.6213 0.0001 4 PPPP 257 0.4713 14.4383 0.0011 5 QQQQ 469 0.3791 9.3576 0.0000 6 RRRR 54 0.8308 27.3640 0.8034 7 SSSS 31 2.0927 40.5582 16.9613 8 SISTEMA 2918 0.0000 . .

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Chilean Sistema Risk Tracking Example (cont.)PRISM (CX) / Risk Diagnostics March 17, 2004 1 Sistema-Relative Tracking Error for 200401 Contribution Active Worst Case to Implied Weight Contribution Tracking Error View Obs AFP Asset Class (%) (%) (%) (%) 1 LLLL PRD_BCD -0.2144 -0.0194 0.0023 -1.0527 2 LLLL PDBC_PRBC -1.1800 -0.0331 -0.0097 0.8206 3 LLLL BCP 0.1830 0.0111 -0.0003 -0.1709 4 LLLL RECOGN_BOND -1.2731 -0.1965 0.1040 -8.1706 5 LLLL CERO -0.2823 -0.0026 0.0002 -0.0617 6 LLLL PRC_BCU -1.1605 -0.0822 0.0193 -1.6626 7 LLLL BCE 0.0385 0.0011 0.0003 0.8206 8 LLLL PCD_PTF -0.0002 -0.0000 -0.0000 0.8206 9 LLLL ZERO -0.0184 -0.0017 0.0002 -1.0363 10 LLLL DEPOSITS_OFFNO 0.4018 0.0041 0.0001 0.0347 11 LLLL DEPOSITS_OFFUF 3.7074 0.1039 0.0304 0.8206 12 LLLL LHF -0.9636 -0.0363 0.0074 -0.7657 13 LLLL BEF -0.0080 -0.0011 0.0002 -2.4299 14 LLLL BSF -0.0395 -0.0029 0.0005 -1.3547 15 LLLL CC2 -0.2398 0.0000 0.0000 0.0000 16 LLLL CFI -0.3106 -0.0212 0.0004 -0.1428 17 LLLL CORPORATE_BOND -0.1932 -0.0084 0.0006 -0.3317 18 LLLL CTC_A -0.3847 -0.0826 0.0148 -3.8358 19 LLLL ENDESA -0.0505 -0.0095 0.0001 -0.2352 20 LLLL COPEC 0.1789 0.0424 0.0031 1.7448 21 LLLL ENTEL -0.0672 -0.0179 0.0015 -2.1698 22 LLLL CMPC 0.2043 0.0436 0.0104 5.1077 23 LLLL SQM-B -0.1966 -0.0356 -0.0010 0.5129 24 LLLL CERVEZAS 0.0069 0.0018 -0.0001 -1.2372 25 LLLL COLBUN 0.0184 0.0032 -0.0001 -0.4791 26 LLLL D&S -0.4237 -0.1334 0.0127 -2.9882 27 LLLL ENERSIS 0.3731 0.1107 0.0286 7.6550 28 LLLL Chile_SMALL_CAP 0.6830 0.0677 0.0039 0.5649 29 LLLL US 0.8959 0.1332 0.0607 6.7785 30 LLLL EUROPE 0.0364 0.0063 0.0022 6.0088 31 LLLL UK -0.0954 -0.0158 -0.0064 6.7521 32 LLLL JAPAN 0.0710 0.0149 0.0029 4.0322 33 LLLL GLOBAL 0.8999 0.1219 0.0518 5.7571 34 LLLL ASIA_EX_JAPAN -2.1238 -0.3475 -0.0165 0.7754 35 LLLL LATIN_AMERICA -0.5469 -0.0898 -0.0272 4.9767 36 LLLL EASTERN_EUROPE -0.1620 -0.0299 -0.0061 3.7790 37 LLLL EMERGING_MARKETS 1.8244 0.2729 0.0638 3.4975 38 LLLL G7_BOND 0.1123 0.0062 0.0008 0.7247 39 LLLL HY 0.0364 0.0041 0.0011 3.0627 40 LLLL EMB 0.2535 0.0304 0.0069 2.7101 41 LLLL MM 0.0094 0.0008 0.0002 2.5109 ========== ============ ============== 0.0000 -0.1870 0.3640

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Chilean Sistema Risk Tracking Example (cont.)PRISM (CX) / Risk Diagnostics March 17, 2004 4 Sistema-Relative Tracking Error for 200401 Contribution Active Worst Case to Implied Weight Contribution Tracking Error View Obs AFP Asset Class (%) (%) (%) (%) 1 PPPP PRD_BCD 0.5674 0.0515 0.0224 3.9475 2 PPPP PDBC_PRBC -1.1800 -0.0331 0.0101 -0.8560 3 PPPP BCP -0.5650 -0.0343 0.0010 -0.1737 4 PPPP RECOGN_BOND -0.2379 -0.0367 0.0009 -0.3819 5 PPPP CERO -0.2119 -0.0020 0.0001 -0.0523 6 PPPP PRC_BCU 0.8570 0.0607 0.0138 1.6098 7 PPPP BCE -0.0128 -0.0004 0.0001 -0.8560 8 PPPP PCD_PTF -0.0002 -0.0000 0.0000 -0.8560 9 PPPP ZERO -0.0184 -0.0017 -0.0006 3.3578 10 PPPP DEPOSITS_OFFNO 6.9872 0.0717 0.0131 0.1876 11 PPPP DEPOSITS_OFFUF -3.3930 -0.0951 0.0290 -0.856 12 PPPP LHF 1.2495 0.0471 0.0084 0.6692 13 PPPP BEF -0.0080 -0.0011 -0.0002 1.9925 14 PPPP BSF -0.0790 -0.0057 -0.0010 1.2991 15 PPPP CC2 -0.2401 0.0000 0.0000 0.0000 16 PPPP CFI -1.2552 -0.0858 0.0207 -1.6488 17 PPPP CORPORATE_BOND -0.6841 -0.0298 -0.0005 0.0663 18 PPPP CTC_A -0.4095 -0.0880 0.0246 -6.0021 19 PPPP ENDESA -0.1138 -0.0213 0.0045 -3.9277 20 PPPP COPEC -0.0081 -0.0019 0.0001 -0.8357 21 PPPP ENTEL 0.0744 0.0198 -0.0017 -2.2350 22 PPPP CMPC -0.0372 -0.0079 0.0007 -2.0070 23 PPPP SQM-B 0.0650 0.0118 -0.0013 -2.0004 24 PPPP CERVEZAS 0.0145 0.0038 0.0001 0.4622 25 PPPP COLBUN 0.1316 0.0232 -0.0009 -0.6567 26 PPPP D&S -0.4237 -0.1334 0.0434 -10.2519 27 PPPP ENERSIS 0.2388 0.0708 -0.0028 -1.1930 28 PPPP Chile_SMALL_CAP 0.8843 0.0876 -0.0165 -1.8696 29 PPPP US -0.1981 -0.0295 0.0147 -7.4319 30 PPPP EUROPE -0.2826 -0.0492 0.0297 -10.5277 31 PPPP UK -0.0419 -0.0069 0.0038 -8.9775 32 PPPP JAPAN 0.0953 0.0200 -0.0090 -9.4234 33 PPPP GLOBAL 0.1448 0.0196 -0.0122 -8.3883 34 PPPP ASIA_EX_JAPAN 0.0479 0.0078 -0.0051 -10.5855 35 PPPP LATIN_AMERICA -0.5673 -0.0931 0.0635 -11.1915 36 PPPP EASTERN_EUROPE -0.3052 -0.0563 0.0318 -10.4358 37 PPPP EMERGING_MARKETS -1.7698 -0.2647 0.2148 -12.135 38 PPPP G7_BOND -0.1264 -0.0070 0.0005 -0.3814 39 PPPP HY 0.3431 0.0386 -0.0167 -4.8702 40 PPPP EMB -0.0563 -0.0068 0.0023 -4.1709 41 PPPP MM 0.5247 0.0460 -0.0150 -2.863 ========== ============ ============== -0.0000 -0.5113 0.4708

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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.

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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:

Lower Partial Moment: The lower partial moment is the sum of the weighted deviations of each potential outcome from a pre-specified threshold level (), 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:

- = R

n p pn

p

)R - ( p = LPM

E(R)

- = R

2 p p

p

E(R)) -(R p = Variance-Semi

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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 Prob. of Return for Prob. of Return for Return Portfolio #1 Portfolio #2 -15% 5% 0% -10 8 0 -5 12 25 0 16 35 5 18 10 10 16 7 15 12 9 20 8 5 25 5 3 30 0 3 35 0 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

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Example of Downside Risk Measures (cont.):

Clearly, however, these portfolios would be viewed differently by different investors. These nuances are best captured by measures of return dispersion (i.e., risk).

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%

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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 semi-standard deviation is smaller.

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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., = 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.