FNCE10002 Lecture 8: Week 8 Lecture 8

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tealcattle862

7 Oct 2018

School

University of Melbourne

Department

Finance

Course

FNCE10002

Professor

Howard

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Document Summary

Assumptions of portfolio theory: the two main assumptions of modern portfolio theory are, returns on assets are normally distributed, returns are measured over one feature time period, return as at the end of this period are normally distri(cid:271)uted. Therefore, a(cid:374) asset"s return distribution can fully described by just two numbers: Portfolio risk and return: two securities: a portfolio"s e(cid:454)pe(cid:272)ted retur(cid:374) is the weighted average of the expected returns of its component securities, note that the (cid:449)eights are per(cid:272)e(cid:374)tages of the i(cid:374)(cid:448)estor"s origi(cid:374)al (cid:449)ealth i(cid:374)(cid:448)ested i(cid:374) each security. T(cid:2925)(cid:2930)a(cid:2922) a(cid:2923)(cid:2925)(cid:2931)(cid:2924)(cid:2930) i(cid:2924)(cid:2932)e(cid:2929)(cid:2930)e: note: (cid:1875)(cid:2869)+(cid:1875)(cid:2870)=(cid:883) and (cid:1875)(cid:2869)=(cid:883) (cid:1875)(cid:2870) (or (cid:1875)(cid:2870)=(cid:883) (cid:1875)(cid:2869), (cid:1853)(cid:1870)((cid:1870))=(cid:2870)=(cid:1875)(cid:2869)(cid:2870)(cid:2869)(cid:2870)+(cid:1875)(cid:2870)(cid:2870)(cid:2870)(cid:2870)+(cid:884)(cid:1875)(cid:2869)(cid:1875)(cid:2870)(cid:2869)(cid:2870, (cid:2869)(cid:2870)=(cid:1829)(cid:1867)(cid:1874)(cid:4666)(cid:1870)(cid:2869),(cid:1870)(cid:2870)(cid:4667)=(cid:1829)(cid:1867)(cid:1874)(cid:1853)(cid:1870)(cid:1861)(cid:1853)(cid:1866)(cid:1855)(cid:1857) (cid:1854)(cid:1857)(cid:1872)(cid:1875)(cid:1857)(cid:1857)(cid:1866) (cid:1871)(cid:1857)(cid:1855)(cid:1873)(cid:1870)(cid:1861)(cid:1872)(cid:1861)(cid:1857)(cid:1871) (cid:883) (cid:1853)(cid:1866)(cid:1856) (cid:884, (cid:1830)((cid:1870)), the sta(cid:374)dard de(cid:448)iatio(cid:374) of the portfolio is . In the above expression, only the third term can be negative. In the special case when (cid:2778)(cid:2779)=(cid:2777), (cid:1875)(cid:2869) si(cid:373)plifies to (cid:1875)(cid:2869) = (cid:2870)(cid:2870) (cid:2869)(cid:2870)+(cid:2870)(cid:2870) (cid:1853)(cid:1866)(cid:1856) (cid:1875)(cid:2870) = (cid:2869)(cid:2870) (cid:2869)(cid:2870)+(cid:2870)(cid:2870: some diversification benefits always exist.

Related Questions

Hi guys could someone help me and explain in detail the two fundamental principles of investments â the risk/return tradeoff and diversification. Also, describe how asset allocation and the correlation of various securities within asset classes impact the riskiness of an investorâs portfolio.

I found an answer but it seems to me very long and not sure about it so can you help me out guys

ANS:

Risk /Return Tradeoff

Risk, return and liquidity are three dimensions of investment decision. Risk/Return trade off considers two dimension ie risk and return. As risk of an investment increases , its expected return will decrease. Risk is measured by volatility of return which can be statistically obtained by standard deviation of return. Higher the standard deviation, higher will be volatility of return and higher the risk

Risk can be also measured as a relative risk compared to the market. This is measured by stock beta with reference to a market index. Higher the beta, higher the value of stock Beta, higher will be the risk .

Investors seeking higher return will need to take higher risks.

Treasury bonds are almost risk free assets. This has the lowest rate of return. Stocks have higher risk , hence have higher expected return. High beta stocks will have higher risk and higher returns.

An investor should understand his risk taking ability and will have to make a trade off between risk and return depending on his risk apetite.

Diversification:

There is a saying, do not put all your eggs in one basket. The reason is that it has higher risk if all eggs are in one basket. If the eggs are distributed in different basket, risk of loss decreases.

Same principle is applied in investment. Diversification reduces risk.

If one invests in different classes of assets like bonds, stock, commodities and properties âhis risk is likely to reduce. Reduction in return in one asset is likely to be compensated by return in other asset.

Within one class of asset also, there should be diversification to reduce risk.

For example one stock âA âmay have expected return of 10% and standard deviation of 5%.This risk can be reduced by diversification in number of assets and the ratio of Risk/return can be decreased.

If we diversify and instead of investing 100% in stock A, we invest 60% in âAâ and 40% in another stock âBâ having expected return15% and standard deviation of 8%.assume the correlation between the two assets are 0.5

Portfolio return=wa*Ra+wb*Rb

Wa=Weight of asset A=0.6

Wb=Weight of asset B=0.4

Ra=Return of A=10%

Rb=Return of B=15%

Sa=Standard deviation of A=5%

Sb=Standard deviation of B=8%

Corr(a,b)=Correlation of A & B=0.2

Portfolio return=0.6*10+0.4*15=12%

Portfolio Variance=(wa^2)*(Sa^2)+(wb^2)*(Sb^2)+2wa*wb*Cov(a,b)

Cov(a,b)=Corr(a,b)*Sa*Sb=0.1*5*8=4

Portfolio Variance=(0.6^2)*(5^2)+(0.4^2)*(8^2)+2*0.6*0.8*4=23.08

Portfolio standard deviation=Square root of Variance=(23.08^0.5)=4.8%

The diversification has increased return and reduced risk

Stock A was giving 10% return with 5% standard deviation. With diversification of portfolio we have 12% return and 4.8% standard deviation. Risk return ratio is decreased.

It may be noted that, higher the correlation between assets in the portfolio, higher will be risk in terms of standard deviation. Hence it is better to diversify with uncorrelated assets.

Diversification reduces unsystematic risk,but it cannot reduce systematic risk of overall market or economy

Problem 1:

You invest in a portfolio of 5 stocks with an equal investment in each one. The betas of the 5 stocks are as follows: .75, -1.2, .90, 1.3, 1.5. The risk free return is 4% and the market return is 9%. (Not a multiple choice problem)

A. Compute the beta of the portfolio

B. Compute the required return of the portfolio

Problem 2:

You are given the following probability distribution for a stock: (Not a multiple choice problem)

Pr. Outcome.5
.4 10%
.1 -6%
.5 12%

A. Compute the expected return of this stock

B. Compute the standard deviation

Problem 3:

You are given the following probability distribution for a stock: (Not a multiple choice problem)

Pr. Outcome.6
.4 -4%
.6 12%

A. Compute the expected return

B. Compute the standard deviation

C. Presuming the stock returns are normally distributed, what do these results indicate?

Problem 4:

A stock has a beta of 0.8. The market return is 14% and the risk free return is 3%. Compute the required return for this stock.

bronzeraven567

Question 1.

Consider a portfolio that has equal amounts of $10 invested in two assets. Suppose returns on the two assets are jointly normally distributed. The annual expected returns and variance or return on the first asset are given by

m₁ = 0.10 s₁^2 = 0.04

And those on the second asset are given by

m₂ = 0.05 s₂^2 = 0.03

Consider three cases:

The correlation between the return is r = 0

The correlation between the return is r = +0.50

The correlation between the return is r = -0.50

For each case, identify the 99% Value-at-Risk of the portfolio. Explain the pattern of dependence of VaR on the correlation.

Question 2.

You are given a portfolio of three assets whose returns are jointly normally distributed with the following mean vector and covariance matrix:

0.20 0.08 0.02 0.02

0.10 0.02 0.06 0.03

0.15 0.02 0.03 0.07

a. Compute the 95% VaR for the portfolio if we invest $1 in the first asset. $2 in the second asset, and $3 in the third asset.

b. How much does each assetâs holding contribute to the overall VaR risk?

Question 3.

You are given a portfolio of two assets whose returns are jointly distributed with the following vector and covariance matrix.

0.20 0.08 0.04

0.10 0.04 0.06

a. Compute the 95% VaR of the portfolio if $1 is invested in the first asset and $1 is invested in the second.

b. Compute the risk-contribution of each asset to the VaR.

c. Is the current portfolio weighting optimal? If not, suggest a better one.

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FNCE10002 Lecture 8: Week 8 Lecture 8

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