– If purchase shares in Company, how might returns on this asset be measures?
– We know price of shares in Company A at point of purchase, i.e. time period t, is Pt .
– Assume price of shares in Company A one period into future is Pt+1 .
– Return on investment will then be proportionate increase in Pt , hence:
rt =Pt+1−Pt Pt
WHAT ABOUT CONTINUOUS SERIES?
– However, this calculation implies share prices move in discrete manner.
– Recall from Lecture 1 that if invest £P at an annual interest rate of i, where interest compounded annually, then future value of investment would be:
F =Pe^it – After 1-year we would have:
– Therefore can generalise for any time period, where r denotes rate of return over period, as:
– Therefore, in terms of share price, we have: Pt+1 =Pter
– Hence rate of return can be derived as
ln P = ln P + ln er = ln P + r
( t+1) ( t ) ( ) ( t )
– Solving for r, we therefore obtain the log-return, i.e.:
r =ln P −ln P
( t+1) ( t )
– From now on, when we refer to returns, we are generally talking about log-returns.
RISK VS. RETURN
– Can capture essential characteristics of these returns via their mean and variance, or standard deviation.
– The standard deviation of returns is used as a measure of the risk of the asset.
– Investors seek to maximise their returns on investments, however, will have to trade-off returns against risk.
– This means that asset that, on average, generates high returns will tend to do so with greater variability.
– Hence, high average return assets will be high risk assets and exhibit a large standard deviation, denoted σ.
HOW DO YOU MEASURE THESE?
– Simplest approach for calculating returns and risk is to use historical data on returns.
– Then use sample statistics of these historical returns as estimates of expected return and risk of asset.
– However, problems with using past data:
– How reliable are past returns as a guide to future returns? – How far back do you have to go?
– What about the impact of past shocks?
THE CONCEPT OF AN INVESTMENT PORTFOLIO
– Assume that an investor has two investments available to him, i.e. to hold shares in Company A and to hold shares in Company B.
– Let rA and rB denote the returns on Company A’s and Company B’s shares, respectively, where, in both cases, these returns are random variables.
– Hence, rA and rB will both have probability distributions, with expected values of E(rA) and E(rB), respectively.
– Assume that standard deviations of these distribution are σA and σB, respectively.
– Assume further that σB > σA, hence, E(rA) > E(rB).
THE INVESTMENT PORTFOLIO
– If investor is risk-averse, then Company A’s shares will be more attractive.
– If investor is risk-lover, then Company B’s shares will be more attractive.
– However, an alternative strategy would be to combine shares in some way, instead of just buying shares in one of the companies.
– This process of combining them is known as constructing an investment portfolio.
– Assume that investor has investment fund and forms an investment portfolio by investing a proportion, denoted α, in Company A’s shares.
– This proportion is known as the portfolio weight for A.
– Investor will then invest the remainder of investment
fund, i.e. 1 – α, in Company B’s shares.
– Now need to determine what the return this portfolio is?
– Return on portfolio will be a weighted average of returns on the individual shares, i.e.:
Er =Eαr+1−αr=αEr +1−αEr (P)A()B(A)()(B)
PORTFOLIO RISK 1
- – What about the risk on the portfolio?
- – Risk on portfolio will be reflected in the variance of the
returns on the portfolio, i.e. Var (rP).
– Important to note that variance of portfolio returns is not a simple weighted average of variances of the returns on individual assets in portfolio.
– In general, portfolio variance is less than the weighted average variances of individual asset returns, thereby indicating that you have effectively reduced risk.
– This is know as portfolio diversification, where by constructing a portfolio, you have reduced overall risk.
PORTFOLIO RISK 2
– The variance of the returns on the portfolio will be:
Varr =Varαr+1−αr (P)A()B
– It can be shown that: Var r =Var αr + 1−α r ( P ) A ( ) B
=α2σ2+1−α σ2+2α 1−α Cov r ;r ABAB ( ) ( ) ( )
– Or, alternatively:
Var r =α2σ2+1−α σ2+2α 1−α ρ σ σ (P) A( )B( )A;BAB
– Assume that an investor is considering constructing a portfolio of two assets, i.e. Asset A and Asset B, where:
rA =0.05;σA =0.04;rB =0.1andσB =0.12
– The portfolio weight for asset A, i.e. α, is assumed to range between 0 and 1, i.e. between 0% and 100%, in increments of 0.05, or 5%.
– Thus, substituting 0.05 for E(rA) and 0.1 for E(rB), and the various assumed values for α, in the portfolio returns equation, will produce various returns on the portfolio.
– Portfolio variance formula is then used to derive corresponding values of portfolio sigma.
– Risk vs. return (assuming ρA;B = 1): Figure 5.3, Correlation=1
CORRELATION VS. PORTFOLIO RISK
– Why does a combination of two perfectly negatively correlated assets produce a riskless portfolio?
– Essentially because any change in returns of one asset is entirely offset by a change in the other.
– In practice, perfectly negatively correlated assets are unlikely to occur, so not all risk can be diversified away.
– However, any negative correlation, and even zero correlation, will allow for some risk diversification.
– Given the possibility of short-selling, even positive correlations can allow for risk diversification.
THE CAPITAL MARKET LINE
THE EFFICIENT FRONTIER
– How do we identify the optimum portfolio?
– Firstly, can reject all portfolios below minimum variance point, denoted MV, on the combination line, as these are inefficient (as can still reduce variance and increase returns further).
– Efficient frontier therefore defined as segment of combination line that excludes inefficient portfolios.
– Optimum portfolio is therefore that portfolio on efficient frontier with preferred risk-return properties.
– This optimum point will essentially reflect an investor’s attitude to risk vs. return.
THE CAPITAL MARKET LINE 1
– However, if we assume that investor can borrow or lend at risk-free rate, then portfolio can be indentified that will form part of investor’s final portfolio.
– This borrowing / lending occurs as follows:
– Lend at risk-free rate by buying government bonds; and
– Borrow at risk-free rate by short-selling government bonds.
– Capital market line (CML) is defined as the line drawn from risk-free interest rate on the y-axis tangential to the efficient frontier.
– Point of tangency between CML and efficient frontier is known as market portfolio, denoted M.
THE CAPITAL MARKET LINE 2
– The CML allows us to either reduce risk beyond MV point or increase returns beyond what would be optimum on efficient frontier:
– If lend at risk-free, i.e. buy government bonds, can shift down CML to risk levels below MV point; while
– If borrow at risk-free rate, i.e. short-sell government bonds, can shift up CML to returns above the efficient frontier.
– Note that when asset are perfectly positively correlated, with possibility of short-selling, risk can be entirely diversified away, however, occurs at rate of return that is less than risk- free rate.
– CML is therefore effective way to reduce risk of positively correlated assets.
– Assume have constructed a portfolio of two assets with following efficient frontier and rF = 3%:
Figure 5.8 Identification of Market Portfolio with the Capital Market Line
– M is market portfolio with rM = 6.25% and σM = 3%.
– What if σM is too high for investor:
– Can’t move around efficient frontier as would increase risk; but
– If σM = 2% is required, this can be achieved by investing 33% in risk-free asset and 67% in market portfolio.
– What if σM is too low for investor:
- – Can move round efficient frontier, and increase risk; but
- – BettersolutionismoveupCMLbyborrowingatrisk-freerate;
– If σM = 6% is required, this can be achieved by giving weight of – 100% to risk-free asset, and therefore a weight of 200% to market portfolio; where
– Results in rM = 9.5%, as opposed to rM = 8%, otherwise.
WHAT HAVE WE LEARNT?
2. We have highlighted the importance of constructing an investment portfolio in the market.
3. We have looked at how to implement the Efficiency frontier for a portfolio.
4. We have looked at how to construct the Capital Market Line for a portfolio.
NEXT WEEK IS GIS!!!
Guided Independent Study Week
Guided Independent “Party” Week Readings to Be Done and May Be Examined
TASKS FOR YOU FOR WEEK 7
– Please ensure that you read:
– Lecture notes for Lecture 6;
– Beninga (Chapters 10 & 11)
– Haugen (Chapters 6, 8 and 9)
– Adams,etal.(Chapter12,pp.245-269,andChapter13) – Elton, et al. (Chapters 4 to 6)
– Jackson & Staunton (Chapter 7)
– For the seminar, please complete the exercises for Seminar 6.
Lecturer by Stefan Van Dellen at the University of Westminster <<<