LEARNING OBJECTIVES Calculating two-asset portfolio expected returns and standard deviations Estimating measures of the extent of interaction – covariance.

LEARNING OBJECTIVES Calculating two-asset portfolio expected returns and standard deviations Estimating measures of the extent of interaction – covariance and correlation coefficients Being able to describe dominance, identify efficient portfolios and then apply utility theory to obtain optimum portfolios Recognise the properties of the multi-asset portfolio set and demonstrate the theory behind the capital market line

Holding period returns
One year: Where: s = semi-annual rate R = annual rate For three year holding period

EXAMPLE: Initial share price = £1.00 Share price three years later = £1.20 Dividends: year 1 = 6p, year 2 = 7p, year 3 = 8p

EXPECTED RETURNS AND STANDARD DEVIATION FOR SHARES
Ace plc A share costs 100p to purchase now and the estimates of returns for the next year are as follows: Event Estimated Estimated Retur n Pr obability selling price, P dividend, D R 1 1 i Economic boom 114p 6p +20% 0.2 Nor mal gr owth 100p 5p +5% 0.6 Recession 86p 4p –10% 0.2 1.0

THE EXPECTED RETURN n  R = R p where R = expected r etur n R = return
i = 1  R = R p i i where R = expected r etur n R = return if event i occurs i p i = pr obability of event i occur ring n = number of events Expected return, Ace plc Event Pr obability of Retur n event pi Ri Ri  pi Boom 0.2 +20 4 Gr owth 0.6 +5 3 Recession 0.2 –10 –2 Expected returns 5 or 5%

STANDARD DEVIATION  =  ( p R – R ) =1 n 2 i i i
Standard deviation, Ace plc Deviation squared probability  Probability Return Expected return Deviation pi Ri Ri Ri – Ri (Ri – R i ) 2 pi 0.2 20% 5% 15 45 0.6 5% 5% 0.2 –10% 5% –15 45 Variance  2 90 Standard deviation  9.49%

Expected return on Bravo
Returns for a share in Bravo plc Event Return Probability Ri pi Boom –15% 0.2 Growth +5% 0.6 Recession +25% 0.2 1.0 Expected return on Bravo (–15  0.2) + (5  0.6) + (25  0.2) = 5 per cent Standard deviation, Bravo plc Deviation squared probability  Probability Return Expected return Deviation pi Ri Ri Ri – Ri (Ri – Ri ) 2 pi 0.2 –15% 5% –20 80 0.6 +5% 5% 0.2 +25% 5% +20 80 1.0 160 Variance  2 Standard deviation  12.65%

COMBINATIONS OF INVESTMENTS
Hypothetical pattern of return for Ace plc 20 – Return % + 5 1 2 3 4 5 6 7 8 Time (years) –10 Hypothetical pattern of returns for Bravo plc 25 – Return % + 5 1 2 3 4 5 6 7 8 Time (years) –15

Returns over one year from placing £571 in Ace and £429 in Bravo
Event Returns Returns Overall returns Percentage Ace Bravo on £1,000 r eturns Boom 571(1.2) = 685 429 – 429(0.15) = 365 1,050 5% Growth 571(1.05) = 600 429(1.05) = 450 1,050 5% Recession 571 – 571(0.1) = 514 429 (1.25) = 536 1,050 5% Hypothetical pattern of returns for Ace, Bravo and the two-asset portfolio Bravo 25 20 – Return % + Portfolio 5 1 2 3 4 5 6 7 8 Time (years) –10 –15 Ace

PERFECT NEGATIVE CORRELATION PERFECT POSITIVE CORRELATION
Annual returns on Ace and Clara Event Pr obability Retur ns Retur ns i p on Ace on Clara i % % Boom 0.2 +20 +50 Growth 0.6 +5 +15 Recession 0.2 –10 –20 Returns over a one-year period from placing £500 in Ace and £500 in Clara Event Returns Returns Overall Percentage i Ace Clara return on r eturn £1,000 Boom 600 750 1,350 35% Growth 525 575 1,100 10% Recession 450 400 850 –15%

50 Clara Portfolio 20 + 5 Ace – 1 2 3 4 5 6 7 8 T ime (years) –10 –15
Hypothetical patterns of returns for Ace and Clara 50 Clara Portfolio 20 + 5 Ace – 1 2 3 4 5 6 7 8 T ime (years) –10 –15 –20

INDEPENDENT INVESTMENTS
Expected returns for shares in X and shares in Y Expected return for shares in X Expected return for shares in Y Return  Probability – 25 0.5 = –12.5 35 0.5 = 5.0% –25  0.5 = –12.5 35 0.5 = 5.0% Return  Probability Standard deviations for X or Y as single investments Deviations squared  probability Return Probability Expected return Deviations Ri pi Ri Ri – R (Ri – R)2 pi –25% 0.5 5% –30 450 35% 0.5 5% 30 450 Variance  900 Standard deviation  30%

Exhibit A mixed portfolio: 50 per cent of the fund invested in X and 50 per cent in Y, expected return Possible Joint Joint Retur n  outcome returns probability pr obability combinations Both firms do badly –25 0.5  0.5 = 0.25 –25  0.25 = –6.25 X does badly Y does well 5 0.5  0.5 = 0.25 5  0.25 = 1.25 X does well Y does badly 5 0.5  0.5 = 0.25 5  0.25 = 1.25 Both firms do well 35 0.5  0.5 = 0.25 35  0.25 = 8.75 1.00 Expected return 5.00% Standard deviation, mixed portfolio Deviations squared  probability Return Probability Expected return Deviations Ri pi R Ri – R (Ri – R)2 pi –25 0.25 5 –30 225 5 0.50 5 35 0.25 5 30 225 Variance  450 Standard deviation  21.21%

A CORRELATION SCALE So long as the returns of constituent assets of a portfolio are not perfectly positively correlated, diversification can reduce risk. The degree of risk reduction depends on: the extent of statistical interdependence between the returns of the different investments: the more negative the better; and the number of securities over which to spread the risk: the greater the number, the lower the risk. –1 +1 Perfect Independent Perfect negative positive correlation correlation Correlation scale

THE EFFECTS OF DIVERSIFICATION WHEN SECURITY RETURNS ARE NOT PERFECTLY CORRELATED
Returns on shares A and B for alternative economic states Event i Pr obability Retur n on A Retur n on B State of the economy pi RA RB Boom 0.3 20% 3% Gr owth 0.4 10% 35% Recession 0.3 0% –5% Company A: Expected return Pr obability Return  RA pi pi RA 0.3 20 6 0.4 10 4 0.3 10% Company A: Standard deviation Deviation squared  probability Expected Probability Return r eturn Deviation pi RA RA (RA – RA) (RA – RA)2 pi 0.3 20 10 10 30 0.4 10 10 0.3 10 –10 30 Variance  60 Standard deviation  7.75%

Company B: Expected return
Probability Return pi RB RB pi 0.3 3 0.9 0.4 35 14.0 0.3 –5 –1.5 13.4% Company B: Standard deviation Deviation squared  probability Probability Return Expected Deviation return pi RB RB (RB – RB) (RB – RB)2 pi 0.3 3 13.4 10.4 32.45 0.4 35 13.4 21.6 186.62 0.3 –5 13.4 –18.4 101.57 Variance  320.64  = 17.91% Standard deviation Summary table: Expected returns and standard deviations for Companies A and B Expected return Standard deviation Company A Company B 10% 13.4% 7.75% 17.91%

A general rule in portfolio theory:
Portfolio returns are a weighted average of the expected returns on the individual investment… BUT… Portfolio standard deviation is less than the weighted average risk of the individual investments, except for perfectly positively correlated investments. Exhibit: Return and standard deviation for shares in firms A and B 20 15 B P Expected return R % 10 A Q 5 5 10 15 20 Standard deviation %

PORTFOLIO EXPECTED RETURNS AND STANDARD DEVIATION
90 per cent of the portfolio funds are placed in A 10 per cent are placed in B Expected returns, two-asset portfolio Proportion of funds in A = a = 0.90 Proportion of funds in B = 1 – a = 0.10 Rp = aRA + (1 – a)RB Rp = 0.90   13.4 = 10.34%  p = a2  2A + (1 – a)2  2B + 2a (1 – a) cov (RA, RB) wher e p = portfolio standard deviation A = variance of investment A B= variance of investment B cov (RA, RB) = covariance of A and B

COVARIANCE The covariance formula is:
cov (RA, RB) = {(RA – RA)(RB – RB)pi}  n i = 1 Exhibit Covariance Event and Expected Deviation of A  pr obability of Returns r etur ns Deviations deviation of B  pr obability event pi R A R R R R A – R R B – R (R – R )(R – R )pi B A B A B A A B B Boom 0.3 20 3 10 13.4 10 –10.4 10  –10.4  0.3 = –31.2 Gr owth 0.4 10 35 10 13.4 21.6  21.6  0.4 = Recession 0.3 –5 10 13.4 –10 –18.4 –10  –18.4  0.3 = 55.2 Covariance of A and B, cov ( R , R ) = +24 A B

STANDARD DEVIATION p = a2  2A + (1 – a)2  2B + 2a (1 – a) cov (RA, RB) p =    0.90  0.10  24 p = p = % Exhibit Summary table: expected return and standard deviation Expected return (%) Standard deviation (%) All invested in Company A 10 7.75 All invested in Company B 13.4 17.91 Invested in a portfolio (90% in A, 10% in B) 10.34 7.49 Exhibit Expected returns and standard deviation for A and B and a 90:10 portfolio of A and B Standard deviation % Expected return R % Portfolio (A=90%, B=10%) 20 15 10 5 B A

CORRELATION COEFFICIENT
cov (RARB) RAB = RAB = = If RAB = then cov (RARB) = p = a22A + (1 – a)2 2B + 2a (1 – a) AB 24 7.75  17.91 cov (RARB) RABAB AB RABAB

Exhibit 7.30 Perfect positive correlation
Returns on G Returns on F Exhibit Perfect negative correlation Returns on G Returns on F Exhibit Zero correlation coefficient Returns on G Returns on F

DOMINANCE AND THE EFFICIENT FRONTIER
Exhibit Returns on shares in Augustus and Brown Event (weather for season) Probability of event Returns on Augustus Returns on Brown pi RA RB Warm 0.2 20% –10% Average 0.6 15% 22% Wet 0.2 10% 44% Expected return 15% 20% Exhibit Standard deviation for Augustus and Brown Probability Returns Returns pi on Augustus (RA – RA)2 pi on Brown (RB – RB)2 pi RA 0.2 20 5 –10 180.0 0.6 15 22 2.4 0.2 10 5 44 115.2 Variance,  10 Variance, B 297.6 Standard deviation,  3.162 Standard deviation, B 17.25

cov (RA, RB) RAB = AB –54 RAB = = –0.99 3.162  17.25 
Exhibit Covariance Deviation of A  Expected deviation of B  probability Probability Returns returns Deviations pi RA RB RA RB RA – RA RB – RB (RA – RA)( RB – RB)pi 0.2 20 –10 15 20 5 –30 5  –30  0.2 = –30 0.6 15 22 15 20 2  2  0.6 = 0.2 10 44 15 20 –5 24 –5  24  0.2 = –24 Covariance (RA RB) –54 cov (RA, RB) RAB = AB –54 RAB = = –0.99 3.162  17.25 

= 3.16 = 1.16 = 0.39 = 1.01 = 7.06 =17.25 =12.2 Standard deviation 0.852    0.85  0.15  –54 0.252    0.25  0.75  –54 0.92    0.9  0.1  –54 0.82    0.8  0.2  –54 0.52    0.5  0.5  –54 Exhibit Risk-return correlations: two-asset portfolios for Augustus and Brown Expected return (%) 15 15.75 15.5 16.0 17.5 18.75 20 Brown weighting (%) 10 15 20 50 75 100 Augustus weighting (%) 100 90 85 80 50 25 Portfolio A J K L M N B

Exhibit 7.37 Risk-return profile for alternative portfolios of
Augustus and Brown 21 B 20 19 N 18 M Efficiency frontier Return Rp % 17 L 16 K J 15 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 p Standard deviation

FINDING THE MINIMUM STANDARD DEVIATION FOR COMBINATIONS OF TWO SECURITIES
If a fund is to be split between two securities, A and B, and a is the fraction to be allocated to A, then the value for a which results in the lowest standard deviation is given by:

INDIFFERENCE CURVES X 14 Z Indifference curve I 105 Return % 10 W Y 16
Exhibit Indifference curve for Mr Chisholm X 14 Z Indifference curve I 105 Return % 10 W Y 16 20 Standard deviation %

North-west I 129 I 121 I 110 I 107 S I 105 Return % T 14 Z 10 W
Exhibit A map of indifference curves North-west I 129 I 121 I 110 I 107 S I 105 Return % T 14 Z 10 W South-west 16 20 Standard deviation %

I105 I101 Return % M I101 I105 Standard deviation %
Exhibit Intersecting indifference curves I105 I101 M Return % I101 I105 Standard deviation %

(a) Moderate risk aversion
Exhibit Varying degrees of risk aversion as represented by indifference curves Return % Return % Return % Standard deviation % Standard deviation % Standard deviation % (a) Moderate risk aversion (b) Low risk aversion (c) High risk aversion

CHOOSING THE OPTIMAL PORTFOLIO
Exhibit Optimal combination of Augustus and Brown I 21 3 I 2 B 20 I 1 19 N Efficiency frontier Return % 18 M 17 L 16 K J 15 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18  Standard deviation % p

THE BOUNDARIES OF DIVERSIFICATION
Exhibit The boundaries of diversification D 22 21 20 RCD = –1 19 H RCD = +1 18 Return % RCD = 0 17 E G 16 F 15 C 1 2 3 4 5 Return % 6 7 8 9 10  Standard deviation % p

EXTENSION TO A LARGE NUMBER OF SECURITIES
Exhibit A three-asset portfolio A 4 1 3 B Return 2 C Standard deviation

IL3 Efficiency frontier IL2 IL1 IH3 V IH2 IH1 Return U
Exhibit The opportunity set for multi-security portfolios and portfolio selection for a highly risk-averse person and for a slightly risk-averse person IL3 Efficiency frontier IL2 IL1 IH3 V IH2 IH1 Return U Inefficient region Inefficient region Standard deviation

THE CAPITAL MARKET LINE
Exhibit Combining risk-free and risky investments M Return F C B rf A Standard deviation

Return M Y X rf Standard deviation
Exhibit Indifference curves applied to combinations of the market portfolio and the risk-free asset M Return Y X rf Standard deviation

Capital market line N T S M Return G H rf Standard deviation
Exhibit The capital market line Capital market line N T S M Return G H rf Standard deviation

Problems with portfolio theory:
relies on past data to predict future risk and return involves complicated calculations indifference curve generation is difficult few investment managers use computer programs because of the nonsense results they frequently produce

LEARNING OBJECTIVES Calculating two-asset portfolio expected returns and standard deviations Estimating measures of the extent of interaction – covariance.

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LEARNING OBJECTIVES Calculating two-asset portfolio expected returns and standard deviations Estimating measures of the extent of interaction – covariance.

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Presentation on theme: "LEARNING OBJECTIVES Calculating two-asset portfolio expected returns and standard deviations Estimating measures of the extent of interaction – covariance."— Presentation transcript:

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