剑桥大学金融学教程session2

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Portfolio standard deviation
This diagram depicts the relation between the number of securities and portfolio’s standard deviation. What can we derive from the diagram?
Diversification
Example continued We add NewCo to existing portfolio
Stocks
σ
% of portfolio
Portfolio (A & B) 28.1 50%
NewCo
30 50%
Ave. Return 17.4% 19%
If the correlation coefficient = 0.3 Revised: Portfolio return = weighted average = 18.2%
No of securities
Measuring risk
Portfolio standard deviation
Unique risk
Market risk No of securities
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Measuring risk
Diversification: strategy designed to reduce risk by spreading the portfolio across many investments Unique risk: risk factors specific only to firms – diversifiable risk Market risk: economy wide sources of risk that affect the overall stock market – systematic risk
2
This lecture
Risk vs. return payoff Portfolio theory Risk diversification Capital Asset Pricing Model (CAPM)
Brief statistical review
Return (r) is measured by expected mean change in value (assuming no dividends)
E(Rp ) = Rf +θσ p
where θ is the slope of the CML
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Capital market line
Return
CML
E(Rp ) = Rf +θσ p
Market return (Rm)
Risk free (Rf)
Efficient Portfolio
Market risk (σm)
Judge Business School
Principles of Finance
2006/2007 Lecture 2 Dr Kim-Hwa Lim
1
Recap on Lecture 1
Asset price = PV of Future cash flow discounted at the risk-adjusted rate
Lending
Risk free (Rf)
Risk
Lending or outside the
borrowing at the efficient frontier
risk
free
rate
(Rf)
allows
us
to
exist
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What we have got so far..
Combining securities into a portfolio might reduce risk Provided there is no transaction & information processing costs, one could always include every tradable security in a portfolio If the risk free asset is available to everyone, then everyone will hold a combination of this risk free asset and the market portfolio
Portfolio risk
For a portfolio of two assets:
Expected portfolio return, E(Rp)
= (x1r1) + (x2r2)
Portfolio variance, Vp
= x12 σ12 + x22σ22 + 2(x1x2ρ12σ1σ2)
Vp= x12 σ12 + x22σ22 + 2(x1x2ρ12σ1σ2) ∂Vp/∂x1 = 2x1σ12 – 2(1–x1)σ22 + 2(1–2x1)ρ12σ1σ2
Set ∂Vp/∂x1 = 0 to minimise
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Diversification
Example Stock σ
A
28
B
42
% of portfolio 60% 40%
Ave. Return 15% 21%
If the correlation coefficient = 0.4 Original: Portfolio return = weighted average = 17.4%
Weighted average standard deviation = 33.6 Portfolio standard deviation = 28.1 NB: portfolio σ < weighted average σ
Portfolio risk
The variance of an equally weighted
portfolio is:
Vp
=
1 N
{Average
var iance}+
1

1 N
{Average
cov ariance }
So when N is sufficiently large, the first term disappears, leaving the portfolio variance equal to the average covariance between stocks.
Application of Concepts
Pension Funds Asset Allocation Given the ideas of diversification (Markowitz) and risk vs. return, how would pension funds allocate their assets? From UBS Pension Funds Indicators 2005
where
xa1ssaentds
xin2viessttheedpsruocphortthioant
of x1
stocki return standard deviation
ρst1o2cisk1caonrrdelsattoiocnk2coefficient between
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Diversification
15
Security market line
Given the CML, we now scale the market risk as the market beta:
Risk (σ)
Capital Asset Pricing Model (CAPM)
Suppose all investors believe in portfolio theory and have the same beliefs It can be shown that the expected return on each stock is related to its market risk The relation is the Security Market Line (SML) See Copeland/Weston Chapters 5 & 6 for derivations
Weighted average standard deviation = 31.8 Portfolio standard deviation = 23.43 NB: Higher return & Lower risk compared to original portfolio
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Measuring risk
The portfolio variance (Vp) is a function of the correlation coefficient (ρ12) Hence Vp is:
Smallest when ρ12 = -1 Largest when ρ12 = 1
Diversification
To minimise the portfolio variance, an investor will choose a value for x1 which is determined by partially differentiating the Vp with respect to x1.
The value of a $1 investment in 1926 (real value)
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Rates of return 1926-1997 Risk and return