程序代写 Lecture 7: Modern Portfolio Theory

Lecture 7: Modern Portfolio Theory
Economics of Finance
School of Economics, UNSW

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We have studied competitive market theories
• An utility function has been specified on the space of consumptions
• Agents make consumption choice to maximize expected utility
• Asset price is the ratio between marginal utilities (values of securities/value of money)
• Trading improves Pareto efficiency
• Financial market plays important social roles:
• consumption smoothing
• risk sharing

• Until now a general framework which captures risk and relevant trade-offs seems absent
• Consider small number of discrete outcomes (Good, Bad) and small number of securities
• In reality there are many securities and their returns are better approximated by continuous variable
• We will discuss situation which involves many possible financial instruments with a more general (continuous) measure of risks.

Measure of risk
Standard deviation is a measure of risk
If investors are risk averse, they would prefer:
• higher expected returns for a given level of standard deviation
• lower standard deviations for a given level of expected return
• Portfolios that provide the maximum expected return for a given standard deviation and the minimum standard deviation for a given expected return are called efficient portfolios.
• All others are inefficient.

First-order Stochastic Dominance
If FA(x) ≤ FB(x), ∀x and FA(x) < FB(x) for at least for some x, random variable A first-order stochastically dominates B. Expected utility: EU(A) ≥ EU(B) for any preferences. First-order Stochastic Dominance: examples • Y has an expected return of 10 and a standard dev. of 15 • X has an expected return of 14 and a standard dev. of 15 • X first-order stochastically dominates Y First-order Stochastic Dominance: examples • Y has an expected return of 10 and a standard dev. of 15 • X has an expected return of 14 and a standard dev. of 15 • X first-order stochastically dominates Y Variance does not have to be the same though: Second-order Stochastic Dominance If 􏰝 x [FA(t) − FB(t)]dt ≥ 0, ∀x and strict inequality at least −∞ for some x, B second-order stochastically dominates A. • When they have the same mean, but A has a higher var than B (A is a mean-preserving spread of B). • B second order stochastically dominates A Second-order Stochastic Dominance If 􏰝 x [FA(t) − FB(t)]dt ≥ 0, ∀x and strict inequality at least −∞ for some x, B second-order stochastically dominates A. • When they have the same mean, but A has a higher var than B (A is a mean-preserving spread of B). • B second order stochastically dominates A Expected utility: EU(B) ≥ EU(A) for risk-averse preferences. Expected utility The mean-variance expected utility takes the form: v s2 Eu = e − t = e − t , • e is the expected return • s is the standard deviation of the expected return, • t = 2/c is the investor’s risk tolerance and c is risk aversion. • t or c can be time-varying and wealth-dependent, but for simplicity we assume they are constant Mean-variance expected utility can be exactly derived from several basic utilities, e.g. from negative exponential (constant absolute risk aversion, CARA) utility u = 1 − e−cr and assuming normality of r ∼ N(e,s2). Expected utility The mean-variance expected utility takes the form: v s2 Eu = e − t = e − t , • e is the expected return • s is the standard deviation of the expected return, • t = 2/c is the investor’s risk tolerance and c is risk aversion. • t or c can be time-varying and wealth-dependent, but for simplicity we assume they are constant Mean-variance expected utility can be exactly derived from several basic utilities, e.g. from negative exponential (constant absolute risk aversion, CARA) utility u = 1 − e−cr and assuming normality of r ∼ N(e,s2). A sketch of proof (optional): from the properties of log-normal distribution E (e−cr) = e−ce+cs2/2. Using monotonic transformation, − ln(1 − Eu)/c, yields the result. Indifference curves Now fix a given expected utility level Eu ≡ Ul e = Ul + s2 This is an upward-sloping, convex curve in terms of s. To maintain a fixed utility level, a higher return is required for a higher risk. Certainty Equivalent e = Ul + s2 t • whens=0,e=Ul,i.e.,afixedamountofreturnwhichis equally satisfying • this is certainty equivalent - certain return which gives the same utility level as the expected utility of risky return Re-arrange, Risk tolerance and premium e−Ul = s2 t is the risk premium required by the individual investor. • given a fixed s, a greater t means a less risk premium; • the more tolerant agent is, the less risk premium she/he would require for taking the risk. Optimal Portfolio Choice • Investor will maximize the utility (blue indifference curves) • Given the e − s opportunities (red) available on the market – efficient frontier of a portfolio Why is efficient frontier concave? Some history A 15-pages PhD thesis by Markowitz in 1954 • titled “Portfolio selection” • a ground-breaking, insightful contribution This inspired a sequence of research • , among others, formulated theories as the CAPM today • Sharpe (1964): “Capital asset prices – a theory of market equilibrium under conditions of risk” • Markowitz, Sharpe and Miller shared 1990 Nobel prize Market opportunities Set of opportunities is presented by portfolio of assets. Example with two assets: • Let R1 and R2 are random returns of two assets • Expected values E (R1) = e1 and E (R2) = e2, • Variances Var(R1) = v1 and Var(R2) = v2 • Correlation Corr(R1, R2) = ρ12 Let x denotes the proportion of asset 1 in the portfolio, then: Market opportunities Set of opportunities is presented by portfolio of assets. Example with two assets: • Let R1 and R2 are random returns of two assets • Expected values E (R1) = e1 and E (R2) = e2, • Variances Var(R1) = v1 and Var(R2) = v2 • Correlation Corr(R1, R2) = ρ12 Let x denotes the proportion of asset 1 in the portfolio, then: • E(xR1 +(1−x)R2)=xe1 +(1−x)e2 • Var(xR1+(1−x)R2)=x2v1+(1−x)2v2+2x(1−x)√v1v2ρ12 • StDev(xR1 + (1 − x)R2) = 􏰞Var(xR1 + (1 − x)R2) Solve for x in Var or StDev equation and substitute to get to e − s opportunities space. Deriving Frontier e=xe1 +(1−x)e2 √ v=x2v1+(1−x)2v2+2x(1−x) v1v2ρ12 • Solve out x: x=e−e2=e· 1 − e2 e1 − e2 e1 − e2 e1 − e2 • on the other hand solve out v as a function of x: v(x)=v1x2 +􏰀1+x2 −2x􏰁v2 +􏰀2x−2x2􏰁√v1v2ρ12 = (v1 + v2 − 2√v1v2ρ12) x2 + 2x (√v1v2ρ12 − v2) + v2; Deriving Frontier e=xe1 +(1−x)e2 √ v=x2v1+(1−x)2v2+2x(1−x) v1v2ρ12 • Solve out x: x=e−e2=e· 1 − e2 e1 − e2 e1 − e2 e1 − e2 • on the other hand solve out v as a function of x: v(x)=v1x2 +􏰀1+x2 −2x􏰁v2 +􏰀2x−2x2􏰁√v1v2ρ12 = (v1 + v2 − 2√v1v2ρ12) x2 + 2x (√v1v2ρ12 − v2) + v2; • substitute x in v(x), we have a functional form of v(e) • notice that (v1 +v2 −2√v1v2ρ12)=Var(R1 −R2)>0;

Deriving Frontier
e=xe1 +(1−x)e2 √ v=x2v1+(1−x)2v2+2x(1−x) v1v2ρ12
• Solve out x:
x=e−e2=e· 1 − e2
e1 − e2 e1 − e2 e1 − e2
• on the other hand solve out v as a function of x: v(x)=v1x2 +􏰀1+x2 −2x􏰁v2 +􏰀2x−2×2􏰁√v1v2ρ12
= (v1 + v2 − 2√v1v2ρ12) x2 + 2x (√v1v2ρ12 − v2) + v2; • substitute x in v(x), we have a functional form of v(e)
• notice that
(v1 +v2 −2√v1v2ρ12)=Var(R1 −R2)>0;
• v is a parabola with an upward opening in v − e space

Deriving Frontier (cont)
v(e)=ae2 +be+c,
where a > 0, b, c are constants depending on
e1 = 0.45,e2 = 0.5,v1 = 0.01,v2 = 0.0225,ρ12 = −0.5.

Mirror it to get to e−v space e (v) is the inverse of quadratic function

“Squeeze” it further to get to e − s space
e (s) = e ( v) nice concave function

Efficient frontier
A higher risk should be rewarded with a higher expected return, if it not the case that is not an efficient investment.

Efficient Frontier: many securities
Create “portfolios of portfolios” and select the most efficient combinations, higher expected return for the same variance (first-order stochastic dominance), lower variance for the same expected return (second-order stochastic dominance).

Example: efficient frontier
Based on historical daily data (two years 2015–2017) of four stocks: Macdonald’s, Disney, Amazon, Microsoft.
Black triangle – example of portfolio with equal shares
Software: module FinQuant for Python.

Risky asset and risk-free asset
When R2 is risk free, e − s frontier is a straight line • E(R1)=e1 andE(R2)=ef 1 is acceptable, S > 2 is very good

Sharpe ratio of the Market portfolio
SM = eM −ef is the slope to the tangent line and therefore the sM
best Sharpe ratio available on the market

Investor’s problem
Investor’s problem: optimal wealth Y0 allocation between the market portfolio and risk-free asset (under CARA). Y0 is often normalised to 1.
maxEu=xeM +(Y0−x)ef −1x2vM, (1) xt
where x is amount of wealth invested in risky asset. Take FOC, we get:
∗ t eM − ef t SM x=2v =2√v
More tolerant investors choose more risky asset, but all investors get the same best Sharpe ratio

Optimal investment
All investors invest in the combination of the risky-free asset and market portfolio. The share of the market portfolio and risk-free asset is determined by their risk tolerance (risk-aversion).

Capital allocation line
A maximum Sharpe ratio is obtained for any portfolio on the straight line from rf tangent with the efficient frontier at M. This line is called capital allocation line (CAL).

Separation theorem
Separation (or two-fund) theorem: an optimal investor’s risky portfolio is identified separately from their risk preferences; investors hold only a combination of two assets (funds): the market portfolio and the risk-free asset.

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