TOC
1. Mean-Variance Frontier
1.1 Mean-Variance Dominance
Mean-Variance Criterion
Choose the asset has both higher mean return and a lower variance. For below example, asset 1 is better than asset 2.
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Mean-Variance Analysis Derivation
Approximation:
where , thus
therefore, the mean and variance are critical to the determination of the expected utility.
As for the conditions for , one kind of assumption is that the Quadratic utility function is used, where and . However, there is a problem using Quadractic utility function: the absolute risk aversion is increasing which is not reasonable. Except for assuming the utility function is Quadratic utility function, assuming returns follow a normal distribution, higher orders of which can be expressed in terms of mean and variance, can also make mean-variance analysis valid. In fact, Elliptical distribution (general multivariate normal distribution) assumptions are also valid.
Minimum Variance Frontier
Mean-variance analysis:
- Given an expected return, to find a portfolio with lowest variance
- Given a variance, to find a portfolio with highest expected return
Minimum variance frontier:
- The combination of assets with minimum variance for all arbitrary levels of expected returns
1.2 Two-Assets Case
Assume there are two risky assets 1 & 2 and . The portfolio with weight in asset 1 has return: . Thus the mean of the return is and the variance of the return is .
Case 1:
the solution is
Substitute into expected return equation, we obtain
Case 2:
, thus or .
When :
When :
Case 3:
, arrange it, we get
Generally: Minimum Variance Portfolio
F.O.C. โ , thus
The effects of correlation is shown as below
1.3 Many-Assets Case
A portfolio consists of N risky assets with weight . The portfolioโs return is
The expected return of the portfolio is
The variance of the portfolio is
Frontier Portfolio
Given expected return , a portfolio , is a frontier portfolio, iff
Other constraints are possible: Elthical, Short sell, Ownership concerns, Home bias puzzle (prefer equities from hometown).
Deonte convariance matrix, portfolio weights and expected return as
A portfolio is a frontier portfolio, iff
An example when is shown as below
The solution to this problem can be characterized as the solution to the Lagrangian
First Order Condition (F.O.C.)
- (1)
- (2)
- (3)
Left multiply (1) by โ
- (4)
Left multiply (4) by โ
- (5)
Left multiply (4) by โ
- (6)
Denote , thus
(5) โ (7)
(6) โ (8)
Left multiply (7) by and (8) by
thus , therefore
Substitute into (8)
Substitute and into (4)
where
For any portfolio on the frontier, we can get
An example is shown as below
Separation Theorem (1)
The portfolio frontier can be described as combinations of any two frontier portfolios.
Proof
Let be any distinct frontier portfolios, let be an arbitrary frontier. Since , there exists a unique , such that
Consider a portfolio of with weight , the new portfolio is with weight
This is the portfolio frontier with return .
2. Efficient Frontier
2.1 Separation Theory
Efficient frontier is the locus of all nondominated portfolios in mean-standard deviation space as shown in below left.
As shown above right, after adding risk-free asset(s), the efficient frontier is a line tangent to the efficient frontier of risky assets.
Assumptions about investors:
- Same universe of securities
- Same time horizon
- Same input list
- All using identical Markowitz analysis
- Desire to hold identical risky Portfolio
Two-Fund Separation Theorem
There is a single fund F of risky assets such that any efficient portfolio can be constructed as a combination of the fund F and the risk-free asset.
Everyone will hold the same combination of risky assets:
- The same relative proportions of risky assets
- Vary the fraction of the riskless asset
2.2 Asset Allocation Puzzle
However, in reality, there are asset allocation puzzle (all investment advisors should recommend the same portfolio of risky assets to all clients) and time horizon puzzle (Popular advisors tend to recommend that an investorโs time horizon should influence the composition of his portfolio).
Explanations:
- Difference in Expectations
- Different Mutual fund will combine risky assets in different proportions
- However, the advisorโs subjective distribution of returns is being held constant across the three recommended portfolio
- Wrong Assumptions (including)
- A riskless asset exists
- Investors are indifferent between any two portfolios with identical means and variance
- Investors can hold long or short positions in all assets
- Investors operate over a one-period planning horizon
- All assets can be freely traded
- All Assets Can Not be Freely Traded
- Human capital. It is more similar to stock. Investor keep this ratio constant:
- The increase rate of the stocks is faster relative to bonds with higher wealth level.
- Problem: human capital is correlated with both stocks and bond returns.
- Nominal Debts. Investors keep this ratio constant:
- The increase rate of the stocks is faster relative to bonds with higher wealth level.
- Problem: Financial advice is same for different investors.
3. CAPM
3.1 Market Equilibrium
Equilibrium Model and CML
In equilibrium, markets have to clear. The market portfolio (representative portfolio) includes all risky assets and the proportion of each stock is equal to the market value of the stock. The price of the asset is exact such that all the supplied assets are demanded by investors.
The Capital Market Line is shown as below
3.2 CAPM
Derivation of CAPM
Consider a portfolio with of wealth invested in security and in market portfolio, thus
As varies, we trace a locus that passes through M and cannot cross the CML (frontier). Thus, the locus must be tangent to the CML at M. Since
The slope of the locus:
The slope of the locus at M = slope of CML means
thus
therefore
CAPM: in equilibrium, the expected return of any asset satisfy
where .
When plotted as a function of , this equation forms the Security Market Line (SML).
Note the difference between CML and SML:
- CML: , align axis is
- SML: , align axis is
In equilibrium, all assets lie on SML while only frontier lie on CML. Only the fraction of the total risk is remunerated by the market. It measures the systematic risk and the remaining fraction is diversified away.
Systematic risk is risk that cannot be diversified away by adding extra securities
Assume , and all variance and covariance are the same, then
as , the influence on the variance approaches 0.
3.3 Valuation
Denote be the payoff next period, and be the asset price today, then the return of asset is
thus
where
Substitute this into the pricing equation above,
thus
that is, price equal to the discount certainty equivalent, which is the expected payoff minus the risk-compensation term.
Performance Measures
Sharpe Ratio:
Assumption here: fund has a zero correlation with the existing portfolio.
Treynor Ratio:
Assumption here: risk is measured by the incremental risk given by beta.
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