A Brief Intro To Modern Portfolio Theory 223i1h

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A Brief Intro Theory to Modern Portfolio

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Michael Donadelli [email protected] 15/06/2011 15/12/2012 Finance Modern Portfolio Theory, Ecient Frontier Additional Info: Additional class notes on portfolio allocation Author

Theory of Finance:

Modern portfolio theory is a theory of investment which tries to maximize return and minimize risk by carefully choosing different assets. Although MPT is widely used in practice in the financial industry and several of its creators won a Nobel prize for the theory, in recent years the basic assumptions of MPT have been widely challenged by fields such as behavioral economics. It is a mathematical formulation of the concept of diversification in investing, with the aim of selecting a collection of investment assets that has collectively lower risk than any individual asset. This is possible, in theory, because different types of assets often change in value in opposite ways. For example, when the prices in the stock market fall, the prices in the bond market often increase, and vice versa. A collection of both types of assets can therefore have lower overall risk than either individually. More technically, Modern Portfolio Theory models an asset's return as a normally distributed random variable, defines risk as the standard deviation return, and models a portfolio as a weighted combination of assets so that the return of a portfolio is the weighted combination of the assets' returns. By combining different assets whose returns are not correlated Modern Portfolio Theory seeks to reduce the total variance of the portfolio. Modern Portfolio Theory also assumes that investors are rational and markets are efficient. This tool was first formulated by Henry Markowitz in 1952 (Markowitz, H.M. (March 1952). "Portfolio Selection". The Journal of Finance 7 (1): 77–91.). In fact a Markowitz Efficient Portfolio is one where no added diversification can lower the portfolio's risk for a given return expectation (alternately, no additional expected return can be gained without increasing the risk of the portfolio). The Markowitz Efficient Frontier is the set of all portfolios that will give the highest expected return for each given level of risk.

Formal Statement of the Model: Modern Portfolio Theory assumes that investors are risk averse, meaning that given two assets that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher returns must accept more risk. The exact trade-off will differ by investor based on individual risk aversion characteristics. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-return profile (i.e., if for that level of risk an alternative portfolio exists which has better expected returns).

Modern Portfolio Theory further assumes that the investor's risk / reward preference can be described via a quadratic utility function. The effect of this assumption is that only the expected return and the volatility (i.e., mean return and standard deviation) matter to the investor. The investor is indifferent to other characteristics of the distribution of returns, such as its skew (measures the level of asymmetry in the distribution) or kurtosis (measure of the thickness or so-called "fat tail"). Thus, investor take care of only the first and second moment of the distribution (mean and variance). Formally the model: • Portfolio return is the weighted combination of the constituent assets' returns • Portfolio volatility is a function of the correlation ρ of the component assets. In general: Expected return: E ( R p ) = ∑ wi E ( Ri ) , i

where Ri is return and wi is the weighting of component asset i. Portfolio variance:

σ p2 = ∑ wi2σ i2 + ∑∑ wi w jσ iσ j ρ i j , i

i

j

where i ≠ j . Alternatively the expression can be written as:

σ p2 = ∑∑ wi w jσ iσ j ρ i j , i

j

where ρ i j = 1 for i = j. Portfolio volatility:-

Suppose to have a portfolio composed by only two asset (asset A and asset B): Portfolio return: E ( R p ) = w A E ( R A ) + (1 − w A ) E ( R B ) = w A E ( R A ) + wB E ( R B ) Portfolio variance:

σ p2 = w A2 σ A2 + wB2 σ B2 + 2w A wBσ Aσ B ρ AB An investor can reduce portfolio risk simply by holding combinations of instruments which are not perfectly positively correlated (note that correlation coefficient is contained between -1 and 1 [-1<(r)<1]). In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification will allow for the same portfolio return with reduced risk.

Markowitz’s Efficient Portfolio Theory: Mathematical Model in Matrix Form Markowitz’s solution is obtained by solving a linear constrained optimization problem where we minimize the variance of the portfolio given an expected return of the portfolio. The problem, for n risky assets, can be formally represented and solved in the following way:

min w ' ∑ w w

subject to : u p = w' z 1 = w '1 where: w = (n ⋅ 1) vector of weights w ' = (1 ⋅ n) vector of weights ≡ transpose vector of weights

∑ = var − cov matrix of dim ension (n ⋅ n) w ' ∑ w ≡ total portfolio var ≡ total portfolio risk z = (n ⋅ 1) vector of returns 1 = (n ⋅ 1) vector of one For simplicity in what follows we will not use the “bar” over the variables. Note also that the form of w ' ∑ w is a quadratic one. Being a linear constrained optimization model we need to build up the Lagrangian Function. Since the goal of the optimization process is the one of finding a set of optimal weights, the above problem has to be solved with respect to w. Thus: L=

1

1 w′ ∑ w + λ (u p − w′z ) + γ (1 − w′1) 1 2

We put 1/2 at beginning of our lagrangian function to simplify the calculus.

and imposing the first order condition (F.O.C.): ∂L 1 =0 ⇒ 2 ∑ w − λz − γ 1 = 0 ⇒ ∑ w = λz + γ 1 ∂w 2 and multipling both side by ∑ −1 we get : (∑ −1 ) ∑ w = (λz + γ 1) ⋅ (∑ −1 ) ⇒ woptimal = λ ∑ −1 z + γ ∑ −1 1 Now we substitute our optimal solution in the constraints:

u p = w′z = (λ ∑ −1 z + γ ∑ −1 1)′ ⋅ z = λz ′ ∑ −1 z + γ 1′ ∑ −1 z 1 = (λ ∑ −1 z + γ ∑ −1 1)′ ⋅ 1 = λ1′ ∑ −1 z + γ 1′ ∑ −1 1 and we simplify by adopting the following definitions:

A = z ′ ∑ −1 z B = 1′ ∑ −1 z C = 1′ ∑ −1 1 where A, B and C are scalars (numbers). Hence:

u p = λA + γB 1 = λB + γC

λ and γ are the Lagrangian multipliers (also called shadow prices). We want now to find their explicit solution. In order to solve for λ and γ it is simpler to represent the above system of equations in the following way: u p   A B  λ   1  =  B C  γ       and using the linear algebra inverse matrix method (Ax = b x = A-1b):

u p   A B   1 B C    

 A B ⇒  B C 

−1

−1

λ  =   ⇒ ∆ matrix = B 2 − AC γ 

 C − B − B A   = 2 B − AC

 C ∆ − B ∆  u p  λ  ⇒   =    − B ∆ A ∆   1  γ 

⇒

λ= γ =

u pC − B ∆ A − upB ∆

⇒

λ= γ =

u pC − B B 2 − AC A − upB B 2 − AC

we find the above values for λ and γ. At this point it is important to stress again that the goal of the mathematical procedure is the one to find out a specific function where the variance of the portfolio depends on the expected return of the portfolio σ p2 = f (u p ). We multiply our optimal solution [ woptimal = λ ∑ −1 z + γ ∑ −1 1 ] on both sides for w′ ∑ .

(w′ ∑ ) ⋅ w = (λ ∑ −1 z + γ ∑ −1 1) ⋅ (w′ ∑ ) where : w′ ∑ w ≡ σ p2 ∑ −1 ∑ = 1 w′z = u p ⇒ 1st constr. w′1 = 1

⇒ 2 st constr.

and we end up with the following solution for σ 2p :

σ p2 = λu p − γ 1 and substituting for λ and γ:

(

)

⇒ w ′ ∑ w = λ ∑ −1 w ′ ∑ z + γ ∑ −1 w ′ ∑ 1

 u pC − B 

 A − upB 

 ⋅ u p +  2  ⋅ 1 σ p2 =  2  B − AC   B − AC  2 p

σ = σp =

u 2p C − u p B + A − u p B B 2 − AC u 2p C − 2u p B + A B 2 − AC

where the risk is a function of the expected return σ p2 = f (u p ) . Graphically it can be well represented by a Hyperbola.

Expected Return, up

Global Min Variance Portfolio This part is dominated because of for the same level of risk we can obtain higher return.

Risk, σ p

Each portfolio lies on the efficient frontier is an optimal portfolio. Each optimal portfolio is chosen as a trade-off between risk and expected return. Combinations along this line represent portfolios (explicitly excluding the risk-free alternative) for which there is lowest risk for a given level of return. Conversely, for a given amount of risk, the portfolio lying on the efficient frontier represents the combination offering the best possible return. Mathematically the Efficient Frontier is the intersection of the Set of Portfolios with Minimum Variance and the Set of Portfolios with Maximum Return. The efficient frontier is a parabola(hyperbola) when expected return is plotted against variance (standard deviation).

Note that points below the frontier are suboptimal. A rational investor will hold a portfolio only on the frontier. The value of the Global Minimum Variance Portfolio can be easily found by setting the first derivative of the portfolio variance function with respect to the expected return, equal to zero. Formally: 2 p

σ = ∂σ p2 ∂u p

u 2p C − u p B + A − u p B B 2 − AC

= 2u p C − 2 B = 0 ⇒ u p C = B

⇒ u p gmv =

B C