Modeling Portfolios that Contain Risky Assets II: Efficient - PowerPoint PPT Presentation

Modeling Portfolios that Contain Risky Assets II: Efficient Frontiers for Various Models C. David Levermore University of Maryland, College Park ICERM Lecture, 16 November 2011 Extracted from Math 420: Mathematical Modeling � 2011 Charles David Levermore c

Lecture II: Efficient Frontiers for Various Models Outline 1 . Efficient-Market Hypothesis 2 . Modeling Portfolios with Risk-Free Assets 3 . Modeling Long Portfolios 4 . Modeling Long Portfolios with a Safe Investment

1. Efficient-Market Hypothesis The efficient-market hypothesis (EMH) was framed by Eugene Fama in the early 1960’s in his University of Chicago doctoral dissertation, which was published in 1965. It has several versions, the most basic is the following. Given the information available when an investment is made, no investor will consistently beat market returns on a risk-adjusted basis over long periods except by chance. This version of the EMH is called the weak EMH. The semi-strong and the strong versions of the EMH make bolder claims that markets reflect information instantly, even information that is not publicly available in the case of the strong EMH. While it is true that some investors react quickly, most investors do not act instantly to every piece of news. Consequently, there is little evidence supporting these stronger versions of the EMH.

The EMH is an assertion about markets, not about investors. If the weak EMH is true then the only way for an actively-managed mutual fund to beat the market is by chance. Of course, there is some debate regarding the truth of the weak EMH. It can be recast in the language of MPT as follows. Markets for large classes of assets will lie on the efficient frontier. You can therefore test the weak EMH with MPT. If we understand “market” to mean a capitalization weighted collection of assets (i.e. an index fund) then the EMH can be tested by checking whether index funds lie on or near the efficient frontier. You will see that this is often the case, but not always. Remark. It is a common misconception that MPT assumes the weak EMH. It does not, which is why the EMH can be checked with MPT!

Remark. It is often asserted that the EMH holds in rational markets . Such a market is one for which information regarding its assets is freely available to all investors. This does not mean that investors will act rationally based on this information! Nor does it mean that markets price assets correctly. Even rational markets are subject to the greed and fear of its investors. That is why we have bubbles and crashes. Rational markets can behave irrationally because information is not knowledge! It could be asserted that the EMH is likely to hold in free markets . Such markets have many agents, are rational and subject to regulatory and legal oversight. These are all elements in Adam Smith’s notion of free market, which refers to the freedom of its agents to act, not to the freedom from any government role. Indeed, his radical idea was that government should nurture free markets by playing the role of empowering individual agents. He had to write his book because free markets do not arise spontaneously, even though his “invisible hand” insures that markets do.

2. Modeling Portfolios with Risk-Free Assets Until now we have considered portfolios that contain only risky assets. We now consider two kinds of risk-free assets (assets that have no volatility associated with them) that can play a major role in portfolio management. The first is a safe investment that pays dividends at a prescribed interest rate µ si . This can be an FDIC insured bank account, or safe securities such as US Treasury Bills, Notes, or Bonds. (US Treasury Bills are most commonly used.) You can only hold a long position in such an asset. The second is a credit line from which you can borrow at a prescribed interest rate µ cl up to your credit limit. Such a credit line should require you to put up assets like real estate or part of your portfolio (a margin ) as collateral from which the borrowed money can be recovered if need be. You can only hold a short position in such an asset.

We will assume that µ cl ≥ µ si , because otherwise investors would make money by borrowing at rate µ cl in order to invest at the greater rate µ si . (Here we are again neglecting transaction costs.) Because free money does not sit around for long, market forces would quickly adjust the rates so that µ cl ≥ µ si . In practice, µ cl is about three points higher than µ si . We will also assume that a portfolio will not hold a position in both the safe investment and the credit line when µ cl > µ si . To do so would effectively be borrowing at rate µ cl in order to invest at the lesser rate µ si . While there can be cash-flow management reasons for holding such a position for a short time, it is not a smart long-term position. These assumptions imply that every portfolio can be viewed as holding a position in at most one risk-free asset: it can hold either a long position at rate µ si , a short position at rate µ cl , or a neutral risk-free position.

Markowitz Portfolios. We now extend the notion of Markowitz portfolios to portfolios that might include a single risk-free asset with return rate µ rf . Let b rf ( d ) denote the balance in the risk-free asset at the start of day d . For a long position µ rf = µ si and b rf ( d ) > 0 , while for a short position µ rf = µ cl and b rf ( d ) < 0 . A Markowitz portfolio drawn from one risk-free asset and N risky assets is uniquely determined by a set of real numbers f rf and { f i } N i =1 that satisfies N � f rf + f i = 1 , f rf < 1 if any f j � = 0 . i =1 The portfolio is rebalanced at the start of each day so that b rf ( d ) n i ( d ) s i ( d − 1) Π( d − 1) = f rf , = f i for i = 1 , · · · , N . Π( d − 1) The condition f rf < 1 if any f j � = 0 states that the safe investment must contain less than the net portfolio value unless it is the entire portfolio.

Its value at the start of day d is N � Π( d − 1) = b rf ( d ) + n i ( d ) s i ( d − 1) , i =1 while its value at the end of day d is N � 1 + 1 � � Π( d ) = b rf ( d ) + n i ( d ) s i ( d ) . D µ rf i =1 Its return rate for day d is therefore r ( d ) = D Π( d ) − Π( d − 1) Π( d − 1) � � N n i ( d ) s i ( d ) − s i ( d − 1) = b rf ( d ) µ rf � Π( d − 1) + D Π( d − 1) i =1 N f i r i ( d ) = f rf µ rf + f T r ( d ) . � = f rf µ rf + i =1

The portfolio return rate mean µ and variance v are then given by D D µ = 1 r ( d ) = 1 � f rf µ rf + f T r ( d ) � � � D D d =1 d =1   D = f rf µ rf + f T  1  = f rf µ rf + f T m , � r ( d ) D d =1 D D � 2 = � 2 1 � 1 � f T r ( d ) − f T m � � v = r ( d ) − µ D ( D − 1) D ( D − 1) d =1 d =1   D T = f T 1 � �� �  f = f T Vf . � r ( d ) − m r ( d ) − m  D ( D − 1) d =1 We thereby obtain the formulas � T f � T f , v = f T Vf . µ = µ rf 1 − 1 + m

Capital Allocation Lines. These formulas can be viewed as describing a point that lies on a certain half-line in the σµ -plane. Let ( σ, µ ) be the point in the σµ -plane associated with the Markowitz portfolio characterized by T f = 1 − f rf > 0 because f � = 0 . the distribution f � = 0 . Notice that 1 Define f ˜ f = T f . 1 � f T V ˜ T ˜ T ˜ ˜ f = 1 . Let ˜ µ = m f and ˜ σ = f . Then (˜ σ, ˜ µ ) is Notice that 1 the point in the σµ -plane associated with the Markowitz portfolio without risk-free assets that is characterized by the distribution ˜ f . Because � T f � T f ˜ T f ˜ µ = 1 − 1 µ rf + 1 σ = 1 µ , σ , we see that the point ( σ, µ ) in the σµ -plane lies on the half-line that starts at the point (0 , µ rf ) and passes through the point (˜ σ, ˜ µ ) that corresponds to a portfolio that does not contain the risk-free asset.

Conversely, given any point (˜ σ, ˜ µ ) corresponding to a Markowitz portfolio that contains no risk-free assets, consider the half-line � � ( σ, µ ) = φ ˜ σ , (1 − φ ) µ rf + φ ˜ where φ > 0 . µ µ ) has distribution ˜ If a portfolio corresponding to (˜ σ, ˜ f then the point on the half-line given by φ corresponds to the portfolio with distribution f = φ ˜ f . T f = 1 − φ of its value to the risk-free asset. This portfolio allocates 1 − 1 The risk-free asset is held long if φ ∈ (0 , 1) and held short if φ > 1 while φ = 1 corresponds to a neutral position. We must restrict φ to either (0 , 1] or [1 , ∞ ) depending on whether the risk-free asset is the safe investment or the credit line. This segment of the half-line is called the capital allocation line through (˜ σ, ˜ µ ) associated with the risk-free asset. We can therefore use the appropriate capital allocation lines to construct the set of all points in the σµ -plane associated with Markowitz portfolios that contain a risk-free asset from the set of all points in the σµ -plane associated with Markowitz portfolios that contain no risk-free assets.