Modeling Portfolios that Contain Risky Assets

Transcription

1 Modeling Portfolios that Contain Risky Assets C. David Levermore University of Maryland, College Park Math 420: Mathematical Modeling January 18, 2012 version c 2011 Charles David Levermore

2 Outline 1. Introduction 2. Markowitz Portfolios 3. Basic Markowitz Portfolio Theory 4. Modeling Portfolios with Risk-Free Assets 5. Modeling Long Portfolios 6. Modeling Long Portfolios with a Safe Investment 7. Stochastic Models of One Risky Asset 8. Stochastic Models of Portfolios with Risky Assets 9. Model-Based Objective Functions 10. Model-Based Portfolio Management 11. Conclusion

3 1. Introduction Suppose you are considering how to invest in N risky assets that are traded on a market that had D trading days last year. (Typically D = 255.) Let s i (d) be the share price of the i th asset at the close of the d th trading day of the past year, where s i (0) is understood to be the share price at the close of the last trading day before the beginning of the past year. We will assume that every s i (d) is positive. You would like to use this price history to gain insight into how to manage your portfolio over the coming year. We will examine the following questions. Can stochastic (random, probabilistic) models be built that quantitatively mimic this price history? How can such models be used to help manage a portfolio?

4 Risky Assets. The risk associated with an investment is the uncertainy of its outcome. Every investment has risk associated with it. Hiding your cash under a mattress puts it at greater risk of loss to theft or fire than depositing it in a bank, and is a sure way to not make money. Depositing your cash into an FDIC insured bank account is the safest investment that you can make the only risk of loss would be to an extreme national calamity. However, a bank account generally will yield a lower return on your investment than any asset that has more risk associated with it. Such assets include real estate, stocks (equities), bonds, commodities (gold, oil, corn, etc.), private equity (venture capital), and hedge funds. With the exception of real estate, it is not uncommon for prices of these assets to fluctuate one to five percent in a day. Any such asset is called a risky asset. Remark. Market forces generally will insure that assets associated with higher potential returns are also associated with greater risk and vice versa. Investment offers that seem to violate this principle are always scams.

5 Here we will consider two basic types of risky assets: stocks and bonds. We will also consider mutual funds, which are managed funds that hold a combination of stocks and/or bonds, and possibly other risky assets. Stocks. Stocks are part ownership of a company. Their value goes up when the company does well, and goes down when it does poorly. Some stocks pay a periodic (usually quarterly) dividend of either cash or more stock. Stocks are traded on exchanges like the NYSE or NASDAQ. The risk associated with a stock reflects the uncertainty about the future performance of the company. This uncertainty has many facets. For example, there might be questions about the future market share of its products, the availablity of the raw materials needed for its products, or the value of its current assets. Stocks in larger companies are generally less risky than stocks in smaller companies. Stocks are generally higher return/higher risk investments compared to bonds.

6 Bonds. Bonds are essentially a loan to a government or company. The borrower usually makes a periodic (often quarterly) interest payment, and ultimately pays back the principle at a maturity date. Bonds are traded on secondary markets where their value is based on current interest rates. For example, if interest rates go up then bond values will go down on the secondary market. The risk associated with a bond reflects the uncertainty about the credit worthiness of the borrower. Short term bonds are generally less risky than long term ones. Bonds from large entities are generally less risky than those from small entities. Bonds from governments are generally less risky than those from companies. (This is even true in some cases where the ratings given by some ratings agencies suggest otherwise.) Bonds are generally lower return/lower risk investments compared to stocks.

7 Mutual Funds. These funds hold a combination of stocks and/or bonds, and possibly other risky assets. You buy and sell shares in these funds just as you would shares of a stock. Mutual funds are generally lower return/lower risk investments compared to individual stocks and bonds. Most mutual funds are managed in one of two ways: actively or passively. An actively-managed fund usually has a strategy to perform better than some market index like the S&P 500, Russell 1000, or Russell A passively-managed fund usually builds a portfolio so that its performance will match some market index, in which case it is called an index fund. Index funds are often portrayed to be lower return/lower risk investments compared to actively-managed funds. In practice index funds tend to outperform most actively-managed funds. Reasons for this include the facts that they have lower management fees and that they require smaller cash reserves.

8 Return Rates. The first thing you must understand that the share price of an asset has very little economic significance. This is because the size of your investment in an asset is the same if you own 100 shares worth 50 dollars each or 25 shares worth 200 dollars each. What is economically significant is how much your investment rises or falls in value. Because your investment in asset i would have changed by the ratio s i (d)/s i (d 1) over the course of day d, this ratio is economically significant. Rather than use this ratio as the basic variable, it is customary to use the so-called return rate, which we define by r i (d) = D s i(d) s i (d 1). s i (d 1) The factor D arises because rates in banking, business, and finance are usually given as annual rates expressed in units of either per annum or % per annum. Because a day is D 1 years the factor of D makes r i(d) a per annum rate. It would have to be multiplied by another factor of 100 to make it a % per annum rate. We will always work with per annum rates.

9 One way to understand return rates is to set r i (d) equal to a constant µ. Upon solving the resulting relation for s i (d) you find that s i (d) = ( 1 + µ D) si (d 1) for every d = 1,, D. By induction on d you can then derive the compound interest formula s i (d) = ( 1 + µ D) d si (0) for every d = 1,, D. If you assume that µ/d << 1 then you can see that whereby ( )D 1 + µ µ D lim(1 + h) 1 h = e, h 0 s i (d) = ( 1 + µ D)D µ µ dd s i (0) e µ d D s i (0) = e µt s i (0), where t = d/d is the time (in units of years) at which day d occurs. You thereby see µ is nearly the exponential growth rate of the share price.

10 We will consider a market of N risky assets indexed by i. For each i you obtain the closing share price history {s i (d)} D d=0 of asset i over the past year, and compute the return rate history {r i (d)} D d=1 of asset i over the past year by the formula r i (d) = D s i(d) s i (d 1) s i (d 1) Because return rates are differences, you will need the closing share price from the day before the first day for which you want the return rate history. You can obtain share price histories from websites like Yahoo Finance or Google Finance. For example, to compute the daily return rate history for Apple in 2009, type Apple into where is says get quotes. You will see that Apple has the identifier AAPL and is listed on the NASDAQ. Click on historical prices and request share prices between Dec 31, 2008 and Dec 31, You will get a table that can be downloaded as a spreadsheet. The return rates are computed using the closing prices..

11 Remark. It is not obvious that return rates are the right quantities upon which to build a theory of markets. For example, another possibility is to use the growth rates x i (d) defined by ( ) si (d) x i (d) = D log. s i (d 1) These are also functions of the ratio s i (d)/s i (d 1). Moreover, they seem to be easier to understand than return rates. For example, if you set x i (d) equal to a constant γ then by solving the resulting relation for s i (d) you find that s i (d) = e 1 D γ s i (d 1) for every d = 1,, D. By induction on d you can then show that s i (d) = e d D γ s i (0) for every d = 1,, D, whereby s i (d) = e γt s i (0) with t = d/d. However, return rates have better properties with regard to porfolio statistics and so are preferred.

12 Statistical Approach. Return rates r i (d) for asset i can vary wildly from day to day as the share price s i (d) rises and falls. Sometimes the reasons for such fluctuations are very clear because they directly relate to some news about the company, agency, or government that issued the asset. For example, news of the Deepwater Horizon explosion caused the share price of British Petroleum stock to fall. At other times they relate to news that benefit or hurt entire sectors of assets. For example, a rise in crude oil prices might benefit oil and railroad companies but hurt airline and trucking companies. And at yet other times they relate to general technological, demographic, or social trends. For example, new internet technology might benefit Google and Amazon (companies that exist because of the internet) but hurt traditional brick and mortar retailers. Finally, there is often no evident public reason for a particular stock price to rise or fall. The reason might be a takeover attempt, a rumor, insider information, or the fact a large investor needs cash for some other purpose.

13 Given the complexity of the dynamics underlying such market fluctuations, we adopt a statistical approach to quantifying their trends and correlations. More specifically, we will choose an appropriate set of statistics that will be computed from selected return rate histories of the relevant assets. We will then use these statistics to calibrate a model that will predict how a set of ideal portfolios might behave in the future. The implicit assumption of this approach is that in the future the market will behave statistically as it did in the past. This means that the data should be drawn from a long enough return rate history to sample most of the kinds of market events that you expect to see in the future. However, the history should not be too long because very old data will not be relevant to the current market. To strike a balance we will use the return rate history from the most recent twelve month period, which we will dub the past year. For example, if we are planning our portfolio at the beginning of July 2011 then we will use the return rate histories for July 2010 through June Then D would be the number of trading days in this period.

14 Suppose that you have computed the return rate history {r i (d)} D d=1 for each asset over the past year. At some point this data should be ported from the speadsheet into MATLAB, R, or another higher level environment that is well suited to the task ahead. The next step is to compute statistical quantities; we will use means, variances, covariances, and correlations. The return rate mean for asset i over the past year, denoted m i, is m i = 1 D D d=1 r i (d). This measures the trend of the share price. Unfortunately, it is commonly called the expected return rate for asset i even though it is higher than the return rate that most investors will see, especially in highly volatile markets. We will not use this misleading terminology.

15 The return rate variance for asset i over the past year, denoted v i, is v i = 1 D(D 1) D d=1 ( ri (d) m i ) 2. The reason for the D(D 1) in the denominator will be made clear later. The return rate standard deviation for asset i over the year, denoted σ i, is given by σ i = v i. This is called the volatility of asset i. It measures the uncertainty of the market regarding the share price trend. The covariance of the return rates for assets i and j over the past year, denoted v ij, is v ij = 1 D(D 1) D d=1 ( ri (d) m i )( rj (d) m j ). Notice that v ii = v i. The N N matrix (v ij ) is symmetric and nonnegative definite. It will usually be positive definite so we will assume it to be so.

16 Finally, the correlation of the return rates for assets i and j over the past year, denoted c ij, is c ij = v ij σ i σ j. Notice that 1 c ij 1. We say assets i and j are positively correlated when 0 < c ij 1 and negatively correlated when 1 c ij < 0. Positively correlated assets will tend to move in the same direction, while negatively correlated ones will often move in opposite directions. We will consider the N-vector of means (m i ) and the symmetric N N matrix of covariances (v ij ) to be our basic statistical quantities. We will build our models to be consistent with these statistics. The variances (v i ), volatilities (σ i ), and correlations (c ij ) can then be easily obtained from (m i ) and (v ij ) by formulas that are given above. The computation of the statistics (m i ) and (v ij ) from the return rate histories is called the calibration of our models.

17 Remark. Here the trading day is an arbitrary measure of time. From a theoretical viewpoint we could equally well have used a shorter measure like half-days, hours, quarter hours, or minutes. The shorter the measure, the more data has to be collected and analyzed. This extra work is not worth doing unless you profit sufficiently. Alternatively, we could have used a longer measure like weeks, months, or quarters. The longer the measure, the less data you use, which means you have less understanding of the market. For many investors daily or weekly data is a good balance. If you use weekly data {s i (w)} 52 w=0, where s i(w) is the share price of asset i at the end of week w, then the rate of return of asset i for week w is r i (w) = 52 s i(w) s i (w 1) s i (w 1) You have to make consistent changes when computing m i, v i, and v ij by replacing d with w and D with 52 in their defining formulas..

18 General Histories and Weights. We can consider a history {r(d)} D h d=1 over a period of D h trading days and assign day d a weight w(d) > 0 such that the weights {w(d)} D h d=1 satisfy D h d=1 w(d) = 1. The return rate means and covariances are then given by where m i = v ij = D h d=1 w(d) r i (d), 1 D(1 w) D h d=1 w(d) ( r i (d) m i )( rj (d) m j ), w = D h d=1 w(d) 2.

19 In practice the history will extend over a period of one to five years. There are many ways to choose the weights {w(d)} D h d=1. The most common choice is the so-called uniform weighting; this gives each day the same weight by setting w(d) = 1/D h. On the other hand, we might want to give more weight to more recent data. For example, we can give each trading day a positive weight that depends only on the quarter in which it lies, giving greater weight to more recent quarters. We could also consider giving different weights to different days of the week, but such a complication should be avoided unless it yields a clear benefit. You will have greater confidence in m i and v ij when they are relatively insensitive to different choices of D h and the weights w(d). You can get an idea of the magnitude of this sensitivity by checking the robustness of m i and v ij to a range of such choices.

20 Exercise. Compute m i, v i, v ij, and c ij for each of the following groups of assets based on daily data, weekly data, and monthly data: (a) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009; (b) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007; (c) S&P 500 and Russell 1000 and 2000 index funds in 2009; (d) S&P 500 and Russell 1000 and 2000 index funds in Give explanations for the values of c ij you computed. Exercise. Compute m i, v i, v ij, and c ij for the assets listed in the previous exercise based on daily data and weekly data, but only from the last quarter of the year indicated. Based on a comparison of these answers with those of the previous problem, in which numbers might you have the most confidence, the m i, v i, v ij, or c ij?

21 2. Markowitz Portfolios A 1952 paper by Harry Markowitz had enormous influence on the theory and practice of portfolio management and financial engineering ever since. It presented his doctoral dissertation work at the Unversity of Chicago, for which he was awarded the Nobel Prize in Economics in It was the first work to quantify how diversifying a portfolio can reduce its risk without changing its expected rate of return. It did this because it was the first work to use the covariance between different assets in an essential way. We will consider portfolios in which an investor can choose to hold one of three positions with respect to any risky asset. The investor can: - hold a long position by owning shares of the asset; - hold a short position by selling borrowed shares of the asset; - hold a neutral position by doing neither of the above. In order to keep things simple, we will not consider derivative assets.

22 Remark. You hold a short position by borrowing shares of an asset from a lender (usually your broker) and selling them immediately. If the share price subsequently goes down then you can buy the same number of shares and give them to the lender, thereby paying off your loan and profiting by the price difference minus transaction costs. Of course, if the price goes up then your lender can force you either to increase your collateral or to pay off the loan by buying shares at this higher price, thereby taking a loss that might be larger than the original value of the shares. The value of any portfolio that holds n i (d) shares of asset i at the end of trading day d is Π(d) = N i=1 n i (d)s i (d). If you hold a long position in asset i then n i (d) > 0. If you hold a short position in asset i then n i (d) < 0. If you hold a neutral position in asset i then n i (d) = 0. We will assume that Π(d) > 0 for every d.

23 Markowitz carried out his analysis on a class of idealized portfolios that are each characterized by a set of real numbers {f i } N i=1 such that N i=1 f i = 1. The portfolio picks n i (d) at the beginning at each trading day d so that n i (d)s i (d 1) Π(d 1) = f i, where n i (d) need not be an integer. We call these Markowitz portfolios. The portfolio holds a long position in asset i if f i > 0 and holds a short position if f i < 0. If every f i is nonnegative then f i is the fraction of the portfolio s value held in asset i at the beginning of each day. A Markowitz portfolio will be self-financing if we neglect trading costs because N i=1 n i (d) s i (d 1) = Π(d 1).

24 Portfolio Return Rate. We see from the self-financing property and the relationship between n i (d) and f i that the return rate r(d) of a Markowitz portfolio for trading day d is r(d) = D = = N i=1 N i=1 Π(d) Π(d 1) Π(d 1) D n i(d)s i (d) n i (d)s i (d 1) Π(d 1) n i (d)s i (d 1) Π(d 1) D s i(d) s i (d 1) s i (d 1) = N i=1 f i r i (d). The return rate r(d) for the Markowitz portfolio characterized by {f i } N i=1 is therefore simply the linear combination with coefficients f i of the r i (d). This relationship makes the class of Markowitz portfolios easy to analyze. We will therefore use Markowitz portfolios to model real portfolios.

25 This relationship can be expressed in the compact form r(d) = f T r(d), where f and r(d) are the N-vectors defined by f = 1. f f N, r(d) = r 1 (d). r N (d) The N-vector of return rate means m and the N N-matrix of return rate covariances V can be expressed in terms of r(d) as V = m = m 1. m N v 1N v v N1 v NN = 1 D = D d=1 1 r(d), D(D 1) D d=1. ( r(d) m ) ( r(d) m ) T.

26 Portfolio Statistics. Markowitz portfolios are easy to analyze because r(d) = f T r(d) where f is independent of d. In particular, for the Markowitz portfolio characterized by f the return rate mean µ and variance v can be expressed simply in terms of m and V as µ = 1 D D d=1 v = 1 D(D 1) r(d) = 1 D D d=1 = 1 D D(D 1) d=1 = f T 1 D(D 1) D d=1 f T r(d) = f T ( r(d) µ ) 2 = 1 D(D 1) 1 D D d=1 D d=1 ( f T r(d) f T m )( r(d) T f m T f ) D d=1 ( r(d) m )( r(d) m ) T r(d) = f T m, ( f T r(d) f T m ) 2 f = f T Vf. Hence, µ = f T m and v = f T Vf. Because V is positive definite, v > 0.

27 Remark. These simple formulas for µ and v are the reason that return rates are preferred over growth rates when compiling statistics of markets. The simplicity of these formulas arises because the return rates r(d) for the Markowitz portfolio specified by the distribution f depends linearly upon the vector r(d) of return rates for the individual assets as r(d) = f T r(d). In contrast, the growth rates x(d) of a Markowitz portfolio are given by ( ) Π(d) x(d) = D log = D log ( Π(d 1) D r(d)) = D log ( 1 + D 1 ft r(d) ) = D log 1 + D 1 = D log 1 + N ( f i i=1 ed 1 ) x i(d) 1 N i=1 = D log f i r i (d) N i=1 f i e 1 D x i(d) Because the x(d) are not linear functions of the x i (d), averages of x(d) over d are not simply expressed in terms of averages of x i (d) over d..

28 Remark. Had we chosen to assign weights {w(d)} D h d=1 to the trading days of a return rate history {r(d)} D h d=1 then we would still obtain the formulas µ = f T m, v = f T Vf, but the N-vector of return rate means m and the N N-matrix of return rate covariances V would be expressed in terms of r(d) as m = V = D h d=1 w(d)r(d), 1 D(1 w) D h d=1 w(d) ( r(d) m ) ( r(d) m ) T. The choices of return rate history {r(d)} D h d=1 and weights {w(d)}d h d=1 thereby determine m and V. They constitute a calibration of our models. Ideally m and V will not be overly sensitive to these choices.

29 Remark. Aspects of Markowitz portfolios are unrealistic. These include: - the fact portfolios can contain fractional shares of any asset; - the fact portfolios are rebalanced every trading day; - the fact transaction costs and taxes are neglected; - the fact dividends are neglected. By making these simplifications the subsequent analysis becomes easier. The idea is to find the Markowitz portfolio that is best for a given investor. The expectation is that any real portfolio with a distribution close to that for the optimal Markowitz portfolio will perform nearly as well. Consequently, most investors rebalance at most a few times per year, and not every asset is involved each time. Transaction costs and taxes are thereby limited. Similarly, borrowing costs are kept to a minimum by not borrowing often. The model can be modified to account for dividends.

30 Remark. Portfolios of risky assets can be designed for trading or investing. Traders often take positions that require constant attention. They might buy and sell assets on short time scales in an attempt to profit from market fluctuations. They might also take highly leveraged positions that expose them to enormous gains or loses depending how the market moves. They must be ready to handle margin calls. Trading is often a full time job. Investors operate on longer time scales. They buy or sell an asset based on their assessment of its fundamental value over time. Investing does not have to be a full time job. Indeed, most people who hold risky assets are investors who are saving for retirement. Lured by the promise of high returns, sometimes investors will buy shares in funds designed for traders. At that point they have become gamblers, whether they realize it or not. The ideas presented in these lectures are designed to balance investment portfolios, not trading portfolios.

31 Exercise. Compute µ and v based on daily data for the Markowitz portfolio with value equally distributed among the assets in each of the following groups: (a) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009; (b) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007; (c) S&P 500 and Russell 1000 and 2000 index funds in 2009; (d) S&P 500 and Russell 1000 and 2000 index funds in Exercise. The volatility of a portfolio is σ = v. In the σµ-plane plot (σ, µ) for the two 2007 portfolios and (σ i, m i ) for each of the 2007 assets in the previous exercise. Exercise. In the σµ-plane plot (σ, µ) for the two 2009 portfolios and (σ i, m i ) for each of the 2009 assets in the first exercise above.

32 3. Basic Markowitz Portfolio Theory The 1952 Markowitz paper initiated what subsequently became known as modern portfolio theory (MPT). Because 1952 was long ago, this name has begun to look silly and some have taken to calling it Markowitz Portfolio Theory (still MPT), to distinguish it from more modern theories. (Markowitz simply called it portfolio theory, and often made fun of the name it aquired.) Any portfolio theory strives to maximize the return of a portfolio for a given risk, or equivalently, to mimimize risk for a given return. Portfolio theories do this by quantifying the notions of return and risk, and by identifying a class of idealized portfolios for which some kind of analysis is tractable. Here we introduce MPT, the first such theory.

33 Markowitz chose to use the return rate mean µ as the proxy for the return of a portfolio, and the volatility σ = v as the proxy for its risk. He also chose to analyze the class that we have dubbed Markowitz portfolios. Then for a portfolio of N risky assets characterized by m and V the problem of minimizing risk for a given return becomes the problem of minimizing σ = f T Vf over f R N subject to the constraints 1 T f = 1, m T f = µ, where µ is given. Here 1 is the N-vector that has every entry equal to 1. Additional constraints can be imposed. For example, if only long positions are to be considered then we must also impose the entrywise constraints f 0, where 0 denotes the N-vector that has every entry equal to 0.

34 Constrained Minimization Problem. Because σ > 0, minimizing σ is equivalent to minimizing σ 2. Because σ 2 is a quadratic function of f, it is easier to minimize than σ. We therefore choose to solve the constrained minimization problem min f R N { 12 f T Vf : 1 T f = 1, m T f = µ }. Because there are two equality constraints, we introduce the Lagrange multipliers α and β, and define Φ(f, α, β) = 1 2 ft Vf α ( 1 T f 1 ) β ( m T f µ ). By setting the partial derivatives of Φ(f, α, β) equal to zero we obtain 0 = f Φ(f, α, β) = Vf α1 βm, 0 = α Φ(f, α, β) = 1 T f + 1, 0 = β Φ(f, α, β) = m T f + µ.

35 Because V is positive definite we may solve the first equation for f as f = αv β V 1 m. By setting this into the second and third equations we obtain the system α1 T V β 1 T V 1 m = 1, αm T V β m T V 1 m = µ. If we introduce a, b, and c by a = 1 T V 1 1, b = 1 T V 1 m, c = m T V 1 m, then the above system can be expressed as ( ) ( ) ( a b α 1 = b c β µ ). Because V 1 is positive definite, the above 2 2 matrix is positive definite if and only if the vectors 1 and m are not co-linear (m µ1 for every µ).

36 We now assume that 1 and m are not co-linear, which is usually the case in practice. We can then solve the system for α and β to obtain ( ) α = 1 ( ) ( ) c b 1 β ac b 2 = 1 ( ) c bµ b a µ ac b 2. aµ b Hence, for each µ there is a unique minimizer given by f(µ) = c bµ ac b 2 V aµ b ac b 2 V 1 m. The associated minimum value of σ 2 is σ 2 = f(µ) T Vf(µ) = ( αv β V 1 m ) T V ( αv β V 1 m ) = ( α β ) ( a b b c = 1 a + a ac b 2 ) ( ) α ( β µ b a) 2. = 1 ac b 2 ( ) ( ) ( c b 1 1 µ b a µ )

37 Remark. In the case where 1 and m are co-linear we have m = µ1 for some µ. If we minimize 1 2 ft Vf subject to the constraint 1 T f = 1 then, by Lagrange multipliers, we find that Vf = α1, for some α R. Because V is invertible, for every µ the unique minimizer is given by f(µ) = The associated minimum value of σ 2 is σ 2 = f(µ) T Vf(µ) = 1 1 T V 1 1 V T V 1 1 = 1 a. Remark. The mathematics behind MPT is fairly elementary. Markowitz had cleverly indentified a class of portfolios which is analytically tractable by elemetary methods, and which provides a framework for useful models.

38 Frontier Portfolios. We have seen that for every µ there exists a unique Markowitz portfolio with mean µ that minimizes σ 2. This minimum value is σ 2 = 1 a + a ac b 2 ( µ b a) 2. This is the equation of a hyperbola in the σµ-plane. Because volatility is nonnegative, we only consider the right half-plane σ 0. The volatility σ and mean µ of any Markowitz portfolio will be a point (σ, µ) in this halfplane that lies either on or to the right of this hyperbola. Every point (σ, µ) on this hyperbola in this half-plane represents a unique Markowitz portfolio. These portfolios are called frontier portfolios. We now replace a, b, and c with the more meaningful frontier parameters σ mv = 1, µ mv = b a a, ν ac b 2 as =. a

39 The volatility σ for the frontier portfolio with mean µ is then given by ) 2. ( σ = σ f (µ) σ 2 µ µmv mv + ν as It is clear that no portfolio has a volatility σ that is less than σ mv. In other words, σ mv is the minimum volatility attainable by diversification. Markowitz interpreted σmv 2 to be the contribution to the volatility due to the systemic risk of the market, and interpreted (µ µ mv ) 2 /νas 2 to be the contribution to the volatility due to the specific risk of the portfolio. The frontier portfolio corresponding to (σ mv, µ mv ) is called the minimum volatility portfolio. Its associated distribution f mv is given by ( ) b f mv = f(µ mv ) = f = 1 a a V 1 1 = σmvv This distribution depends only upon V, and is therefore known with greater confidence than any distribution that also depends upon m.

40 The distribution of the frontier portfolio with mean µ can be expressed as f f (µ) f mv + µ µ mv ν 2 as V 1( m µ mv 1 ). Because 1 T V 1( m µ mv 1 ) = b µ mv a = 0, we see that f T mvv ( ) f f (µ) f mv = σ 2 µ µ mv mv νas 2 1 T V 1( m µ mv 1 ) = 0. The vectors f mv and f f (µ) f mv are thereby orthogonal with respect to the V-inner product, which is given by (f 1 f 2 ) V = f T 1 Vf 2. In the associated V-norm, which is given by f V 2 = (f f) V, we find that ) 2. ( f mv V 2 = σ mv 2, f f (µ) f mv V 2 µ µmv = ν as Hence, f f (µ) = f mv + (f f (µ) f mv ) is the orthogonal decomposition of f f (µ) into components that account for the contributions to the volatility due to systemic risk and specific risk respectively.

41 Remark. Markowitz attributed σ mv to systemic risk of the market because he was considering the case when N was large enough that σ mv would not be significantly reduced by introducing additional assets into the portfolio. More generally, one should attribute σ mv to those risks that are common to all of the N assets being considered for the portfolio. Definition. When two portfolios have the same volatility but different means then the one with the greater mean is said to be more efficient because it promises greater return for the same risk. Definition. Every frontier portfilio with mean µ > µ mv is more efficient than every other portfolio with the same volatility. This segment of the frontier is called the efficient frontier. Definition. Every frontier portfilio with mean µ < µ mv is less efficient than every other portfolio with the same volatility. This segment of the frontier is called the inefficient frontier.

42 The efficient frontier is represented in the right-half σµ-plane by the upper branch of the frontier hyperbola. It is given as a function of σ by µ = µ mv + ν as σ 2 σ 2 mv, for σ > σ mv. This curve is increasing and concave and emerges vertically upward from the point (σ mv, µ mv ). As σ it becomes asymptotic to the line µ = µ mv + ν as σ. The inefficient frontier is represented in the right-half σµ-plane by the lower branch of the frontier hyperbola. It is given as a function of σ by µ = µ mv ν as σ 2 σ 2 mv, for σ > σ mv. This curve is decreasing and convex and emerges vertically downward from the point (σ mv, µ mv ). As σ it becomes asymptotic to the line µ = µ mv ν as σ.

43 General Portfolio with Two Risky Assets. Consider a portfolio of two risky assets with mean vector m and covarience matrix V given by m = ( m1 m 2 ), V = ( v11 v 12 v 12 v 22 The constraints 1 T f = 1 and m T f = µ uniquely determine f to be ( ) ( ) f1 (µ) 1 m2 µ f(µ) = =. f 2 (µ) m 2 m 1 µ m 1 Because there is exactly one portfolio for each µ, every portfolio must be a frontier portfolio. Therefore there is no optimization problem to solve and f f (µ) = f(µ). These portfolios trace out the hyperbola σ 2 = f(µ) T Vf(µ) = v 11(m 2 µ) 2 + 2v 12 (m 2 µ)(µ m 1 ) + v 22 (µ m 1 ) 2 (m 2 m 1 ) 2. ).

44 The frontier parameters µ mv, σ mv, and ν as are given by µ mv = (v 22 v 12 )m 1 + (v 11 v 12 )m 2 v 11 + v 22 2v 12, σ 2 mv = v 11v 22 v 2 12 v 11 + v 22 2v 12, ν 2 as = (m 2 m 1 ) 2 v 11 + v 22 2v 12. The minimum volatility portfolio is σ 2 mv ( v22 v 12 f mv = σmvv = v 11 v 22 v12 2 v 12 v 11 ( ) 1 v22 v = 12. v 11 + v 22 2v 12 v 11 v 12 ) ( 1 1 ) Remark. Each (σ i, m i ) lies on the frontier of every two-asset portfolio. Typically each (σ i, m i ) lies strictly to the right of the frontier of a portfolio that contains more than two risky assets.

45 Remark. The foregoing two-asset portfolio example also illustrates the following general property of frontier portfolios that contain two or more risky assets. Let µ 1 and µ 2 be any two return rate means with µ 1 < µ 2. The distributions of the associated frontier portfolios are then f 1 = f f (µ 1 ) and f 2 = f f (µ 2 ). Because f f (µ) depends linearly on µ we see that f f (µ) = µ 2 µ µ 2 µ 1 f 1 + µ µ 1 µ 2 µ 1 f 2. This formula can be interpreted as stating that every frontier portfolio can be realized by holding positions in just two mutual funds with the portfolio distributions f 1 and f 2. When µ (µ 1, µ 2 ) both funds are held long. When µ > µ 2 the first fund is held short while the second is held long. When µ < µ 1 the second fund is held short while the first is held long. This observation is often called the Two Mutual Fund Theorem, which is a label that elevates it to a higher status than it deserves. We will call it simply the Two Mutual Fund Property.

46 Simple Portfolio with Three Risky Assets. Consider a portfolio of three risky assets with mean vector m and covarience matrix V given by m = m 1 m 2 = m 3 m d m m + d, V = s 2 1 r r r 1 r r r 1 Here m R, d, s R +, and r ( 1 2,1). The last condition is equivalent to the condition that V is positive definite given s > 0. This portfolio has the unrealistic properties that (1) every asset has the same volatility s, (2) every pair of distinct assets has the same correlation r, and (3) the return rate means are uniformly spaced with difference d = m 3 m 2 = m 2 m 1. These simplifications will make it easier to follow the ensuing calculations than it would be for a more general three-asset portfolio. We will return to this simple portfolio in subsequent examples..

47 It is helpful to express m and V in terms of 1 and n = ( ) T as m = m1 + dn, V = s 2 (1 r) ( I + r ) 1 r 11T. Notice that 1 T 1 = 3, 1 T n = 0, and n T n = 2. It can be checked that ( V 1 1 = s 2 I r (1 r) 1 + 2r 11T V 1 1 n = s 2 (1 r) n, V 1 m = ), V 1 1 = The parameters a, b, and c are therefore given by a = 1 T V 1 1 = m s 2 (1 + 2r) s 2 (1 + 2r), b = 1T V 1 m = c = m T V 1 m = 3m2 s 2 (1 + 2r) + 2d2 s 2 (1 r). 1 s 2 (1 + 2r) 1, d s 2 (1 r) n. 3m s 2 (1 + 2r),

48 The frontier parameters are then r σ mv = a = s, µ 3 mv = b a = m, ν as = c b2 a = d 2 s 1 r, whereby the frontier is given by σ = σ f (µ) = s 1 + 2r r 2 ( ) 2 µ m for µ (, ). Notice that each (σ i, m i ) lies strictly to the right of the frontier because 1 + 2r 5 + r σ mv = σ f (m) = s < σ 3 f (m ± d) = s < s. 6 Notice that as r decreases the frontier moves to the left for µ m < 2 3 3d and to the right for µ m > 2 3 3d. d

49 The minimum volatility portfolio has distribution f mv = σ 2 mvv 1 1 = 1 3 1, while the distribution of the frontier portfolio with return rate mean µ is f f (µ) = f mv + µ µ mv ν 2 as V 1( m µ mv 1 ) = 1 3 µ m 2d µ m 2d Notice that the frontier portfolio will hold long postitions in all three assets when µ (m 3 2 d, m + 2 3d). It will hold a short position in the first asset when µ > m+ 3 2 d, and a short position in the third asset when µ < m 2 3 d. In particular, in order to create a portfolio with return rate mean µ greater than that of any asset contained within it you must short sell the asset with the lowest return rate mean and invest the proceeds into the asset with the highest return rate mean. The fact that f f (µ) is independent of V is a consequence of the simple forms of both V and m. This is also why the fraction of the investment in the second asset is a constant..

50 Remark. The frontier portfolios in the above examples are independent of all the parameters in V. While this is not generally true, it is generally true that they are independent of the overall market volatility. In particular, they are independent of the factor D(D 1) that was chosen to normalize V. Said another way, the frontier portfolios depend only upon the correlations c ij, the volatility ratios σ i /σ j, and the means m i. Moreover, the minimum volatility portfolio f mv depends only upon the correlations and the volatility ratios. Because markets can exhibit periods of markedly different volatility, it is natural to ask when correlations and volatility ratios might be relatively stable across such periods. Remark. The efficient frontier quantifies the relationship between risk and return that we mentioned earlier. A portfolio management theory typically assumes that investors prefer efficient frontier portfolios and will therefore select an efficient frontier portfolio that is optimal given some measure of the risk aversion of an investor. Our goal is to develop such theories.

51 Exercise. Consider the following groups of assets: (a) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009; (b) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007; (c) S&P 500 and Russell 1000 and 2000 index funds in 2009; (d) S&P 500 and Russell 1000 and 2000 index funds in On a single graph plot the points (σ i, m i ) for every asset in groups (a) and (c) along with the frontiers for group (a), for group (c), and for groups (a) and (c) combined. Do the same thing for groups (b) and (d) on a second graph. (Use daily data.) Comment on any relationships you see between the objects plotted on each graph. Exercise. Compute the minimum volatitly portfolio f mv for group (a), for group (c), and for groups (a) and (c) combined. Do the same thing for group (b), for group (d), and for groups (b) and (d) combined. Compare these results for the analogous groupings in 2007 and 2009.

52 Application: 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 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.

53 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!

54 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! Remark. 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.

55 Exercise. Select at least one index fund for each of the following asset classes: 1. stocks of large U.S. companies, 2. stocks of small U.S. companies, 3. stocks of large non-u.s. companies, 4. stocks of small non-u.s. companies, 5. U.S. corperate bonds, 6. U.S. government bonds, 7. U.S. municipal bonds, 8. non-u.s. bonds. Compute the efficient frontier for the years 2005, 2007, 2009, and What is the relationship of each of these market indices to the efficient frontier for these years?

56 4. 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 shares in a fund that holds only safe securities such as US Treasury Bills, Notes, or Bonds. 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.

57 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.

58 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 f rf + N i=1 f i = 1, f rf < 1 if any f j 0. The portfolio is rebalanced at the start of each day so that b rf (d) Π(d 1) = f n rf, i (d) s i (d 1) = 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.

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

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

61 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 the distribution f 0. Notice that 1 T f = 1 f rf > 0 because f 0. Define f = f 1 T f. Notice that 1 T f = 1. Let µ = m T f and σ = f T V f. Then ( σ, µ) is the point in the σµ-plane associated with the Markowitz portfolio without risk-free assets that is characterized by the distribution f. Because µ = ( 1 1 T f ) µ rf + 1 T f µ, σ = 1 T f 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.

62 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 µ ) where φ > 0. If a portfolio corresponding to ( σ, µ) has distribution f then the point on the half-line given by φ corresponds to the portfolio with distribution f = φ f. This portfolio allocates 1 1 T f = 1 φ of its value to the risk-free asset. 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.

63 Efficient Frontier. We now use the capital allocation line construction to see how the efficient frontier is modified by including risk-free assets. Recall that the efficient frontier for portfolios that contain no risk-free assets is given by µ = µ mv + ν as σ 2 σ 2 mv for σ σ mv. Every point ( σ, µ) on this curve has a unique frontier portfolio associated with it. Because µ rf < µ mv there is a unique half-line that starts at the point (0, µ rf ) and is tangent to this curve. Denote this half-line by µ = µ rf + ν tg σ for σ 0. Let (σ tg, µ tg ) be the point at which this tangency occurs. The unique frontier portfolio associated with this point is called the tangency portfolio associated with the risk-free asset; it has distribution f tg = f f (µ tg ). Then the appropriate capital allocation line will be part of the efficient frontier.

64 The so-called tangency parameters, σ tg, µ tg, and ν tg, can be determined from the equations µ tg = µ rf + ν tg σ tg, µ tg = µ mv + ν as σ 2 tg σ 2 mv, ν tg = ν as σ tg σtg 2 σ mv 2. The first equation states that (σ tg, µ tg ) lies on the capital allocation line. The second states that it also lies on the efficient frontier curve for portfolios that contain no risk-free assets. The third equates the slope of the capital allocation line to that of the efficient frontier curve at the point (σ tg, µ tg ). By using the last equation to eliminate ν tg from the first, and then using the resulting equation to eliminate µ tg from the second, we find that µ mv µ rf ν as = σ 2 tg σ 2 tg σ 2 mv σ 2 tg σ 2 mv = σ 2 mv σ 2 tg σ 2 mv.

65 We thereby obtain ( σ tg = σ νas σ mv 1 + mv µ mv µ rf ν tg = ν as ) 2, µ tg = µ mv + ν 2 as σ 2 mv µ mv µ rf, 1 + ( µmv µ rf ν as σ mv ) 2. The distribution f tg of the tangency portfolio is then given by f tg = f f (µ tg ) = f mv + µ tg µ mv ν 2 as V 1( m µ mv 1 ) = σmvv σ mv 2 V 1 µ mv µ rf = σ mv 2 V 1 ( m µrf 1 ). µ mv µ rf ( m µmv 1 )

66 Remark. The above formulas can be applied to either the safe investment or the credit line by simply choosing the appropriate value of µ rf. Example: One Risk-Free Rate Model. First consider the case when µ si = µ cl < µ mv. Set µ rf = µ si = µ cl and let ν tg and σ tg be the slope and volatility of the tangency portfolio that is common to both the safe investment and the credit line. The efficient frontier is then given by µ ef (σ) = µ rf + ν tg σ for σ [0, ). Let f tg be the distribution of the tangency portfolio that is common to both the safe investment and the credit line. The distribution of the associated portfolio is then given by f ef (σ) = σ σ tg f tg for σ [0, ), These portfolios are constructed as follows:

67 1. if σ = 0 then the investor holds only the safe investment; 2. if σ (0, σ tg ) then the investor places σ tg σ of the portfolio value in the safe investment, σ tg σ of the portfolio value in the tangency portfolio f σ tg ; tg 3. if σ = σ tg then the investor holds only the tangency portfolio f tg ; 4. if σ (σ tg, ) then the investor places σ σ tg of the portfolio value in the tangency portfolio f tg, by borrowing σ σ tg σ tg of this value from the credit line.