Sharpe-optimal SPDR portfolios

Transcription

1 Sharpe-optimal SPDR portfolios or How to beat the market and sleep well at night by Vic Norton Bowling Green State University Bowling Green, Ohio USA

2 Abstract The Sharpe Ratio of an investment portfolio is, loosely speaking, the ratio of its reward to its risk. We seek portfolios of maximum Sharpe Ratio from a fixed universe of Exchange Traded Funds (Select Sector SPDRs). It is convenient to look at this problem in a geometric setting. Then a portfolio is identified with its risk vector in a high-dimensional Euclidean space, and the Sharpe Ratio of the portfolio is proportional to the cosine of the angle between the risk vector and an expected-reward axis. Now we seek to maximize this cosine (and thus the Sharpe Ratio) over a convex polytope of portfolios. The maximization is accomplished by a simplex-type algorithm using updated QR-factorizations.

3 Sharpe Ratio The Sharpe Ratio of an investment is the ratio of its expected reward to its risk. The higher the Sharpe ratio the better. William F. Sharpe introduced the Sharpe Ratio (by another name) in In 1990 Sharpe won the Nobel Prize for Economics for his Capital Asset Pricing Model, sharing the prize with Harry Markowitz and Merton Miller. Together, their work revolutionized the financial/business industries New York Times Almanac

4 Select Sector SPDRs The Select Sector SPDRs (pronounced spiders ) are ETFs (Exchange Traded Funds) that partition the S&P 500 (U.S. large-cap stocks) into 9 categories: XLY Consumer Discretionary XLP Consumer Staples XLE Energy XLF Financial XLV Health Care XLI Industrial XLB Materials XLK Technology XLU Utilities

5 Investment Strategy How to beat the market and sleep well at night. One investment strategy: Invest your money in the current ex post Sharpe-optimallong SPDR portfolio, SOLNG. Check your investment portfolio at the end of each week. When its Sharpe Ratio diverges from the Sharpe Ratio of the ex post SOLNG portfolio by more than 30%, reinvest in the ex post SOLNG portfolio. Other strategies might include deleveraging with the base fund or reinvesting in the Sharpe-optimal-long-short portfolio SOLS0, especially when the prospects for pure long investment are suspect. Sharpe-Optimal SPDR Portfolios Website:

6 Investment Portfolio SOLNG

7 Investment Portfolio SOLNGB

8 Investment Portfolio SOMIX

9 Overwrought Portfolio SOLNGW

10 Investment Portfolio Statistics For the 418 week period from 3-Jan-2000 to 7-Jan-2008 Portfolio FinalVal CmpdRtn StdvRtn ShrpRat Reinv SOLNG % 20.00% SOLNGB % 11.15% SOMIX % 14.10% SOLNGW % 20.17% SPY (S&P 500) % 16.84% Base (3M Treas) % 0.24% undef 1

11 Portfolio Calculator (Saturday, 29-Dec-2007) XLE: $80.37, XLI: $39.34, XLB: $

12 Portfolio Calculator (Saturday, 29-Dec-2007)

13 Mathematics Begins (more or less) >Agewmètrhtoc mhdeèc eêsðtw Let no one ignorant of geometry enter here inscription above the gateway to Plato s Academy

14 Returns risky fund return vector: base fund return vector: r R m r 0 R m Examples r i = p i p i+13 1 r 0,i = k=0 y 3M i+k 13-week simple return. Here p i is the adjusted closing price of the risky fund at the end of week i (with week indices increasing toward the past). 13-week average return. Here y 3M i is the average annualized yield on a 3-month Treasury bill (secondary market, discount basis) over week i.

15 Data Historical Adjusted Closing Prices Historical Interest Rates

16 Reward and the Sharpe Ratio weights: µ i (i = 1,..., m), µ i > 0, µi = 1 (fixed) reward vector: w = r r 0 R m expected reward: w = µ i w i variance of reward: v = µ i (w i w) 2 standard deviation of reward: σ = v (risk) Sharpe Ratio: s = w σ ( ) expected reward risk

17 Weights µ i (i = 1,..., m), µ i > 0, µi = m = 39 1/

18 Root-Weighted Reward root-weights: β i = µ i (i = 1,..., m) root-weight vector: β = [β 1,..., β m ] T R m, β = 1 root-weight matrix: B = diag β R m m root-weighted reward vector: z = B w expected reward: w = β T z risk vector: f = z β w variance of reward: v = f 2 standard deviation of reward: σ = f (risk) Sharpe Ratio: s = w σ ( ) expected reward risk

19 (Root-Weighted) Reward Space Expected Reward w= β T z Risk Hyperplane 1 β 0 z=b w f=z βw σ = f Sharpe Ratio: s = w σ ( ) expected reward risk

20 Risk Space and Expected Reward Risky fund universe: F 1,..., F n (9 Select Sector SPDRs) Fund risk vectors: f 1,..., f n R m (linearly independent) Risk matrix: F = [f 1,..., f n ] R m n (rank n) Risk space: F = span(f 1,..., f n ) = range(f) (dimension n) Fund expected rewards: Expected reward vector: satisfies w 1,..., w n w 1 e = F(F T F) 1. F w n e T [f 1,..., f n ] = [w 1,..., w n ] Thus expected reward is a linear function of risk: w = e T f ( f = F p )

21 Portfolios of Risky Funds Portfolio: p = [p 1,..., p n ] T, n j=1 p j = 1 p j is the signed proportion of fund F j in the portfolio: p j > 0: p j < 0: p j = 0: Example shares of fund F j bought long shares of fund F j sold short no investment in fund F j Fund Shares Price Value Portfolio F $20.10 $10, % F $37.40 $11, % F 3 0.0% F $32.50 $26, % Nominal Value $47, %

22 Simple Necessity Portfolio: p = [p 1,..., p n ] T, n j=1 p j = 1 Simple rates work well with portfolios: 1 + r t = n j=1 p j [1 + sign(p j )r j t] where r = n j=1 p j r j Compound rates do not: exp(r t) = n j=1 p j exp[sign(p j )r j t] where r = 1 { n } t log j=1 p j exp[sign(p j )r j t] Here r is the rate of return on the portfolio (simple or compound) resulting from the individual fund rates r 1,..., r n.

23 Portfolio in Risk Space w= e T f ER=0 Expected Reward e 0 non-productive risk θ f=fp total risk P productive risk Sharpe Ratio: Productivity quotient: s = w σ = S(p) = et f f Q(p) = cos θ = = e cos θ productive risk total risk

24 Sharpe-Optimal Portfolios The Sharpe-optimal long-short portfolio SOLS1: SOLS1 = x / w 1 n j=1 x j, x = (F T F) 1. Moreover S(SOLS1) = e. w n The Sharpe-optimal long portfolio SOLNG: Maximize cos θ = et f e f n for f = F p, j=1 p j = 1, p j 0 (j = 1,..., n). Then SOLNG = p max, S(SOLNG) = e max(cos θ), and Q(SOLNG) = max(cos θ) = S(SOLNG)/S(SOLS1).

25 Productivity Quotient of SOLNG

26 Caveat Our definition of reward implies that short money received is invested in the base fund, just as base fund money is used for long investments. The corresponding Sharpe Ratio has been denoted by S, with Sharpe-optimal long-short portfolio SOLS1. This situation does not generally apply to small investors, who receive no interest on short money received. We denote the corresponding Sharpe Ratio by S 0, with Sharpeoptimal long-short portfolio SOLS0. Then S(SOLNG) S(SOLS0) S(SOLS1) S 0 (SOLS1) S 0 (SOLS0)

27 How to maximize a cosine by Vic Norton Bowling Green State University Bowling Green, Ohio USA mailto:vic@norton.name

28 Abstract Given a unit m-vector u and a nonzero m n matrix F, we describe a simplex-type algorithm, using updated QR factorizations, to maximize the cosine function cos u (P) = (u T P)/ P on the convex hull of the columns of F. The maximizing point P max is returned in the form P max = F J p, where J is a sequence of n J distinct indices from {1,..., n}, the n J columns of F J are linearly independent, and p is an n J -vector satisfying p = 1, p > 0.

29 Picture of the Algorithm Scenario 1

30 Picture of the Algorithm Scenario 2

31 End of Talk This is the end of my talk (except for some technical mumbo jumbo that I might gloss over). Thanks for listening!

32 QR-Decomposition QR = F J = [A, B, C] the QR-decomposition of F J : Q T Q = I, R upper-triangular, nonsingular u + = F + J u = R 1 Q T u the F J -coordinates of the projection of u onto the range of F J Q = F J (u + / u + ) the unique critical point of cos u restricted to the affine subspace spanned by the columns of F J

33 To Add a Vertex (e.g., at P 1 ) QR = F J = [A, B]. To add vertex (column) C: Set r = Q T C, x = C Qr, s = x, q = r/s. Then replace [ ] Q := Q q, R := To get QR = F J = [A, B, C]. [ ] R r 0 T, J := [J, index of C], s

34 To Remove a Vertex (e.g., at P 2 ) QR = F J = [A, B, C]. To remove vertex (column) A: Let Q 0 = Q and set J := [index of B, index of C]. Then R 0 R 1 R 2 F J = [B, C] = Q 0 = Q 1 0 = Q 2 0, where Q j = Q j 1 G T j, R j = G j R j 1, and G T j G j = I (j = 1, 2). (The G j are called Givens rotations. ) Set Q := Q 2 with the last column removed and R := R 2 with the last row removed. Then QR = F J = [B, C].

35 Octave Code for Removal of a Vertex ## remove column i of nj i f i < nj R( 1 : nj, i : nj 1) = R( 1 : nj, i +1: nj ) ; J ( i : nj 1) = J ( i +1: nj ) ; ## do Givens rotations to remove subdiagonal ## elements of R and adjust Q accordingly for j = i : nj 1 [ cs, sn ] = givens (R( j, j ), R( j +1, j ) ) ; G = [ cs, sn ; sn, cs ] ; # Givens rotation R( j : j +1, j : nj 1) = G * R( j : j +1, j : nj 1); Q( :, j : j +1) = Q( :, j : j +1) * G ; endfor endif nj = 1;

36 Really the End Goodbye!!!