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- 2. The Five Pillars 2 Nobel Prize winner and former Univ. of Chicago professor, Merton Miller, published a paper called the “The History of Finance” Miller idenBfied five “pillars on which the field of finance rests” These include 1. Miller-‐Modigliani ProposiBons • Merton Miller 1990 and Franco Modigliani 1985 2. Capital Asset Pricing Model • William Sharpe 1990 3. Efficient Market Hypothesis • (Eugene Fama, Paul Samuelson, …) 4. Modern Por+olio Theory • Harry Markowitz 1990 5. OpBons • Myron Scholes and Robert Merton 1997
- 3. Learning ObjecBves ¨ Build a por[olio an opBmal por[olio of securiBes consistent with your expected risk and return requirements ¤ DiversificaBon is key ¤ Single, not mulBperiod, investment horizons n So can use r & d or α and δ for simple ¨ Understand ¤ Random variables with cross correlaBons ¤ matrix algebra and ¤ quadraBc opBmizaBon ¨ Note ¤ r and σ are used as generic symbols to represent expected (mean) return rate and standard deviaBon over the planning period n Can be conBnuously or discretely compounded, but must be consistent 3
- 4. 4 Por[olio of M Risky Assets ¨ Each asset has returns expected to be normally distributed ¨ The portfolio’s expected returns are also normally distributed ¨ A stock’s expected return might come from the CAPM model ¨ A bond’s expected return come from a similar model ¤ bexpected = rforecast + ( bhistorical - rhistorical ) Mi1 )σ,(r ii ≤≤ )σ,(r PP )rr(rr FMFE −⋅β+=
- 5. 5 Por[olio of M Risky Assets ¨ Expected variance for an asset is o`en assumed to be the historical variance ¨ In this topic we will also assume that the expected return is the long term historical average return ¨ What is the proper length of the historical record and the sampling frequency?
- 6. 6 A Por[olio With Two Risky Assets 0.50% 0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Std Dev Return (rA,σA) (rB,σB)
- 7. 7 A Por[olio With Two Risky Assets ¨ rP = wA·∙rA + wB·∙rB ¤ wA + wB =1 n requires that the por[olio is fully invested in the 2 assets A and B ¤ wA ≥ 0, wB ≥ 0 n prohibits short selling or borrowing an asset ¤ 1 ≥ wA, 1 ≥ wB n Restricts buying an asset on margin ABBABA 2 B 2 B 2 A 2 A 2 p ABBA 2 B 2 B 2 A 2 A 2 p ABBABB 2 BAA 2 A 2 p ρσσw2wσwσwσ σw2wσwσwσ σw2wσwσwσ ++= ++= ++= AAAA 2 A σσσσ ≡≡
- 8. 8 Por[olios With Two Risky Assets ¨ σA= 8.3% ¨ σB= 16.3% ¨ σAB = .004 ¨ rA =0.9% ¨ rB = 2.3% ¨ ρAB = .28 A AVBV AB 2 B 2 A AB 2 B AV w-‐1w 2σσσ )σ(σ w = −+ − = ABBABA 2 B 2 B 2 A 2 A 2 p ρσσw2wσwσwσ ++= 0.50% 0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Std Dev Return A B Minimum variance portfolio
- 9. 9 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 1.8% 2.0% 2.2% 2.4% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% Portfolio Std Dev Portfolio Return Por[olios With Two Risky Assets ρAB=1 ρAB=0 ρAB=-‐.5 ρAB=-‐1 A B ABBABA 2 B 2 B 2 A 2 A 2 p ρσσww2σwσwσ ⋅⋅⋅⋅⋅+⋅+⋅=
- 10. 10 Por[olios With Two Risky Assets 0.00% 0.25% 0.50% 0.75% 1.00% 1.25% 1.50% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Portfolio Std Dev Portfolio Return EFA AGG SPY DJP
- 11. 11 Two Risky and One Risk Free Asset 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 1.8% 2.0% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% Std Dev Return Asset B Min Variance Portfolio V risk free asset F Tangent Portfolio T Asset A ABA TT ABFBFA 2 AFA 2 BFA ABFB 2 BFA T w-‐1w σ)]r(r)r[(rσ)r(rσ)r(r σ)r(rσ)r(r w = ⋅−+−−⋅−+⋅− ⋅−−⋅− =
- 12. 12 Now Determine Your OpBmal Por[olio 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Std Dev Return Indifference curves A=2 , 4, 7 T: OpBmal Risky Por[olio F P: Your opBmal por[olio A B V
- 13. 13 Por[olio with 2 Risky Assets 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Std Dev Return Indifference curves A=4 T: OpBmal Risky Por[olio F P: Your opBmal por[olio A B V rCE
- 14. 14 Now Consider M > 2 Risky Assets 0.25% 0.50% 0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% Extected Std Dev %/mo. Expeced Return %/mo Now where is the opBmal risky por[olios ? Symbol ri σ i IBM 1.07% 9.03% TM 0.92% 7.82% XOM 1.21% 5.25% BRK-‐B 1.06% 5.94% GE 0.79% 6.42% WMT 0.99% 7.30% C 0.96% 8.35% ORCL 2.36% 16.07%
- 15. 15 Compute rP and σP with M risky assets 1w0 i ≤≤ 1w M 1i i =∑= i M 1i iP rwr ∑= ⋅= ij M 1j ji M 1i 2 P σwwσ ⋅⋅= ∑∑ == ∑∑∑ ≠ === ⋅⋅+⋅= M ij 1j ijji M 1i M 1i 2 i 2 i 2 P σwwσwσ
- 16. 16 Now Use Array NotaBon For rP and σP ⎣ ⎦[ ]{ }jiji T2 P wσwwCwσ =⋅⋅= ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 2 MM2M1 2M 2 221 1M12 2 1 MMM2M1 2M2221 1M1211 σσσ σσσ σσσ σσσ σσσ σσσ C { }i σ=σ ⎣ ⎦i T σ=σ { }i r r = ⎣ ⎦i T rr = ij M 1j ji M 1i 2 P σwwσ ⋅⋅= ∑∑ ==
- 17. 17 Compute Covariance – Variance Matrix ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = NMN2N1 2M2221 1M1211 rrr rrr rrr R stocks 1 to M returns 1 to N N AA C T = ji ij ij σσ σ ρ ⋅ = N r r N 1k ki i ∑= = N )r(r σ N 1k 2 iki 2 i ∑= − = N )r)(rr(r σ N 1k jkjiki ij ∑= −− = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ −−− −−− −−− = MNM2N21N1 M2M222121 M1M212111 rrrrrr rrrrrr rrrrrr A
- 18. Compute Por[olio Return 18 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ = NMMN22N11 2MM222211 1MM122111 rwrwrw rwrwrw rwrwrw R ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑∑ = = = = = = = == ⋅= ⋅= ⋅= ⋅= =⋅ ⋅ = M 1i N 1k kii M 1i N 1k kii M 1i N 1k kiiP M 1i iiP N 1k ki N 1k kii i i rw N 1 r N 1 w r N 1 wr rwr r N 1 rw wN 1 r
- 19. 19 Example Matrices Covariance matrix CorrelaBon matrix Visualize IBM TM XOM BRK-‐B GE WMT C ORCL IBM 0.00815 0.00162 0.00149 0.00046 0.00226 0.00150 0.00394 0.00483 TM 0.00162 0.00612 0.00054 0.00084 0.00224 0.00146 0.00205 0.00341 XOM 0.00149 0.00054 0.00276 0.00053 0.00056 0.00010 0.00111 0.00052 BRK-‐B 0.00046 0.00084 0.00053 0.00353 0.00139 0.00151 0.00174 -‐0.00066 GE 0.00226 0.00224 0.00056 0.00139 0.00412 0.00185 0.00237 0.00416 WMT 0.00150 0.00146 0.00010 0.00151 0.00185 0.00533 0.00270 0.00299 C 0.00394 0.00205 0.00111 0.00174 0.00237 0.00270 0.00697 0.00231 ORCL 0.00483 0.00341 0.00052 -‐0.00066 0.00416 0.00299 0.00231 0.02582 IBM TM XOM BRK-‐B GE WMT C ORCL IBM 1.00 0.23 0.31 0.09 0.39 0.23 0.52 0.33 TM 0.23 1.00 0.13 0.18 0.45 0.26 0.31 0.27 XOM 0.31 0.13 1.00 0.17 0.17 0.03 0.25 0.06 BRK-‐B 0.09 0.18 0.17 1.00 0.37 0.35 0.35 -‐0.07 GE 0.39 0.45 0.17 0.37 1.00 0.39 0.44 0.40 WMT 0.23 0.26 0.03 0.35 0.39 1.00 0.44 0.26 C 0.52 0.31 0.25 0.35 0.44 0.44 1.00 0.17 ORCL 0.33 0.27 0.06 -‐0.07 0.40 0.26 0.17 1.00 ji ij ij σσ σ ρ ⋅ =
- 20. 20 More CorrelaBon Examples Yahoo Finance Frequency Number of samples SPX -‐ VIX Correlation Daily 2512 -‐0.75 Weekly 520 -‐0.75 Monthly 120 -‐0.69 DJC TYX IRX SPX DJC -‐0.02 1.00 0.05 0.25 TYX 1.00 -‐0.02 0.11 0.10 IRX 0.11 0.05 1.00 0.09 SPX 0.10 0.25 0.09 1.00 USO DBA GLD SPX USO 1.00 0.27 0.40 -‐0.28 DBA 0.27 1.00 0.51 0.07 GLD 0.40 0.51 1.00 -‐0.21 SPX -‐0.28 0.07 -‐0.21 1.00
- 21. 21 More CorrelaBon Examples XLE BBH XLV XLF IGW UTH XLP IYR SPX Energy 1.00 0.11 0.18 0.26 0.36 0.65 0.13 0.31 0.57 Biotech 0.11 1.00 0.61 0.35 0.27 0.09 0.35 0.31 0.43 Healthcare 0.18 0.61 1.00 0.65 0.37 0.29 0.62 0.52 0.70 Financial 0.26 0.35 0.65 1.00 0.48 0.45 0.70 0.60 0.86 Semiconductors 0.36 0.27 0.37 0.48 1.00 0.32 0.22 0.28 0.67 Utilities 0.65 0.09 0.29 0.45 0.32 1.00 0.39 0.52 0.60 Consumer Staples 0.13 0.35 0.62 0.70 0.22 0.39 1.00 0.54 0.70 Real Estate 0.31 0.31 0.52 0.60 0.28 0.52 0.54 1.00 0.63 SPX 0.57 0.43 0.70 0.86 0.67 0.60 0.70 0.63 1.00 EWH EWQ EWG EWJ EWZ EWD EWC EWA SPX Hong Kong 1.00 0.62 0.63 0.53 0.45 0.60 0.50 0.58 0.67 France 0.62 1.00 0.89 0.54 0.54 0.81 0.62 0.63 0.79 Germany 0.63 0.89 1.00 0.54 0.54 0.81 0.63 0.60 0.79 Japan 0.53 0.54 0.54 1.00 0.36 0.53 0.47 0.50 0.55 Brazil 0.45 0.54 0.54 0.36 1.00 0.49 0.58 0.55 0.53 Sweden 0.60 0.81 0.81 0.53 0.49 1.00 0.61 0.61 0.73 Canada 0.50 0.62 0.63 0.47 0.58 0.61 1.00 0.66 0.64 Australia 0.58 0.63 0.60 0.50 0.55 0.61 0.66 1.00 0.60 United States 0.67 0.79 0.79 0.55 0.53 0.73 0.64 0.60 1.00
- 22. 22 CorrelaBon Between Por[olios A & B wT = ⎣ wIBM wTM wXOM wBRK-‐B wGE wWMT wC wORCL ⎦ rT = ⎣ rIBM rTM rXOM rBRK-‐B rGE rWMT rC rORCL ⎦ Example: Por[olio A has weight vector a and is half TM and half GE aT = ⎣ .0 .5 .0 .0 .5 .0 .0 .0 ⎦ ij M 1j ji M 1i AB σbaσ ⋅⋅= ∑∑ == i M 1i iA rar ⋅= ∑= i M 1i iB rbr ⋅= ∑=
- 23. 23 Diversifiable Risk 2 σ σ ρ ρσ2 ⋅ is the avg var of the M assets is the avg std dev of the M assets is the avg corr between the M assets is the avg cov between the M assets 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 35 40 M M 1 M 1M− ∑∑∑ ≠ === ⋅⋅+⋅= M ij 1j ijji M 1i M 1i 2 i 2 i 2 P σwwσwσ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⋅+⋅= ∑∑∑ ≠ === M ij 1j 2 ij M 1i M 1i 2 i2 P M σ M 1 M σ M 1 σ ρσ M 1)(M σ M 1 σ 222 P ⋅⋅ − +⋅= ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ −⋅ ⋅ − +⋅= ∑∑∑ ≠ === M ij 1j 2 ij M 1i M 1i 2 i2 P 1)(MM σ M 1M M σ M 1 σ
- 24. 0% 5% 10% 15% 20% 25% 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 Number of Assets, M Por[olio Std Dev -‐0.50 -‐0.25 0.00 0.25 0.5 0.75 1.00 Avg Std Dev = 20% 24 Diversifiable Risk 10%1%σ 1%.25.2.2σ ρσσ P 2 P 22 P =⇒ =⋅⋅⇒ ⋅⇒ ρσσσ M ρσ M 1)(M σ M 1 σ 222 P 222 P ⋅⋅+⋅⇒ ∞→ ⋅⋅ − +⋅= 10 Diversifiable risk for ρ=0.25 Non-‐diversifiable risk for ρ=0.25 ρ
- 25. 25 OpBmal Por[olios of M Risky Assets 0.25% 0.50% 0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% Expected Std Dev %/mo. Expeced Return %/mo IBM TM XOM BRK-‐B GE WMT C ORCL
- 26. 26 Find the Minimum Risk Por[olio via QuadraBc OpBmizaBon ¨ Minimize this quadraBc objecBve funcBon ¨ Subject to these linear constraints ¨ Solve Using Excel Solver 1w 0 1w i M 1i i ≥≥ =∑= ij M 1j ji M 1i 2 V σwwσ ⋅⋅= ∑∑ == Symbol r σ Equal 1.17% 5.04% Min Risk 1.09% 3.81% SPX 0.38% 4.25% IBM TM XOM BRK-‐B GE WMT C ORCL 1.4% 9.4% 43.1% 23.3% 8.6% 13.3% 0.0% 0.9% i M 1i iV rwr ⋅= ∑=
- 27. 27 Find the Minimum Risk Por[olio via QuadraBc OpBmizaBon 0.25% 0.50% 0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% Expected Std Dev % Expected Return %. IBM TM XOM BRK-‐B GE WMT C ORCL Equal Min Risk SPX V
- 28. 28 0.25% 0.50% 0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% Expected Return %. Expected Std Dev % IBM TM XOM BRK-‐B GE WMT C ORCL Equal Min Risk SPX ¨ Determine the other por[olios ¨ Minimize ¨ Subject to these constraints Find the other opBmal risky por[olios %36.2r 1.09% * P << ij M 1j ji M 1i 2 P σwwσ ⋅⋅= ∑∑ == 1w 0 1w i M 1i i ≥≥ =∑= i M 1i i * P rwr ⋅= ∑=
- 29. 29 0.25% 0.50% 0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% Expected Return %. Expected Std Dev % IBM TM XOM BRK-‐B GE WMT C ORCL Equal Min Risk SPX ¨ Determine the other por[olios ¨ Minimize ¨ Subject to these constraints Por[olio with more than 2 risky assets ij M 1j ji M 1i 2 P σwwσ ⋅⋅= ∑∑ == 1w 0 1w i M 1i i ≥≥ =∑= i M 1i i * P rwr ⋅= ∑=
- 30. 30 Find the other opBmal risky por[olios Port Mean Port Std Dev IBM TM XOM BRK-‐B GE WMT C ORCL 1.09% 3.81% 1.4% 9.4% 43.1% 23.3% 8.6% 13.3% 0.0% 0.9% 1.24% 4.00% 0.0% 4.4% 48.1% 29.1% 0.0% 8.9% 0.0% 9.4% 1.44% 5.00% 0.0% 0.0% 49.7% 27.1% 0.0% 0.0% 0.0% 23.1% 1.55% 6.00% 0.0% 0.0% 48.7% 19.4% 0.0% 0.0% 0.0% 31.9% 1.64% 7.00% 0.0% 0.0% 16.8% 0.0% 0.0% 0.0% 83.2% 0.0% 1.73% 8.00% 0.0% 0.0% 46.7% 6.8% 0.0% 0.0% 0.0% 46.4% 1.82% 9.00% 0.0% 0.0% 45.8% 1.1% 0.0% 0.0% 0.0% 53.1% 0.25% 0.50% 0.75% 1.00% 1.25% 1.50% 1.75% 2.00% 2.25% 2.50% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% Expected Return %. Expected Std Dev % IBM TM XOM BRK-‐B GE WMT C ORCL Equal Min Risk SPX
- 31. 31 One Risk Free Asset & M Risky Assets ¨ The tangency por[olio is the opBmal risky por[olio (asset). ¨ The opBmal risky asset is dependent on the return of the risk free asset, but is independent of the investor’s risk preference ¨ The slope of the CAL line is the called the “Sharpe raBo” and has the steepest slope of any line connecBng the risk free asset and a tangency por[olio on the efficient fronBer ¨ A por[olio containing the risk free asset and the opBmal risky asset is opBmal for the investor ¨ The allocaBon of investor funds between the risk free and risky asset depends on the investor’s astude towards risk. ¨ Extension of the CAL beyond the opBmal risky asset requires the investor to short or borrow the risk free asset. ¤ In this case the risk free asset weight will be negaBve and the weight for the opBmal risky asset will be greater than 1. ¤ For the CAL to be straight beyond the opBmal risky asset, the borrowing rate must equal the risk free rate.
- 32. 32 EssenBal Concepts ¨ Asset and por[olio returns other than the risk free asset are modeled as normally distributed random variables ¨ This topic uses historical staBsBcs as expected staBsBcs for simplicity; however, this is not always a good assumpBon. ¤ However, historical variances and covariances are quite stable unless a firm undergoes significant changes to its business or financial model. ¨ Lack of correlaBon between asset returns reduces por[olio risk. ¨ In the case of more than two risky assets, opBmal por[olios lie along a curve called the efficient fronBer (of opBmal risky por[olios) ¨ When M is large, covariance terms dominate the calculaBon of por[olio variance and thus consBtute non-‐diversifiable risk ¨ Por[olio risk can be reduced by diversificaBon i.e., by including non-‐correlated assets ¨ The efficient fronBer is computed by sequenBal applicaBon of quadraBc programming