Upcoming SlideShare
×

# Mpt pdf

476 views

Published on

This slide set is a work in progress and is embedded in my Principles of Finance course site (under construction) that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/

Published in: Economy & Finance, Business
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
476
On SlideShare
0
From Embeds
0
Number of Embeds
156
Actions
Shares
0
6
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Mpt pdf

1. 1. Por\$olio  Theory
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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