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This training material is the property of the International Monetary Fund (IMF) and is intended for use in IMF Institute for Capacity Development (ICD) courses. Any reuse requires the permission of the ICD. Financial Market Analysis (FMAx) Module 6 “Asset Allocation and Diversification”

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Page 1: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

This training material is the property of the International Monetary Fund (IMF) and is intended for use in IMF Institute for Capacity Development (ICD) courses.Any reuse requires the permission of the ICD.

Financial Market Analysis (FMAx)Module 6

“Asset Allocation and Diversification”

Page 2: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Preamble:  Why  should  you  care  about  portfolio  allocation?§ You might be an investor.

§ Your institution might be an investor.

§ As a policymaker, you’ll be interested in investor behavior.

§ Your country may be interested in what drives international investment.

Page 3: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Preamble:In  this  module  we  will…1. Review statistical concepts related to return and risk.

2. Emphasize the importance of correlation.

3. Explore how to choose an “optimal portfolio”.Of 2 assetsOf n > 2 assets

4. Do the same for an international portfolio.

Page 4: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

An  Introduction-­ Concepts  -­

Portfolio Theory - Harry Markowitz (1952)

Investments are compared in terms of trade-off between…§ Risk (variance)§ Return (expected reward)

An investor who cares only about risk and return, will always prefer…§ Highest mean return for given amount of risk; and§ Lowest risk given the mean return.

Page 5: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

An  Introduction-­ Concepts  -­Insights from H. Markowitz:

§ Diversification does not rely on returns being uncorrelated, but rather on having them be imperfectly correlated.

§ Risk reduction from diversification is limited by the extent to which returns are correlated.

Page 6: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

An  Introduction-­ Review  of  Statistics  -­Expected Returns:

Using historical data:

( ) ( )( )( ) ( )[ ] 11.....1111

321 -++++== TTg rrrrrrE

( )T

rrrrrrE Ta

++++==

....321

Page 7: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

An  Introduction-­ Review  of  Statistics  -­Risk: (Variance or Standard Deviation of Returns)

Using historical data:

( ) ( )( ) ( )( ) ( )( )T

rErrErrErrrVar Tg

222

21 ... -+-+-

==

Page 8: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

An  Introduction-­ Review  of  Statistics  -­Correlation: (Degree of co movement between two variables.)

( )( )( ) ( )( )

( )

, ,1,

,

T

A t A B t Bt

A B

A BAB

A B

r E r r E rCov r r

TCov r r

rs s

=

- -=

=

åCorrelation coefficient ranges from:

+ 1 (perfect co-movement)0 (variables are independent)- 1 (perfect negative correlation)

Page 9: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Introducing  Two  Risky  Assets

Define Portfolio P containing two risky assets (portfolios):Bonds (D) and Equity (E), Weights wD and wE

Expected rate of return:

Correlation coefficient rDE :

( ) ( ) ( )EEDDP rEwrEwrE +=

( )ED

EDDE

rrCovss

r ,=

Page 10: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Introducing  Two  Risky  Assets

Risk of P:

Question: What is the largest possible value for rDE?When rDE =1 (perfect correlation):

( ) ( )[ ]EDDEEDEEDD

PPP

wwwwrErEPVar

ssrss

s

22222

22

++=

-==

( )( )

EEDDP

EEDD

EDEDEEDDP

wwww

wwwwPVar

sssss

sssss

+=Þ+=

++==2

22222 2

Page 11: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Introducing  Two  Risky  Assets

Unless correlation between D and E is perfect:

Note that:§ Expected return of P is unaffected by the correlations§ We often call the difference between the weighted average of individual s’s

and sp the “gains from diversification”.

§ Hedge asset: negative correlation with other assets (rDE < 0)

EEDDP ww sss +<

Page 12: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Introducing  Two  Risky  Assets

In the extreme case of a perfect hedge (rDE = -1):

Can construct a zero risk portfolio:

)()( 22

EEDDp

EEDDp

wwabsww

sss

sss

-=

-=

0, 1

, 1

p D D E E D E

E DD E D

D E D E

w w w w

w w w

s s s

s ss s s s

= - = + =

Þ = = = -+ +

Page 13: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Graphical  Representation-­ Two-­Assets  Portfolio  -­

Example:

E(rD) = 2%, E(rE) = 6%, sD = 5%, sE = 10%, rDE = 0.2

Return and Risk:

2 2 2 2 2

2 2

( ) 0.02 0.06

(0.05) (0.10) 2 0.05 0.10 0.2

0.025 0.01 0.002

p D E

p D E D E

D E D E

E r w w

w w w w

w w w w

s

= +

= + + × × × ×

= + +

Page 14: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Graphical  Representation-­ Two-­Assets  Portfolio  -­

Minimizing Risk:

Solving the above for wD, wE

2 2 2 2 2 2. . : 1,or 1

p D D E E D E D E

D E E D

Min Min w w w ws t w w w w

s s s rs s= + +

+ = = -

2

min 2 2

min min

( )2

( ) 1 ( )

E DE D E

E D DE D E

w D

w E w D

s r s ss s r s s

-=

+ -= -

Page 15: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Graphical Representation- Two-Assets Portfolio -

Portfolio  100%  in  D

s

Portfolio  100%  in  E

r = 0.20

r = 1.0

r = -1.0

E(rp)

Page 16: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Graphical  Representation-­ Two-­Assets  Portfolio  -­

Benefits from Diversification:

§ Come from imperfect correlation between returns.

§ The smaller r, the greater the benefits from diversification.

§ If r = 1, no risk reduction is possible.

§ Adding extra assets with lower correlation with the existing ones decreases total risk of the portfolio.

§ Diversification can eliminate some, but not all risk.

Page 17: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Comparing  Different  Portfolios-­ Two-­Assets  -­

The Risk-Free Asset:IF it is possible to borrow/lend at the risk-free rate rf,

THEN the portfolio selection problem is to maximize the excess return over the risk-free rate, for a given amount of risk:

( ) ( )2 2 2 2

( ) ( )( )

2. . : 1

D D f E E fp f

p D D E E D E D E

D E

w E r r w E r rE r rMax

w w w ws t w w

s s s rs s

- + --=

+ +

+ =

Page 18: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Comparing  Different  Portfolios-­ Two-­Assets  -­

Our Numerical Example:§ Assume rf =0.9%

§ Define A as the minimum-variance portfolio obtained earlier(wD = 0.84, wE = 0.16)

§ “Capital allocation line” (CAL): combinations of A and the risk-free asset.

§ Slope of CAL = “reward-to-variability”, or Sharpe ratio:

( ) 2.57 0.9 0.3504.78

A fA

A

E r rS

s- -

= = =

Page 19: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Comparing  Different  Portfolios-­ Two-­Assets  -­

Consider an alternative portfolio B:WD = 0.65, WE = 0.35Is it better than portfolio A?

The optimal portfolio will be such that the reward-to-variability ratio is maximized (depends on rf):

( ) 5.2 0.9 0.4783.4

B fB

B

E r rS

s- -

= = =

1

( ). . 1

i

Np f

p iw ip

E r rMax S s t w

s =

-= =å

Page 20: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Comparing Different Portfolios- Two-Assets -

E(rp)

sP

rfA

B

CALB

CALA

Page 21: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Selecting  the  Best  Portfolio-­ Two-­Assets  -­

The Optimization Problem:

After some algebra, and using wE = 1 - wD :

( ) ( )2 2 2 2

( ) ( )( )

2. . : 1

D D f E E fp f

p D D E E D E D E

D E

w E r r w E r rE r rMax

w w w ws t w w

s s s rs s

- + --=

+ +

+ =

( ) ( )( ) ( ) ( )

2*

2 2

( ) ( ) cov( , )( ) ( ) ( ) ( ) cov( , )

D f E E f D ED

D f E E f D D f E f D E

E r r E r r r rw

E r r E r r E r r E r r r r

s

s s

- - -=

- + - - - + -

Page 22: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Selecting  the  Best  Portfolio-­ Two-­Assets  -­

Expressing the Numerator in Matrix Notation:

Let’s call the numerator of (E1) a column vector z, which will be:

( ) ( ) ( )

2

2*

* 2 2

( )cov( , )( )cov( , )

( ) ( ) ( ) ( ) cov( , )

D fE D E

E fD E DD

E D f E E f D D f E f D E

E r rr rE r rr rw

w E r r E r r E r r E r r r r

ss

s s

-é ùé ù-ê úê ú -é ù -ë û ë û=ê ú - + - - - + -ë û

( )

1

2

1

2

2

2

1

( )cov( , )( )cov( , )D fE D E

E fD E D

f

E r rz r rE r rz r r

zS R R

z

ss

-

-é ùé ù-é ù= ê úê úê ú --ë û ë û ë û

é ù= -ê ú

ë û

(E1)

(E2)

Page 23: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Selecting  the  Best  Portfolio-­ Two-­Assets  -­

You can verify that the denominator of (E1) is equal to the sum of the z’s.

Therefore, the solution to the weights of the optimal portfolio P* is:

1*

2*

1 2

D

E

zzw

z zw

é ùê úé ù ë û=ê ú +ë û

(E3)

Page 24: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Selecting  the  Best  Portfolio-­ Two-­Assets  -­

Example:

This is the “tangency portfolio” P*; no other portfolio achieves a higher reward-to-variability (Sharpe ratio) with respect to the risk-free rate.

* *

* * *

0.34, 0.66( ) 4.7%, 7.2%, 0.524D E

p p p

w wE r Ss

= == = =

Page 25: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Selecting  the  Best  Portfolio-­ Two-­Assets  -­

Questions:

If the risk-free rate declines (monetary loosening), what happens to…§ the composition of the optimal portfolio (wD and wE)?§ its mean return?§ its risk?§ its Sharpe ratio?

Page 26: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Selecting  the  Best  Portfolio-­ Two-­Assets  -­

Answers:

If the risk-free rate declines (monetary loosening), then…

§ Optimal portfolio P* shifts toward the Southwest: ↑wD and ↓wE

§ Both mean return and risk decline§ Sharpe ratio increases

Page 27: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Recap  of  Optimal  Portfolio-­ Two-­Assets  -­

Main Ideas:§ Investor chooses the combination of D, E that maximizes reward-to-variability

relative to the risk-free rate.

§ This reward to variability = Sharpe ratio = slope of the CAL.

§ Two methods…

§ Algebraic (matrix solution)

§ Numerical (using Solver)

§ Recall: benefits to diversification depend on (imperfect) correlation between Dand E.

( )1

2

1f

zS R R

z-é ù

= -ê úë û

1*

2*

1 2

D

E

zzw

z zw

é ùê úé ù ë û=ê ú +ë û

Page 28: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Recap  of  Optimal  Portfolio-­ Two-­Assets  -­

Questions:

If the correlation between D and E increases, what happens to…§ the composition of the optimal portfolio (wD and wE)?§ its mean return?§ its risk?

Page 29: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Recap  of  Optimal  Portfolio-­ Two-­Assets  -­

Answers:

If the correlation between D and E increases, then:§ the optimal portfolio P* shifts to the Northeast: ↓wD and ↑wE

§ (in this case, it means even shorting D) § Both mean return and risk increase§ Sharpe ratio increases slightly (from 0.450 to 0.454)

Page 30: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Completing  the  Investor  Decision-­ Two-­Assets  -­

What next?

We have (1) the tangency portfolio P* and (2) the riskless asset.But each investor must now decide how much to invest in each.

§ Will depend on personal preference (risk aversion or tolerance).

This can be expressed by a utility function:25.0)( CC ArEU s-=

where A = degree of risk aversion;if A = 0, risk-neutral

Page 31: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Completing  the  Investor  Decision-­ Two-­Assets  -­

Individual Investor Choice:Maximize utility subject to the tangency portfolio and the riskless asset, to obtain the proportions to be invested in each (wP* and wrf).

The resulting portfolio will be called C.

In our example, assuming risk aversion A = 9.4, solve for wP* :

( ) ( )

2

2 2

( ) 0.51 0.5

C C

P P f P P P

MaxU E r Aw E r r w Aw

s

s

= -

= + - -

0.0466 0.009 78%2 0.5 9.4 0.0051Pw

-= =

´ ´ ´

Page 32: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Completing the Investor Decision- Two-Assets -

E(rp)

sP

rf

P*

CAL

C

wP*=100%

wP*=78%

U

Page 33: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Completing  the  Investor  Decision-­ Two-­Assets  -­

Separation Property:§ Technical information guides the decision to choose the optimal portfolio

of risky assets P*:§ Mean returns, relative to risk-free rate§ Volatilities§ Correlation

§ The choice of ultimate investor position (how much in P*, how much in rf) depends on individual preferences (A).

§ The two decisions are separate.

Page 34: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Generalizing  to  n  Assets-­ Part  1  -­

Maximize reward-to-variability, subject to all weights summing to 1.The solution, wi

*, is the same matrix as before, now generalized to N assets:

N: # risky assets

R: The column vector: expected returns

rf: risk-free rate

w: The column vector: portfolio sharesS: NxN variance-covariance matrix

{1

( , )( ,1) ( ,1)

*

[ ]fN NN N

ii

z S R r

zwz

-ì = -ïïíï =ïî å

1 2 3 14 2 43

Page 35: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Generalizing  to  n  Assets-­ Part  1  -­

Finding the optimal portfolio, step-by-step:#1 - Construct an Excess Return (ER) matrix for the N assets.

,1 ,1 ,1

,2 ,2 ,2

, , ,

(T rows, N columns)

A A B B N N

A A B B N N

A T A B T B N T N

r r r r r rr r r r r r

ER

r r r r r r

- - -é ùê ú- - -ê ú=ê úê ú- - -ê úë û

LL

M M O ML

1 4 4 4 4 4 4 2 4 4 4 4 4 43T: # observations

N: # risky assets

Page 36: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Generalizing  to  n  Assets-­ Part  1  -­

Finding the optimal portfolio, step-by-step:#2 - Multiply ER by its transpose, divide by the number of time observations, to obtain the Variance-Covariance Matrix (S).

{ {

2, ,

2, ,

( , ) ( , )2( , )

, ,

(N rows, N columns)

( ) ( )( ) ( )

/

( ) ( )

rA A B A N

A B rB B NT

N T T N

N NA N B N rN

Cov r r Cov r rCov r r Cov r r

S ER ER T

Cov r r Cov r r

ss

s

é ùê úê ú= ´ = ê úê úê úë û

LL

M M O M1 4 2 43L

1 4 4 4 4 4 4 44 2 4 4 4 4 4 4 4 43

Page 37: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Generalizing  to  n  Assets-­ Part  1  -­

Finding the optimal portfolio, step-by-step:#3 - Find the inverse of S and multiply it by the difference between the mean returns and the risk-free rate (R - rf). This gives us the z vector.#4 - The optimal portfolio weights wi

* will be equal to the z vector divided by the sum of z’s.

{1

( , )( ,1) ( ,1)

*

[ ]fN NN N

ii

z S R r

zwz

-ì = -ïïíï =ïî å

1 2 3 14 2 43

Page 38: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Generalizing  to  n  Assets-­ Part  2  -­

With the optimal portfolio obtained (P*), compute the (1) expected or mean return, (2) variance, and (3) standard deviation.

{ { {* *( ,1)(1,1) (1, )

( ) TP P

NN

E r W R=

{ { { {2* * *

( , )(1,1) (1, ) ( ,1)

TP P P

N NN N

W S Ws =

{ {2

* *(1,1) (1,1)P Ps s=

Expected  or  Mean  Return:

Variance:

Standard  Deviation

Page 39: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Building  the  Entire  Frontier-­ n  Assets  -­

§ You have computed the optimal portfolio P*, which holds for a certain level of the risk-free rate.

§ Vary rf and compute a second optimal portfolio P**.§ Can choose any arbitrary rf

§ The frontier can be generated as a series of linear combinations of P* and P**.

Page 40: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Building  the  Entire  Frontier-­ n  Assets  -­

The mean return and standard deviation of each combination of P* and P**can then be computed:

Mean  Return:

Standard  Deviation:

Covariance between  two  market  portfolios:

*, ** * **

2 2 2 2 * *** **

( ) ( ) (1 ) ( )

(1 ) 2 (1 ) ( , )

P P P P

EP P P

E r E r E r

Cov P P

a a

s a s a s a a

= + -

= + - + -

{ { {* **

* **( , )(1, ) ( ,1)(1,1)

( , ) TP P

N NN N

Cov P P W S W=1 44 2 4 43

Page 41: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Building the Entire Frontier- n Assets -

E(r)

sP

Individual  Assets

rf1

rf2

Min  variance

Aggressive  investor

Conservative  investor

Page 42: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

International  Diversification-­ Part  1  -­

The main principles related to diversification within the domestic market continue to apply when diversifying internationally:

§ Including additional (international) assets that are imperfectly correlated with domestic assets will produce gains (risk reduction).

§ But not all risk can be eliminated.

§ Currency fluctuations introduce an additional element (Part 2).

Page 43: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio
Page 44: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

International  Diversification-­ Part  2  -­

Investing across international borders introduces additional risk-return elements coming from currency fluctuations.

What should an investor do?

1. Nothing.

2. Hedge their foreign currency exposure (in different ways).

Page 45: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

International  Diversification-­ Part  2  -­

Example: A Japanese investor wants to buy U.S. stock [Apple, Inc.].Must buy dollars today at the spot exchange rate (S0, ¥/$ ) in order to buy the shares at today’s price (PA, 0).

Today’s (Nov 30, 2015) cost (in ¥) is therefore: § PA,0 x S0

§ PA,0 = $118.88, S0 = 123.26 ¥/$

Cash flow in one year: if unhedged, can sell the shares and exchange US$ for ¥ at the spot rates:

§ PA,1 x S1

Page 46: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

International  Diversification-­ Part  2  -­

Example: A Japanese investor wants to buy U.S. stock [Apple, Inc.].Suppose Apple’s stock price rises by 5% in one year:

§ PA,1 = $118.88 x (1+ 0.05) = $124.82

Regarding the exchange rate, suppose there are two possible scenarios (6% depreciation or appreciation):

§ S1,d = 123.26 x (1 + 0.06) = 130.66 ¥/$§ S1,a = 123.26 x (1 - 0.06) = 115.86 ¥/$

Page 47: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

International  Diversification-­ Part  2  -­

Unhedged Return:

( )

,1 1 ,0 0

,0 0

1 0

0

: return on US stock in $: % change in the spot exchange rate (1$= Yen)

=

A AU

A

U A FX

A

FX

FX

P S P Sr

P S

r r rrr S

S SrS

æ ö´ - ´= ç ÷ç ÷´è ø» +

-

,124.82 130.66 118.88 123.26 11.3%

118.88 123.26U dr ´ - ´æ ö =ç ÷´è ø , ?U ar =

Page 48: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

International  Diversification-­ Part  2  -­

Full currency hedging:Each investment in a foreign stock is fully hedged by a forward position; the investor agrees today to sell at the 1-year forward rate.

F1,0 = 121.23 ¥/$

Cash flow (per share) one year from now:CF1,d = (118.88 x 121.23) + (124.82 – 118.88) x 130.66 = ¥15,188

CF1,a = (118.82 x 121.23) + (124.82 – 118.88) x 115.86 = ¥15,101

Page 49: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

International  Diversification-­ Part  2  -­

Hedged Return: Value in Japanese Yen of the American Stock at time 1 Initial Investment

,0 1,0 ,1 ,0 1 ,0 0

,0 0

Initial Investment

1,0 1

0

( )

Forward return

A A A AH

A

H U

P F P P S P Sr

P S

F Sr r

S

é ù´ + - ´ - ´ë û=´

-- = »

6 4 4 4 44 7 4 4 4 4 48 64 7 48

14 2 43

( ) ( ) ( ),

,

118.88 121.23 124.82 118.88 130.66 118.88 123.263.7%

118.88 123.26?

H d

H a

r

r

´ + - ´ - ´æ ö= =ç ÷´è ø=

Page 50: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

International  Diversification-­ Part  2  -­

Two Strategy Comparison:§ As expected, much less variation under

hedging (although exchange rate risk is not eliminated entirely).

§ Hedging misses out on potentially large returns.

§ Risk vs Reward yet again.

§ Alternative hedging strategies: using Markowitz approach, incorporating rs,p

Page 51: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Wrap-­Up

Portfolio allocation main messages:

§ Diversification benefits come from imperfect correlation (r<1)

§ Var(portfolio) < Weighted average of Var(individual assets)§ If rf is available, then portfolio choice maximizes excess return (over rf)

relative to the additional risk: P*§ In other words, maximizes the Sharpe ratio or Reward-to-variability (slope

of CAL)

§ Finally, investor chooses a combination of P* and rf according to risk attitude

§ Separation Property

Page 52: Financial Market Analysis (FMAx) Module 6 - edXIMF+FMAx+1T2017... · Module 6 “Asset Allocation ... 0.34, 0.66 ()4.7%, 7.2%, 0.524 DE pp p ww Er S! == == = Selecting)theBest)Portfolio

Wrap-­Up

Portfolio allocation problem, in a nutshell:

s

rf A

B

P*

CAL( ( ) )P f

P

E r rSlope

s-

=

Return

This  continues  until  the  CAL  is  tangent  to  the  investment  opportunity  set.

The  optimal  portfolio  P*  maximizes  the  Sharpe  Ratio

By  choosing  another  portfolio  (B  vs.  A),  the  CAL  becomes  steeper.

The  Sharpe  Ratio  of  the  portfolio  increases.