financial market analysis (fmax) module 6 - edximf+fmax+1t2017... · module 6 “asset allocation...
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Financial Market Analysis (FMAx)Module 6
“Asset Allocation and Diversification”
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.
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.
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.
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.
An Introduction- Review of Statistics -Expected Returns:
Using historical data:
( ) ( )( )( ) ( )[ ] 11.....1111
321 -++++== TTg rrrrrrE
( )T
rrrrrrE Ta
++++==
....321
An Introduction- Review of Statistics -Risk: (Variance or Standard Deviation of Returns)
Using historical data:
( ) ( )( ) ( )( ) ( )( )T
rErrErrErrrVar Tg
222
21 ... -+-+-
==
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)
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 ,=
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
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 +<
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 = = = -+ +
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
= +
= + + × × × ×
= + +
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
-=
+ -= -
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)
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.
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
- + --=
+ +
+ =
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- -
= = =
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
Comparing Different Portfolios- Two-Assets -
E(rp)
sP
rfA
B
CALB
CALA
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
- - -=
- + - - - + -
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)
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)
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
= == = =
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?
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
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
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?
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)
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
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
-= =
´ ´ ´
Completing the Investor Decision- Two-Assets -
E(rp)
sP
rf
P*
CAL
C
wP*=100%
wP*=78%
U
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.
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
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
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
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
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
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**.
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
Building the Entire Frontier- n Assets -
E(r)
sP
Individual Assets
rf1
rf2
Min variance
Aggressive investor
Conservative investor
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).
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).
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
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 ¥/$
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 =
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
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=
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
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
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.