Download PORTFOLIO THEORY : KEY CONCEPTS

11/9/2001
LECTURE :
PORTFOLIO THEORY AND RISK
Note that only a selection of these slides will be dealt with in detail, in
the lecture
All other slides are there to guide you towards the key points in
Cuthbertson/Nitzsche “Investments” and in the end of chapter questions
Revise your elementary stats before the lecture
Copyright K. Cuthbertson and D. Nitzsche
1
TOPICS
Basic Ideas
Efficient Frontier
Transformation Line, Capital Market Line
and the Market Portfolio
Practical Issues in Portfolio Allocation
Self-Study Slides
Copyright K. Cuthbertson and D. Nitzsche
2
READING
Investments:Spot and Derivative Markets,
K.Cuthbertson and D.Nitzsche
CHAPTER 10:
Section 10.1: Overview
Section 10.2: Portfolio Theory
Note:
Chapter 18 also contains much useful material
for those who wish to learn more !
Copyright K. Cuthbertson and D. Nitzsche
3
Basic Ideas
Copyright K. Cuthbertson and D. Nitzsche
4
PORTFOLIO THEORY
Portfolio theory works out the ‘best combination’ of
stocks to hold in your portfolio of risky assets.
You like return but dislike ‘risk’
We assume the investor is trying to ‘mix’ or combine
stocks to get the best return relative to the overall
riskiness of the chosen portfolio.
As we shall see ‘Best’ has a very specific meaning.
Copyright K. Cuthbertson and D. Nitzsche
5
PORTFOLIO THEORY
Question 1
What proportions of your own $100 should you put in
two different stocks
(e.g. ‘weights’ = 25%, 75% which implies $25, $75)
Different ‘weights’ give rise to different ‘risk-return’
combinations and this is the ‘efficient frontier’
Question 2
We now allow you to borrow or lend (from the bank),
How does this alter your choice of ‘weights’ and the
amount you actually choose to borrow or lend?
Latter depends on your ‘love of risk’
Copyright K. Cuthbertson and D. Nitzsche
6
Statistics: Some Definitions
Expected Return of Portfolio
E(RP) =
w1 ER1 + w2 ER2
Variance of Portfolio
s2P
=
w21 s21+ w22 s22 + 2 w1 w2 s12
s2P
=
w21 s21+ w22 s22 + 2 w1 w2(  s1 s2)
Also, ‘proportions’ are:
w1 + w2 = 1.
Note s12 =  s1 s2 - from statistics
Copyright K. Cuthbertson and D. Nitzsche
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Some Intuition: Domestic Assets
Risk of a single asset is the variance (SD = s1 )
of its return
( eg. Man.Utd share)
Risk of a portfolio of shares depends crucially
on covariance (correlation) between the
returns.
(Eg. Man Utd and Arsenal)
Copyright K. Cuthbertson and D. Nitzsche
8
Random selection of shares
Increasing the size (=n) of the portfolio
(each asset has ‘weight’ wi = 1/n)
Standard
Deviation
Note: 100%=risk when holding
only one asset
100%
Diversifiable /
Idiosyncratic Risk
Market /
Non-Diversifiable Risk
1 2...
20
40
No. of shares in portfolio
Copyright K. Cuthbertson and D. Nitzsche
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Some Intuition: International Diversification
US resident invests $100 in UK Stock index (FTSE100)
Suppose whenever FTSE100 goes up by 1% the sterling
exchange rate always goes down by 1% - perfect
negative correlation between the two returns
Then the US resident has zero US dollar risk
Hence negative correlations (strictly any  < +1) reduces
risk
(True, she also has zero expected USD return but
seeing as she is holding zero risk, that seems OK.
‘It’s the 1st rule of finance, stupid!’)
Copyright K. Cuthbertson and D. Nitzsche
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Random Selection: International Portfolio
Standard
Deviation
Note: 100%=risk when holding
only one asset
100%
Domestic Only
International
1 2...
20
40
No. of shares in portfolio
Copyright K. Cuthbertson and D. Nitzsche
11
Efficient Frontier
Copyright K. Cuthbertson and D. Nitzsche
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Can we do better than “random selection” ?
Consider ‘Return’ together with ‘Risk’
Assumptions
You like return and dislike portfolio risk (variance/ SD).
Assume everyone has the same view of future returns
ERi and correlations s12 , 12 .
2-Stage Decision Process
STAGE 1
Use only “own wealth” of $100 and work out the riskreturn combinations which are open to you by
distributing this $100 in different combinations
(proportions, wi ) in the available stocks. This gives
the “efficient frontier”
Copyright K. Cuthbertson and D. Nitzsche
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Efficient Frontier: Diversification
Expected
wi = (50%, 50%)
Return
wi = (25%,75%)
.
B
.
A
Own wealth of $100 split
between 2 assets in proportions
wi. As you alter the proportions
you move around ABC
Individual variances and
correlation coefficients are held
constant in this graph
C
RISK, s
Copyright K. Cuthbertson and D. Nitzsche
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Figure 10.4 : Risk Reduction Through Diversification
25
Corr = + 1
Corr = +0.5
Expected return
20
Corr = 0
15
Corr = -1
Corr = -0.5
10
5
0
0
5
10
15
20
Std. dev.
25
Copyright K. Cuthbertson and D. Nitzsche
30
35
15
Transformation Line
the
Capital Market Line CML
and the
Market Porfolio
Copyright K. Cuthbertson and D. Nitzsche
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Borrowing and Lending, ‘safe rate’= r
STAGE 2
You are now allowed to borrow and lend at risk free
rate, r while still investing in any SINGLE ‘risky bundle’
on the efficient frontier .
For each SINGLE risky bundle, this gives a new set of
risk-return combinations =“transformation line, TL”
~ which is a ‘straight line’
Each risky asset bundle has its ‘own’ TL
You can move along this TL by altering your
borrowing/lending
Copyright K. Cuthbertson and D. Nitzsche
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Transformation Line(s) TL
TL = Combination of ANY SINGLE ‘risky bundle’ and the safe asset
+
B
ER
.
M
ERm
. .Z
A
rr
This is TL to
point Z
.
This is TL to
point M =‘CML’
Point M corresponds to fixed wi (e.g.
50%, 50%)
Point Z corresponds to fixed wi
(e.g. 25%, 75%)
Everyone would choose the ‘highest’ TL
= point M and proportions 50-50.
sm
Copyright K. Cuthbertson and D. Nitzsche
s
18
CML: Some Properties
NO BORROWING OR LENDING (ONLY USE OWN $100)
You are then at point M
LEND SOME OF $100 (e.g lend $90 at r and $10 in risky bundle)
You are then at point like A
BORROW (say $50 ) and put all $150 in risky assets
You are then at point like B
Surprisingly the proportions at A and B are the same as at M
(I.e. 50%,50%) - but the $ amounts are NOT the same! (Tricky !)
Copyright K. Cuthbertson and D. Nitzsche
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CML and Market Portfolio (M)
.
M/s-B less risk
averse than M/s-A
CML
ER
B
ERm
M
ERm - r
A
r
wi - optm proportions at M
sm
wi maximises “reward to risk ratio”
- “Sharpe Ratio”
sm
Copyright K. Cuthbertson and D. Nitzsche
s
20
Market Portfolio = Passive Investment Strategy
Optimal wi maximises “reward to risk ratio” - “Sharpe Ratio”.
At the time you choose your optimal proportions you expect to
obtain a ‘reward to risk ratio’ of
S = ( ERm - r ) / sm
Note that both M/s-A and M/s-B have the same Sharpe ratio
Of course the ‘out-turn’ for the Sharpe ratio could be very different to
what you envisaged (because your forecasts turned out to be
poor).
Ball park estimate for Sharpe ratio for S&P500 (annual)
= 0.4 [= (12-4)/20]
Copyright K. Cuthbertson and D. Nitzsche
21
Practical Issues
in
Portfolio Allocation
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22
‘Active’ versus ‘Passive’ Strategy
Sharpe Ratio for any portfolio-k
Sk = ( ERk - r ) / sk
Active portfolio managers must try and beat the Sharpe ratio of the
‘passive’ investment strategy (I.e. holding the market portfolio,
month in-month-out ).
ERk = average of ‘out-turn’ values for monthly portfolio returns (net
of transactions costs) over say 3 years, for any portfolio-k and
any ‘strategy’ (e.g. trying to pick winners)
sk= sample SD of these monthly returns (over 3 years)
Compare investment strategies:
The investor with the highest value of Sk is the ‘winner’
Copyright K. Cuthbertson and D. Nitzsche
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Practical Issues
1) Suppose all investors do not have the same views about
expected returns and covariances.
~ we can still use our methodology to work out optimal
proportions/weights for for each individual investor.
2) The optimal weights will change as forecasts of returns and
correlations change - the ‘passive’ portfolio needs ‘some
rebalancing’ - ‘Tracking Error’
3)The method can be easily adopted to include transactions costs
of buying and selling, and investing “new” flows of money.
4) Lots of weights might be negative, which implies short-selling,
possibly on a large scale. If this is ‘impractical’ you can recalculate, where all the weights are forced to be positive.
Copyright K. Cuthbertson and D. Nitzsche
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No-Short Sales Allowed
(ie. All ‘weights’ > 0 )
ERP
‘Unconstrained’ Efficient Frontier - allows short sales
Efficient Frontier - no short sales
1) Always lies ‘within’ or ‘on’ frontier which allows short sales
2) Deviates more at ‘high’ levels of expected return and sP
sP (=SD)
Copyright K. Cuthbertson and D. Nitzsche
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Practical Issues
5) The optimal weights depend on estimate/forecasts of
expected returns and covariances.
If these forecasts are incorrect, the actual risk-return
outcome may be very different from that envisaged
when you started out
Put another way a small change in expected returns
can radically alter the optimal weights - ie. Extreme
sensitivity to the” inputs”.
The optimal weights are relatively insensitive to errors in
forecasts of correlations and variances - hence some
investors choose weights to min. SD only.
Copyright K. Cuthbertson and D. Nitzsche
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Forecast Errors, (ER, sP) Error in ‘proportions’
MIN
VARIANCE
PORTFOLIO,Z
ERP
Confidence
band around Z
may be
relatively small
- because it
does not use
‘poor’
forecasts of
ERi
M = mathematical optimum = (50%, 50%) say
x
x
x
xxx
X
xx
z x
xx
.
90% S&P500 +
10% Europe.
Optimal for US
investor ?
A
x
x
x
x
x
x
Each ‘cross’
represents a
different set of
‘weights’ wi
It is possible that
(90%,10%) lies within a
95% CONFIDENCE
BAND
C
Copyright K. Cuthbertson and D. Nitzsche
sP (=SD)
27
Practical Issues
6) To overcome this “sensitivity problem” try:
a) Choose the weights to minimise portfolio variance - the weights
are then independent of the “badly measured” expected returns.
(Note:does not imply a zero expected return - see fig).
b) Choose “new proportions” which do not deviate from existing
proportions by more than 2%.
c) Choose “new proportions” which do not deviate from “index
tracking proportions” (eg. S&P500) by more than 2%.
d) Do not allow any short sales of risky assets ( All wi >0).
e) Limit the analysis to investment in say 5 sectors, so sensitivity
analysis can be easily conducted (A sophisticated version of
which is Monte Carlo Simulation).
Copyright K. Cuthbertson and D. Nitzsche
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International Diversification
Tries to take advantage of “lower” (own) return correlations
compared to solely domestic investments.
-this can arise because of different timing of business cycles. (eg.
US is booming, Japan is in recession)
Diversification benefits can also arise because of exchange rate
correlations.
e.g.Suppose whenever FTSE100 goes up by 1% the sterling
exchange rate goes down by 1% (perfect negative correlation).
Then a US based investor faces no risk in dollar terms from his
UK investments.
Above extreme case is unlikely in practice so the issue of currency
hedging arises (via forwards, futures and options).
Copyright K. Cuthbertson and D. Nitzsche
29
International Diversification
‘HOME BIAS’ PROBLEM
It appears that investors, invest too much in the home
country relative to the results given by “optimal”
portfolio weights
BUT
- actual weights may not be statistically different
from the optimal weights, given that the latter are
subject to (large ? ) estimation error.
- actual weights might reflect “ a long view” of returns,
including the fact that purchases of goods (when
investments are cashed in) are largely made the
“home currency”.
Copyright K. Cuthbertson and D. Nitzsche
30
International Diversification
INVESTMENT COMMITTEES usually make STRATEGIC ASSET
ALLOCATION decisions based on a long term view of risk and
return (including political risk). This gives them their ‘baseline’
asset allocation between countries.
(e.g. no more than 10% portfolio in S.America over next 3 years)
- conventional portfolio theory largely ignores
political/default risk but could in principle incorporate
this in forecast of expected returns, variances etc but usually done on an ad-hoc basis.
The ‘international portfolio’ may then be ‘fine tuned’
using portfolio theory, but the weights will be heavily
constrained (to not move far from those set by the
Investment committee).
Copyright K. Cuthbertson and D. Nitzsche
31
International Diversification
Within a particular country, either portfolio theory will be used to
guide proportions in each industrial sector, or they will try just
‘track’ the respective domestic indices (e.g. the S&P500,
FTSE 100).
There is some evidence that INVESTMENT COMMITTEES are
moving towards choosing industrial sector weights, subject to
limits on the resulting country proportions. This is to ‘gain’
from the disparate business cycles between industries (e.g.
world car industry has different cycle to world chemicals)
This is because ‘country indices’ are beginning to have ‘high
correlations’ (e.g. US and UK aggregate business cycles are
now more highly correlated.
Copyright K. Cuthbertson and D. Nitzsche
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International Diversification
TACTICAL ASSET ALLOCATION
Use part of funds for market timing’ the business cycle’
(e.g. switch 10% of speculative funds out of US and
into SE Asia )
-might use a macro-economic model for forecasts
-does not easily ‘fit’ into portfolio theory because
usually little or no formal estimate of risk is made
Copyright K. Cuthbertson and D. Nitzsche
33
LECTURE ENDS HERE
Copyright K. Cuthbertson and D. Nitzsche
34
SELF STUDY SLIDES
The following slides provide a simple
numerical example to construct
the efficient frontier,the capital market
line and the market portfolio
These slides will NOT be covered in the
lectures
Copyright K. Cuthbertson and D. Nitzsche
35
STATISTICS REVISION: Some Definitions
Expected Return of Portfolio
E(RP) =
w1 ER1 + w2 ER2
Variance of Portfolio
s 2P
=
w21 s21+ w22 s22 + 2 w1 w2 s12
s 2P
=
w21 s21+ w22 s22 + 2 w1 w2(  s1 s2)
Also, ‘proportions’ are:
w1 + w2 = 1.
Note s12 =  s1 s2 - from statistics
The above are used to derive the EFFICIENT
FRONTIER by (arbitrarily) altering the w’s
Copyright K. Cuthbertson and D. Nitzsche
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STAGE 1: 2 Risky Assets:
Real world data (statistician)
Risky Assets
Equity-1
Equity-2
Mean, ERi
8.75
21.25
sSD) .
10.83
19.80
Correlation (Equity-1, Equity-2): - 0.9549
Cov(Equity-1, Equity-2) :
-204.688
Copyright K. Cuthbertson and D. Nitzsche
37
STAGE 1: Construct Efficient Frontier
Choose different w’s and calculate ERp and sp
combinations)
State
Shares of
Equity-1 Equity-2
Portfolio
ERp
sp
w1
w2
1
1
0
8.75
10.83
2
0.75
0.25
11.88
3.70
3
0.5
0.5
15
5
4
0
1
21.25
19.80
Now plot values of ERp and sp and construct
the Efficient Frontier
Copyright K. Cuthbertson and D. Nitzsche
38
Efficient Frontier
Expected Return
30
25
0, 1
20
0.5, 0.5
15
10
0.75, 0.25
(1, ,00 )
1
5
0
0
5
10
15
20
25
Standard deviation
Copyright K. Cuthbertson and D. Nitzsche
39
Efficient Frontier with ‘n’ - Risky Assets
You require EXCEL ‘SOLVER’ to ‘draw’ the
EFFICIENT FRONTIER (=A-B)
ERP
ERz
ERZ
End of Excel
minimisation
wz = 25%,75%,
say
x
Z
X
X
x
x
xA
‘Start’ Excel (50%,50%, say)
x
B x
sz ‘Finish Excel’
Each ‘cross’
represents a
different set of
‘weights’ wi
x x
Excel solver changes
the weights to minimise
risk (SD) for any
arbitrarily chosen level
expected return, ERz
So, Z moves to the left
Copyright K. Cuthbertson and D. Nitzsche
xC
sP (=SD)
40
STAGE 2: Transformation Line
We have ‘constructed’ the efficient frontier
Now introduce a “safe asset”
What does the risk-return “trade-off” look like when we
allow borrowing or lending at the safe rate
and
we combine this with any ‘single bundle’ of risky assets?
‘New Portfolio’=1-safe asset + 1 “bundle of risky assets”
Answer = Straight Line relationship between ER and s
Copyright K. Cuthbertson and D. Nitzsche
41
STAGE 2: Transformation Line
What is a ‘Risky Asset bundle’ ?:
Keep (arbitrary) fixed weights in risky assets
eg. 20% in asset-1, 80% in asset-2
So, if you have W0 = $100 you will hold $20 in
asset-1 and $80 in asset-2
Assume this gives rise to a fixed “bundle” of risky
assets” called “q” with ERq=22.5% and sq= 24.8%
Now combine ‘fixed risky bundle’ with the safe asset
by borrowing/lending different $ amounts of safe asset
Copyright K. Cuthbertson and D. Nitzsche
42
Construct ‘One’ Transformation Line
Return
Data
Mean
T-bil l (safe)
Equity (Risky)
r = 10
Rq = 22.5
0
sq24.87
Std. Dev.
FORMULAE FOR EXPECTED RETURN AND SD OF ‘NEW’
PORTFOLIO
N= “new” portfolio of: ‘safe + risky ‘bundle’
sq 2 = variance of the risky ‘bundle’
x
= proportion held in ‘risky asset’
(1-x) = proportion held in safe asset(with s = 0 )
Expected Return:
E(RN) = (1- x) . r + x ERq
THEN: Variance (SD) of NEW PORTFOLIO of “ 1-safe + 1 risky asset”
sN 2 = x2 sq 2
or sN = x sq
Copyright K. Cuthbertson and D. Nitzsche
43
“New Portfolio (N) :
“Arbitrarily alter ‘x’ to give different Expected
Return ERN and risk combinations sN
This gives a straight line = Transformation Line
State
Shares of
Wealth in
“New” portfolio
ERN
sN (SD)
0
10
0
0.5
0.5
16.25
12.4373
3
0
1
22.5
24.8747
4
-0.5
1.5
28.75
37.312
T-bill
Equity
1-x
-x)
1
1
2
Copyright K. Cuthbertson and D. Nitzsche
44
Variance of ( 1-safe + 1 risky asset BUNDLE)
Note: Borrowing:
When proportion (1-x)= - 0.5 is in the safe asset, this implies
x = 1.5 held in risky asset
Suppose ‘own’ initial wealth W0 =$100
Hence above implies borrowing 50% of “own wealth” (=$50) to
add to your initial $100 and putting all $150 into the bundle of
risky assets (in the fixed proportions 20%, 80%, I.e $30 and
$120 in each risky asset)
- this is referred to as ‘leverage’ and involves a higher expected
return but also higher risk (SD). ‘Its the first law of finance
again!
Now plot the combinations ER and s in the previous slide
Copyright K. Cuthbertson and D. Nitzsche
45
Transformation Line
1 safe asset + 1 risky "bundle"
Exp. Return
30
22.5
Borrow -0.5,
put all 1.5 in
risky bundle
25
20
No Borrowing/ No lending
15
10
5
All lending
0
0
10
0.5 lending +
0.5 in risky bundle
20
24.87
30
40
Standard deviation
Note: At “no borrow/lend” position, ER and s of “new” portfolio equals that
for the risky asset alone (not surprisingly)
Copyright K. Cuthbertson and D. Nitzsche
46
Transformation Lines
Safe asset plus ANY ONE ‘arbitrary’ risky bundle,
gives a specific transformation line (which is straight
line) between r and the s.d of the risky bundle
Every single, risky bundle has its own transformation
line
Which transformation line is “best”?
“THE HIGHEST ACHIEVABLE” = Capital Market Line
Copyright K. Cuthbertson and D. Nitzsche
47
Transformation Lines
Exp. Return
L’
1 safe + risky "bundles"
30
sq =24.87
25
L
20
15
sk = 10
r =10
5
0
0
10
20
30
40
Standard deviation
q and k are both ‘points’ on the efficient frontier. So q might represent(20%,80%) in risky assets
and k might represent (70%,30%). Each “fixed weight” risky bundle has its own transformation
line
Copyright K. Cuthbertson and D. Nitzsche
48
“B” is highest attainable transformation line, while still remaining on the
efficient frontier. ‘B’ represents the optimal weights (50%,50%) for the risky
bundle.
Efficient Frontier
L’
CMLand CML
Expected Return
30
A
25
20
L
B
15
C
D
10
5
0
0
5
10
15
20
25
Standard deviation
Copyright K. Cuthbertson and D. Nitzsche
49
Market Portfolio
Point-B is therefore a rather special portfolio and
hence is known as the “Market Portfolio” (as
indicated by the subsript ‘m’ in the next slide)
IF everyone has the same expectations about
returns, standard deviation and correlations then:
Everyone chooses point-B (which here gives 50%,
50% held in each risky asset)
Copyright K. Cuthbertson and D. Nitzsche
50
CML and Market Portfolio (M)
+
ER
M/s-B less risk
averse than M/s-A
ERm
M
CML
wi - optm proportions at M
ERm - r
A
r
B
sm
wi maximises “reward to
risk ratio” - “Sharpe Ratio”
sm
Copyright K. Cuthbertson and D. Nitzsche
s
51
How Much Should an Individual Borrow or Lend?
~while still maintaining the 50:50 proportions in the 2-RISKY assets ?
This depends on the individual’s “preferences” for risk versus return
M/s-A is VERY “risk averse” (=dislike risk)
implies uses e.g. $90 of her $100 “own wealth” to invest in the
safe asset and puts only V= $10 in the risky “bundle” thus holding
$5 in each risky asset (5/10 = 50%)
M/s-B is LESS “risk averse” (=not too worried about risk)
She borrows say $60 and invests the whole V= $160 in the risky
bundle thus holding $80 in each risky asset (80/160 =50%)
Hence both A and B invest the same PROPORTIONS in the risky
assets but DIFFERENT $-amounts. The latter implies A and B hold
different DOLLAR risk (For the ‘experts’: $-RISK = V x sm )
Copyright K. Cuthbertson and D. Nitzsche
52
END OF SLIDES
Copyright K. Cuthbertson and D. Nitzsche
53