copyright k. cuthbertson and d. nitzsche 1 lecture : portfolio theory and risk note that only a...

Copyright K. Cuthbertson and D. Nitzsche 1

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

11/9/2001


Basic Ideas

Efficient Frontier

Transformation Line, Capital Market Line

and the Market Portfolio

Practical Issues in Portfolio Allocation

Self-Study Slides

TOPICS


Investments:Spot and Derivative Markets,

K.Cuthbertson and D.Nitzsche

CHAPTER 10:CHAPTER 10:

Section 10.1: OverviewSection 10.1: Overview

Section 10.2: Portfolio TheorySection 10.2: Portfolio Theory

Note:Note:

Chapter 18 also contains much useful materialChapter 18 also contains much useful material

for those who wish to learn more !for those who wish to learn more !

READING


Basic Ideas


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.


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’


Statistics: Some Definitions

Expected Return of Portfolio

E(RP) = w1 ER1 + w2 ER2

Variance of Portfolio

P = w2

1 1+ w2

2 2 + 2 w1 w2 12

P = w2

1 1+ w2

2 2 + 2 w1 w2( 1 2)

Also, ‘proportions’ are: w1 + w2 = 1.

Note 12 = 1 2 - from statistics


Some Intuition: Domestic Assets

Risk of a single asset is the variance (SD = 1 ) 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)


Random selection of shares

Standard Deviation

20 40

Diversifiable / Idiosyncratic Risk

Market / Non-Diversifiable Risk

No. of shares in portfolio1 2...

Note: 100%=risk when holding only one asset

100%

Increasing the size (=n) of the portfolio(each asset has ‘weight’ wi = 1/n)


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!’)


Random Selection: International Portfolio

Standard Deviation

20 40

Domestic Only International

No. of shares in portfolio1 2...

100%

Note: 100%=risk when holding only one asset


Efficient Frontier


Can we do better than “random selection” ?

AssumptionsYou like return and dislike portfolio risk (variance/ SD).

Assume everyone has the same view of future returns ERi and correlations 12 , 12 .

2-Stage Decision ProcessSTAGE 1

Use only “own wealth” of $100 and work out the risk-return combinations which are open to you by distributing this $100 in different combinations (proportions, wi ) in the available stocks. This gives the “efficient frontier”

Consider ‘Return’ together with ‘Risk’


Efficient Frontier: Diversification

Expected

Return

RISK,

Awi = (50%, 50%)

wi = (25%,75%)

..

B

C

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


Figure 10.4 : Risk Reduction Through Diversification

0

5

10

15

20

25

0 5 10 15 20 25 30 35

Std. dev.

Expe

cted

retu

rnCorr = + 1

Corr = +0.5

Corr = 0

Corr = -1

Corr = -0.5


Transformation Line

the

Capital Market Line CML

and the

Market Porfolio


Borrowing and Lending, ‘safe rate’= r

STAGE 2You are now allowed to borrow and lend at risk freerate, r while still investing in any SINGLE ‘risky bundle’on the efficient frontier .

For each SINGLE risky bundle, this gives a new set ofrisk-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


+

Transformation Line(s) TL

ER

ERm

r

This is TL to point M =‘CML’

M

..

m

Point M corresponds to fixed wi (e.g. 50%, 50%)

Point Z corresponds to fixed wi (e.g. 25%, 75%)

Z

Everyone would choose the ‘highest’ TL = point M and proportions 50-50.

A.

.B

TL = Combination of ANY SINGLE ‘risky bundle’ and the safe asset

This is TL to point Z

r


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 !)


.

CML and Market Portfolio (M)

ER

ERm

r

CML

M

m

ERm - r

wi - optm proportions at M

m

A

B

M/s-B less risk averse than M/s-A

wi maximises “reward to risk ratio”

- “Sharpe Ratio”


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 ) / m

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]


Practical Issues in

Portfolio Allocation


‘Active’ versus ‘Passive’ Strategy

Sharpe Ratio for any portfolio-k

Sk = ( ERk - r ) / k

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)

= sample SD of these monthly returns (over 3 years)

Compare investment strategies:

The investor with the highest value of Sk is the ‘winner’


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 re-calculate, where all the weights are forced to be positive.


No-Short Sales Allowed

P (=SD)

‘Unconstrained’ Efficient Frontier - allows short salesERP

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 P

(ie. All ‘weights’ > 0 )


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.

Practical Issues


Forecast Errors, (ER, P) Error in ‘proportions’

A

MIN VARIANCE PORTFOLIO,Z

Confidence band around Z may be relatively small - because it does not use ‘poor’ forecasts of ERi

C

It is possible that (90%,10%) lies within a 95% CONFIDENCE BAND

P (=SD)

xxxxX x

x

z

x

x

M = mathematical optimum = (50%, 50%) say

Each ‘cross’ represents a different set of ‘weights’ wi

ERP

xxx

x90% S&P500 + 10% Europe. Optimal for US investor ?

.xx

xxx


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).

Practical Issues


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).

International Diversification


‘HOME BIAS’ PROBLEMIt 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”.



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).



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.



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



LECTURE ENDS HERE


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


STATISTICS REVISION: Some Definitions

Expected Return of Portfolio

E(RP) = w1 ER1 + w2 ER2

Variance of Portfolio

P = w2

1 1+ w2

2 2 + 2 w1 w2 12

P = w2

1 1+ w2

2 2 + 2 w1 w2( 1 2)

Also, ‘proportions’ are: w1 + w2 = 1.

Note 12 = 1 2 - from statistics

The above are used to derive the EFFICIENT FRONTIER by (arbitrarily) altering the w’s


STAGE 1: 2 Risky Assets: Real world data (statistician)

Risky Assets

Equity-1 Equity-2

Mean, ERi 8.75 21.25

SD) . 10.83 19.80

Correlation (Equity-1, Equity-2): - 0.9549 Cov(Equity-1, Equity-2) : -204.688


STAGE 1: Construct Efficient Frontier

State Shares of Portfolio

Equity-1 Equity-2 ERp p

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 p and construct the Efficient Frontier

Choose different w’s and calculate ERp and p combinations)


Efficient Frontier

0

5

10

15

20

25

30

0 5 10 15 20 25

Standard deviation

Exp

ecte

d R

etu

rn

(1 , 0 )1 , 00.75, 0.25

0.5, 0.5

0, 1


Efficient Frontier with ‘n’ - Risky Assets

ERZ

A

B

C

Excel solver changes the weights to minimise risk (SD) for any arbitrarily chosen level expected return, ERz

So, Z moves to the left P (=SD)

xxx

xx

Xx

x

End of Excel minimisation

wz = 25%,75%, say

x

x

Z

Each ‘cross’ represents a different set of ‘weights’ wi

ERP

‘Start’ Excel (50%,50%, say)

z ‘Finish Excel’

X

You require EXCEL ‘SOLVER’ to ‘draw’ the EFFICIENT FRONTIER (=A-B)

ERz


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


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


Construct ‘One’ Transformation Line

Return

T-bill (safe) Equity (Risky)

Mean r =10 Rq = 22.5

Std. Dev. 0 q24.87

FORMULAE FOR EXPECTED RETURN AND SD OF ‘NEW’ PORTFOLIO

N= “new” portfolio of: ‘safe + risky ‘bundle’ q 2 = variance of the risky ‘bundle’ x = proportion held in ‘risky asset’ (1-x) = proportion held in safe asset(with = 0 )

Expected Return: E(RN) = (1- x) . r + x ERq

THEN: Variance (SD) of NEW PORTFOLIO of “ 1-safe + 1 risky asset”

N 2 = x2 q 2 or N = x q

Data


“New Portfolio (N) :

State Shares ofWealth in

“New” portfolio

T-bill Equity ERN N (SD)

1-x - x )

1 1 0 10 0

2 0.5 0.5 16.25 12.4373

3 0 1 22.5 24.8747

4 -0.5 1.5 28.75 37.312

“Arbitrarily alter ‘x’ to give different Expected Return ERN and risk combinations N

This gives a straight line = Transformation Line


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 in the previous slide


Transformation Line1 safe asset + 1 risky "bundle"

0

5

10

15

20

25

30

0 10 20 30 40

Standard deviation

Exp

. Ret

urn

Efficient Frontier and CLM

0

5

10

15

20

25

30

0 5 10 15 20

Standard deviation

Exp

ecte

d R

etur

nBorrow -0.5, put all 1.5 inrisky bundle

0.5 lending +0.5 in risky bundleAll lending

No Borrowing/ No lending

24.87

22.5

Note: At “no borrow/lend” position, ER and of “new” portfolio equals that

for the risky asset alone (not surprisingly)


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


Transformation Lines1 safe + risky "bundles"

0

5

10

15

20

25

30

0 10 20 30 40

Standard deviation

Exp

. Ret

urn

Efficient Frontier and CLM

0

5

10

15

20

25

30

0 5 10 15 20 25

Standard deviation

Exp

ecte

d R

etu

rnq =24.87

k = 10r =

L

L’

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


Efficient Frontier and CML

0

5

10

15

20

25

30

0 5 10 15 20 25

Standard deviation

Ex

pe

cte

d R

etu

rn

A

CD

B

CML L’

“B” is highest attainable transformation line, while still remaining on the efficient frontier. ‘B’ represents the optimal weights (50%,50%) for the risky bundle.

L


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)

Market Portfolio


+

CML and Market Portfolio (M)

ER

ERm

r

CML

M

m

ERm - r

wi - optm proportions at M

m

A

BM/s-B less risk averse than M/s-A

wi maximises “reward to risk ratio” - “Sharpe Ratio”


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 m )


END OF SLIDES

copyright k. cuthbertson and d. nitzsche 1 lecture : portfolio theory and risk note that only a...

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