stanford2008.pageout week post… · 15-2 Levich Chap 15 Scan Read Pages 554-597 Equity Portfolios Luenberger Solnik Eun Chap 8, 9, 12, 13 Chap Pages Pages 317-84, 385-432, 523-627

http://stanford2008.pageout.net

Handout #20 as of 0808, 2008

International Asset PortfoliosEquity Portfolios

+Foreign Exchange Market Intervention

MWF 3:15-4:30 Gates B01Final Exam MS&E 247S

Fri Aug 15 2008 12:15PM-3:15PM Gates B03 (Alternate Time)Or Saturday Aug 16 2008 7PM-10PM Terman Auditorium (Official Time)Remote SCPD participants will also take the exam on Friday, 8/15

Please Submit Exam Proctor’s Name, Contact info as SCPD requires.C.c. the info to [email protected], preferably a week before the exam.

Local SCPD students please come to Stanford to take the exam. Light refreshments will be served.

15-2

Levich Chap 15

Scan Read

Pages 554-597 Equity Portfolios

Luenberger

Solnik

Eun

Chap

8, 9, 12, 13

Chap Pages

Pages 317-84, 385-432, 523-627

Pages

Chap Pages

Wooldridge

+ Chap 17 Foreign Exchange Market Intervention (pp. 656-71)

Reading Assignments for this Week

International Financial Management 4E:Chapter 13 International Equity Markets

Alternate Investments, Int’l Diversification, Performance Measurement, Global Asset Allocation: Structuring and Quantifying the Process

International Asset PortfoliosEquity Portfolios

MS&E 247S International InvestmentsYee-Tien Fu

15-4

Introduction to Equities• While they share a long history, equities and

bonds are very different financial instruments.• The owner of a £ bond is entitled to a set

amount of £ at periodic intervals. All £ bonds (excluding those with equity-like features) are claims on some nominal amount of £.

• In comparison, the owner of an equity share in a British firm may receive dividends denominated in £. But what the shareholder truly owns is a claim on the real assets of the firm and all the cash flows that accrue once the firm has paid all of its creditors.

15-5

Introduction to Equities• Both bonds and shares may be exposed to

similar market forces. If a £ bond was issued in London and trades in London, it will be subject to British exchange control risk, expropriation risks, and withholding taxes when viewed by non-British investors.

• The fundamental difference is this:¤ The valuation of a £ bond is based on a stream

of nominal £ cash flows that we can enumerate.¤ The valuation of a British equity is linked to the

firm’s real assets (regardless of location) and the cash flows (in all currencies) associated with the firm’s operations.

15-6

Introduction to Equities• The performance of equity investments is

usually evaluated in 2 dimensions - expected return and risk. These dimensions describe the 2 basic incentives for international investment: ¤ to enhance portfolio returns for the same level

of risk, or¤ to reduce the riskiness of a portfolio without

sacrificing expected return.

15-7

Introduction to Equities• Expected value gains could occur

¤ if foreign equity markets are inefficient, such that foreign equity prices do not reflect all available information, or

¤ if foreign equity markets may be segmented from other capital markets, such that investors in the foreign market receive a different compensation for bearing equity risk than in other markets.

• Diversification gains could occur if the correlation of returns across countries is low.

15-8

Introduction to Equities

• Diversification gains are available even when domestic and foreign capital markets are integrated, so that risk bearing in different markets is rewarded in a similar fashion.

• When investors are risk averse, international equity investment offers an opportunity for welfare gains through superior “sharing” of international equity risks.

15-9

Introduction to Equities• A pioneering study by Herbert Grubel (1968)

showed that investors from 11 developed economies could have enjoyed substantially more favorable risk-return opportunities had they diversified their portfolios internationally in the 1960s.

• However...¤ The analysis overstated the gains.¤ Capital restrictions would have made some

markets “off-limits” to foreign investors..¤ The efficient frontier could not have been

obtained by all investors.

15-10

Return and Risk in World Equity Marketsand Efficient Frontiers, 1959-1966

Figure 15.1 Pg 524

14

12

6

16

4

2

00 20 40

% R

ate

of R

etur

n (%

per

ann

um)

Risk: Standard Deviation of Returns

8

10

18

60 80 100 120

++++

+

++

xx

x

xxxx

Australia

Japan

ItalyGermany

Netherlands

Belgium

FranceCanada

USA

U.K.S. AfricaA

A

BB

Efficient frontier labeled AA includes all 11 industrial countries, while BB includes only 8 European and North American countries.

15-11

Introduction to Equities

¤ The portfolios in segment AA of the efficient frontier call for roughly a 40% weight on Australia. Since Australia represents less than 1% of the world equity market, these portfolios were inconsistent with the overall valuation of equity markets.

• Despite the shortcomings, Grubel’s insight has been verified by many subsequent studies.

• Yet, despite promotion by investment advisors as a prudent strategy, most investor portfolios reflect a home country bias in equities as well as bonds, as shown in Table 14.1.

15-12Table 14.1

Holdings of Foreign Securitiesby Residence of Investor

CanadaGermanyJapanNetherlandsU.K.U.S.

1983

$ 94257128883

1985

$ 1269

18521

148112

1988

$ 16176431

46267157

1983

513

42922

1

1985

510

82420

2

1988

515

72520

2

1983

174

1016

3

1985

210

81524

2

1988

215

72125

2

Residenceof investor

US$ Valueof Holdings(in billions)

Equities % of EquityPortfolio

Bonds %of BondPortfolio

15-13

Introduction to Equities• This investment pattern is a puzzle that has

rekindled research into international equity markets.¤ Have home country investors been inexcusably

slow to diversify their portfolios internationally?¤ Or have important aspects of the international

investment process been left out of the theoretical and empirical analysis thus far?

15-14

Size and Institutional Features of Global Equity Markets

Market Capitalization Measures• From Figure 15.2, we see that U.S. market

capitalization, while growing in absolute terms, has fallen in relative size from 54.1% of the world market in 1984 to 38.6% in 1995.

• Higher economic growth rates in smaller economies is the primary contributor to this long-term trend which is most likely to continue.

• Also note the substantial growth of emerging markets.

15-15

World Market Capitalization

05

1015

20

2530

3540

1984 1988 1993 1999

U.S. U.K. Japan Other Developed Emerging

US$

Tril

lions

$3.4

$9.7$14.1

$36.0

46.2%

8.1%12.6%

24.6%

8.4%

5.0%18.2%40.2%7.9%

28.7%

4.2%15.2%19.4%7.1%

54.1%

11.6%21.9%21.3%8.2%

37.0%

Figure 15.2 15-15

15-16Market Capitalization of Equity Markets in Developed Countries (in Billions of U.S. Dollars)

15-17Market Capitalization of Equity Markets in Selected Developing Countries (in Billions of U.S. Dollars)

15-18


• The pattern for Japan is unusual on 2 accounts:1 Japan’s share of world stock market

capitalization more than doubled between 1984 and 1988, and then dropped by half in 1995. This reflects the surge in Japanese equity prices in the late 1980s, which many label a speculative bubble, and the collapse of those prices in 1990.

2 The value of Japanese equities surpassed the U.S. in 1987 to become (for 3 years) the world’s “largest” equity market. Note that the GNP of U.S. exceeds that of Japan by about 75%.

15-19

How Large is the Japanese Stock Market?

• Cross-holding of securities (the practice of firm A owning equity shares in firm B) complicates the calculation of market capitalization values.

• Cross-holding is common in Japan, Germany, etc. where banks are permitted to hold substantial and sometimes controlling interests in non-banking firms. Cross-holding is fairly rare in the United States.

• Suppose firm A has $100 of net productive assets and 100 shares outstanding, each valued at $1. Firm B is similar. The market value of these 200 shares of firms A and B is $200.

• To introduce a cross-holding effect, let A issue 50 new shares at $1 each and use the proceeds to purchase shares in B.

15-20

How Large is the Japanese Stock Market?

• As conventionally measured (taking the number of shares outstanding and multiplying by the price per share) the market value of firms A and B is now $250.

• Yet the value of productive physical assets is unchanged at $200, and $200 is sufficient to purchase all of A’s and B’s stock. It takes $150 to buy all of A’s stock, and only $50 to buy the remaining shares of B not already acquired by purchasing A.

• Hence, to measure market value properly, we must adjust for the cross-holding effect by netting out the value of the cross-held shares.

• This adjustment reduced the 1988 market capitalization weight for Japan from 44% to 29.5%, a figure very close to Japan’s GDP weight in the world portfolio.

15-21


Institutional Aspects of Global Equity Markets• Investors are unlikely to invest abroad if

restrictions and limitations affect the repatriation of their capital.

Number of Firms Listed• In 1994, less than 7% of the firms listed on U.S.

exchanges are foreign firms. In comparison, foreign firms make up about 18% of the total firms listed in United Kingdom, the center for trading in foreign stocks. The requirements for listing shares are more stringent in the U.S. than elsewhere.

15-22


Market Concentration• Market concentration, measured by the

percentage of market capitalization within the 10 largest firms, is another statistic with wide variation across countries. ¤ In the U.S., Japan, and India, the top 10 firms

account for 15-20% of the overall market capitalization.

¤ In all other countries, market concentration is higher, averaging close to 30%.

¤ In the Netherlands, New Zealand, and some smaller emerging markets, market concentration exceeds 60%.

15-23Percentage of Market Capitalization Represented by the 10 Largest Stocks: Emerging Equity Markets in Selected Developing Countries

15-24


Trading Volume• Market turnover, measured as the annual

volume of trading as a percentage of market capitalization, also varies substantially across countries.

• Statistics suggest that liquidity varies considerably across markets, as high trading volume tends to reduce liquidity risks and trading costs. But liquidity could vary as well within a market, with greater liquidity for a small number of high capitalization stocks, and much lower liquidity otherwise.

15-25

Turnover Ratio of Equity Markets in Developed Countries (Transactions in US $ / Year-End Market Capitalization in US $)

15-26Turnover Ratio of Emerging Equity Markets in Selected Developing Countries (Transactions in US $ / Year-End Market Capitalization in US $)

15-27


Transaction Taxes, Transaction Costs, Clearing and Settlement

• A long settlement period for making payment and obtaining delivery of securities (on the buy side) and delivering securities and obtaining cash settlement (on the sell side) is a deterrent to investment.

15-28Trad

ing

Prac

tices

and

Cos

ts o

f Maj

or E

quity

Mar

kets

15-29Trading Practices and Costs of Major Equity Markets

15-30

International Investment Vehicles• Direct Purchase of Foreign Shares• American Depositary Receipts (ADRs)

¤ In order to issue an ADR, a U.S. bank takes custody of foreign shares in its foreign office. Then an ADR can be issued as a claim against these foreign shares. This can be especially valuable when there are doubts about the authenticity of foreign shares.

¤ Owners of the ADR have the right to redeem their ADR and obtain the true underlying foreign shares. Arbitrage of this sort ensures that the price of the ADR and the underlying shares will be nearly identical.

15-31

Mec

hani

cs o

f Iss

uanc

e an

d C

ance

llatio

n of

AD

Rs

15-32

International Investment Vehicles¤ The issuing bank services the ADR by collecting

all dividends, rights offerings, and so forth in foreign currency, and distributing the proceeds in US$ to the ADR owner.

¤ Rights offering - When a corporation is about to issue additional stock, it is customary to offer the stock first to its existing shareholders at special rate.

¤ U.S. investors can trade ADRs with each other without recourse to the foreign equity market, without using the foreign exchange market, and without relying on foreign clearing and settlement.

15-33Types of ADRs

15-34

International Investment Vehicles¤ In a sponsored ADR, the foreign firm pays a fee

to the depositary bank to cover the cost of the ADR program. In an unsponsored ADR, the issuance of the ADR is “demand driven” in response to a security firm’s desire to facilitate trading in a popular foreign issue.

• Closed-End and Open-End Mutual Funds¤ Mutual funds that invest in foreign stocks can be

grouped into several categories - global, international, regional, country, specialty.

¤ In addition, foreign stock funds are classified as either open-end or closed-end.

15-35

International Investment Vehicles¤ An open-end fund stands ready to issue and

redeem shares at prices reflecting the net-asset-value of the underlying foreign shares.

¤ A closed-end fund issues a fixed number of shares against an initial capital offering. The shares of the fund then trade in a secondary market (usually listed on an exchange) at prices reflecting a premium or discount relative to the net-asset-value of the underlying foreign shares.

¤ Closed-end country funds were the fastest growing segment of the public investment funds during the late 1980s. At the end of 1992, there were 42 closed-end country funds listed in the U.S., representing $4.3 billion in equity.

15-36

Global Depository Receipt Tombstone

15-37Example of Dow Jones Country Stock Market Indexes

15-38Major National Stock Market IndexesM

ajor

Nat

iona

l Sto

ck M

arke

t Ind

exes

15-39

International Investment Vehicles¤ A 1994 paper by Gikas Hardouvelis, et al.

analyzed the behavior of closed-end country fund discounts and premiums. They concluded that:

– Such discounts varied widely and are a significant factor in the variability of country fund returns.

– On average, the variance of country fund returns is 3 times larger than the variance on the underlying foreign assets.

– Discounts tend to be mean reverting, implying that unusually large discounts and premiums tend back toward their average value.

– Thus, by selecting a closed-end country fund, the investor also takes a position on an additional unobserved factor - the “local sentiment” about world events and country-specific events.

15-40

Calculating the Unhedged Returns on Foreign Equity in US$ Terms

• Let Et represent the initial purchase price of the equity in foreign currency terms.

• Let St represent the spot exchange rate, in $/FC terms, on the purchase date.

• The product EtSt is the US$ purchase price of the foreign equity.

• After one period, the value of the equity is Et+1, representing the initial equity price plus the price change over the period (Δt+1) plus dividends Dt+1:

Risk and Return inInternational Equity Markets

111~~

+++ +Δ+≡ tttt DEE

~

~

15-41

• The value of the equity after one period in US$ terms is Et+1St+1.

• The continuous rate of return on the foreign equity measured in US$ and on an unhedged basis is:

The equation shows that the unhedged US$ return has 2 components: the return on the equity shares in foreign currency terms plus the return on the foreign currency used to buy the shares. Both terms may be greater than or less than zero.


~ ~

FCUS$,FCU SES

SE

ESESER

t

t

t

t

tt

tt ~~~

ln~

ln~~

ln~ 1111

$, +=+== ++++

(15.1)

15-42

• The variance of the returns reflects the variance of each term and the covariance between the returns on the foreign equity and the returns on spot foreign exchange:

The covariance term represents the sensitivity of share returns to exchange rate changes, and can be either positive or negative. A positive covariance implies that the value of foreign equity tends to fall or rise along with the value of foreign currency as shown in cells A and B in Table 15.5.


( ) ( ) ( ) ( )FCUS$,FCFCUS$,FCU SECovSER~

;~

2~~~ 22

$,2 ++= σσσ

(15.2)

15-43

Currency Market Returnand Stock Market Return Combinations

StockMarketReturns

Stock Market Prices ↓Spot FX ↓ (A)

Stock Market Prices ↑Spot FX ↓ (D)

Stock Market Prices ↑Spot FX ↑ (B)

Stock Market Prices ↓Spot FX ↑ (C)

Negative

Positive

Negative PositiveCurrency Market Return

Table 15.5

15-44

• The Mexican peso devaluation in late 1994 and early 1995 is an example of cell A, where capital flight and a loss in confidence in the Mexican economy brought the Mexican stock market down as well.With the peso overvalued and the country running a large current account deficit, Mexican policy makers allowed the peso to depreciate. Interest rates rose dramatically, as did import prices; the Mexican stock market dropped sharply in anticipation of a fall in Mexican GDP and corporate profits.


15-45

Pricing Determinants

• The analysis of international equity prices requires us to confront several challenging problems:1 Are national equity markets integrated or

segmented?2 Are national equity markets efficient or

inefficient?3 Does purchasing power parity hold or not?4 Do the assumptions of the capital asset pricing

model (CAPM) apply or is arbitrage pricing theory (APT) more appropriate?

$$

15-46


• The traditional CAPM hypothesizes that returns for an individual equity (Ri) in excess of the risk-free rate (RF) are proportional to the systematic risk of the equity (βiM) times the expected market risk premium :

where E(RM) is the expected return on the market portfolio.

$$

( )[ ]FMiMFi RRERR −=− β

15-47


• The assumptions of the traditional CAPM are:¤ Investors maximize their utility which depends

only on expected return (+) and risk (-).¤ Investors have homogeneous expectations,

agreeing about expected return and risk for all assets.

¤ Returns are expressed in nominal terms.¤ A risk-free interest rate exists and unlimited

borrowing and investing is possible at this rate.¤ No transaction costs or taxes exist.

$$

15-48

Pricing Determinants• A security’s β is related to its covariance with the

return on the market portfolio.

regression coefficient

• β tells us how much the security’s rate of return tends to change when the return on the market portfolio changes.

• Thus, for a security with a β of 2, if the market goes up by 10% more than what was expected, the return on the security will tend to go up by 20% more than what was expected.

$$

2W

iW

σσβ ≡

15-49


• The CAPM leads to a separation, or mutual fund theorem, which claims that all investors will hold some combination of two assets: The risk-free asset and the market portfolio of all risky assets.

$$

15-50

Is Investment in MNCs a Close Substitutefor International Investment?

• If portfolios exhibit a home country bias, can investors argue that the shares of multinational corporations (MNCs) offer a close substitute for international diversification?

• The shares of an MNC could reflect real assets and/or cash flows from, say, 20 countries. So, the MNC could offer “ready-made”diversification and an inexpensive proxy for the purchase of 20 firms, each one based in a different country.

15-51

Is Investment in MNCs a Close Substitutefor International Investment?

• While this strategy sounds reasonable, the data reject the hypothesis that MNCs are a proxy for foreign markets or international diversification. ¤ A study by Jacquillat and Solnik (1978)

examined the returns of MNCs from 9 countries by regressing their returns against all 9 market indexes.

¤ In each case, the returns on MNCs were most closely connected with the domestic market index. In the case of U.S. and U.K. MNCs, the addition of foreign markets offered virtually no improved explanation of MNC share returns.

15-52

Policy Matters - Private Enterprises

• After 15 years, Let’s ask the same question: “Can investors create “homemade”international diversification?”¤ Using data from 1973 to 1993 for seven

developed and nine emerging markets, a study found that a set of domestically traded assets, including market indices, industry portfolios, 30 MNCs, closed-end country funds and ADRs, was successful at mimicking the gains from international portfolio diversification.

15-53Tota

l, D

omes

tic, a

nd F

orei

gn C

ompa

ny L

istin

gs o

n M

ajor

Nat

iona

l Sto

ck E

xcha

nges

for 2

003

15-54

Assignments from Chapter 15Exercises 3, 4, 5.

(no need to hand in)

15-55

Overview• Foreign Exchange Market Intervention

¤ Intervention as a Policy Instrument¤ The Objectives of Central Bank Intervention¤ The Mechanics of Intervention¤ Empirical Evidence on Intervention¤ The Effectiveness of Central Bank Intervention¤ Security Transaction Taxes: Should We Throw

Sand in the Gears of Financial Markets?

Chap 17 Foreign Exchange Market Intervention (pp. 656-71)

15-56

Foreign Exchange MarketIntervention

• Many government actions (such as monetary policy, interest rate policy, fiscal spending, and taxation policies) can have an impact on the foreign exchange rate.

• The central bank may also intervene officially by directly purchasing or selling currency.

• Intervention is an essential part of a pegged exchange rate system.

15-57


• The modern experience of floating exchange rates is better described as a period of managed floating exchange rates.

• Note that acknowledging the importance of exchange rates and the potentially adverse effects of exchange rate misalignments or volatility does not automatically establish a valid case for central bank intervention.

15-58


• Under floating exchange rates, an active intervention policy presumes that:

markets are at times inefficient, thus permitting misaligned or excessively volatile rates,policymakers can identify such market inefficiencies,intervention techniques can correct the misalignments and excess volatility, andthe benefits from the correction exceed the costs of conducting the intervention.

15-59

The Objectives ofCentral Bank Intervention

• Shortly after the breakdown of the BrettonWoods Agreement in 1973, the International Monetary Fund (IMF) enacted a set of guidelines designed to limit the use of intervention and the potential for conflicts among nations.

15-60

The Objectives ofCentral Bank Intervention

• The guidelines, which are still in effect, specify that member nations of the IMF:

Have an obligation to intervene to prevent “disorderly conditions” in the foreign exchange market.Should avoid manipulating exchange rates to prevent balance of payments adjustment or gain an unfair competitive advantage in trade.Should take into account the interests and policies of other members when setting their own intervention policies.

15-61

Eun: International Financial Management Chapter 4: The Market for Foreign Exchange

http://highered.mcgraw-hill.com/sites/dl/free/0072521279/91312/eun21279_ch04_dr.pdf

15-62

Eun: International Financial Management Chapter 4: The Market for Foreign Exchange

http://highered.mcgraw-hill.com/sites/dl/free/0072521279/91312/eun21279_ch04_dr.pdf

15-63

Stephen G. Cecchetti on Central Banks, Monetary Policy, and

Financial Stability

(great optional reading)Chapter 15 Central Banks in the World Today

http://highered.mcgraw-hill.com/sites/dl/free/0072452692/238721/cec5269

2_ch15.pdf

15-64

Stephen G. Cecchetti on Central Banks, Monetary Policy, and

Financial Stability

(great optional reading)Chapter 16 The Structure of Central Banks:

The Federal Reserve and the European Central Bankhttp://highered.mcgraw-

hill.com/sites/dl/free/0072452692/238721/cec52692_ch16.pdf

15-65

The Mechanics of Intervention

• Central bank interventions typically occur in the spot foreign exchange market.¤ If the domestic currency is stronger than

desired, the central bank sells domestic currency, and vice versa.

• Central bank interventions may generate direct effects associated with the changed quantities of money and/or bonds.¤ The magnitude of the effects depends on

whether the intervention was sterilized or unsterilized.

15-66


• An unsterilized intervention is simply a foreign exchange market sale or purchase. ¤ The money supplies in both countries are

affected.• A sterilized intervention includes an offsetting

transaction in the domestic money market (such as the purchase or sale of government securities) that reverses, or sterilizes, the impact of the initial intervention transaction.¤ The money supplies remain unchanged, but the

bond supplies are affected.

15-67


• According to the monetary approach, sterilized interventions have no direct impact on the exchange rate.

• However, according to the portfolio balance approach, the relative supply of government bonds helps to determine the exchange rate.

15-68


• Central bank interventions may also generate indirect effects:

They may signal the market about future monetary and fiscal policies.They may interrupt short-term patterns in rates and reduce the profitability and incidence of noise trading.

15-69

Empirical Evidence on Intervention

• From 1982 to 1991, the U.S. Federal Reserve sold $35.8 billions and purchased $15.8 billions.¤ Note that the interventions were small

compared with the daily foreign exchange trading volume.

• There was also evidence of coordinated interventions with other central banks, such as the German Bundesbank and the Swiss National Bank.

15-70

The Effectiveness ofCentral Bank Intervention

• Does intervention have any effect - beneficial or detrimental - on the course of exchange rates and the ability of policymakers to achieve their larger macroeconomic goals?

• The debate hinges on whether a market failure has occurred and whether official intervention can correct this failure.

15-71

The Effectiveness ofCentral Bank InterventionPrivate Speculation Official Intervention

A CStabilizing Efficient markets view Official intervention

smoothes the marketCredible signals of future

policy remove uncertaintyEncourages stabilizing

private speculatorsB D

Destabilizing Inefficient markets: Stabilization policy gamedbandwagons, bubbles, by market and becomesnoise traders destabilizing

Intervention is inconsistentwith underlying economicpolicies

15-72

The Effectiveness ofCentral Bank Intervention

• Evidence suggests that intervention may stabilize exchange rates by lowering the daily volatility, as well as cause the rates to move in the intended direction.

• It seems that interventions send the strongest signals and have the highest chance of success when the conditions of surprise,

publicity, and coordination with other central banks, are met.

15-73

Excel Finance Gallery

Available at Companion Website forOxford Handbook of Financial Modeling by Ho and Lee

Oup.org (Oxford University Press Website)

94 downloadable Excel Templates for Finance Modelers.

15-74

Stock fund Bond FundRate of Squared Rate of Squared

Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00%Normal 12% 0.01% 7% 0.00%Boom 28% 2.89% -3% 1.00%Expected return 11.00% 7.00%Variance 0.0205 0.0067Standard Deviation 14.3% 8.2%

Ross / Corporate Finance 7E / CAPMCapital Asset Pricing Model

10.3 The Return and Risk for Portfolios

Note that stocks have a higher expected return than bonds and higher risk. Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks.

15-75


Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.160%Normal 12% 7% 9.5% 0.003%Boom 28% -3% 12.5% 0.123%

Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08%

The rate of return on the portfolio is a weighted average of thereturns on the stocks and bonds in the portfolio:

SSBBP rwrwr +=

%)17(%50%)7(%50%5 ×+−×=

15-76




The rate of return on the portfolio is a weighted average of thereturns on the stocks and bonds in the portfolio:

%)3(%50%)28(%50%5.12 −´+´=

SSBBP rwrwr +=

15-77




The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio.

%)7(%50%)11(%50%9 ´+´=

)()()( SSBBP rEwrEwrE +=

15-78




The variance of the rate of return on the two risky assets portfolio is

BSSSBB2

SS2

BB2P )ρσ)(wσ2(w)σ(w)σ(wσ ++=

where ρBS is the correlation coefficient between the returns on the stock and bond funds.

15-79




Observe the decrease in risk that diversification offers.

An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than stocks or bonds held in isolation.

15-80

Portfolo Risk and Return Combinations

5.0%6.0%7.0%8.0%9.0%

10.0%11.0%12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)

Por

tfol

io R

etur

n

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%

50.00% 3.08% 9.00%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%

10.4 The Efficient Set for Two Assets

We can consider other portfolio weights besides 50% in stocks and 50% in bonds …

100% bonds

100% stocks

15-81


5.0%6.0%7.0%8.0%9.0%

10.0%11.0%12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%


Por

tfol

io R

etur

n


We can consider other portfolio weights besides 50% in stocks and 50% in bonds …

100% bonds

100% stocks

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%50% 3.1% 9.0%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%


100% 14.3% 11.0%

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%

10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%50% 3.1% 9.0%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%


100% 14.3% 11.0%

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%50% 3.1% 9.0%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%100% 14.3% 11.0%


100% 14.3% 11.0%

15-82


5.0%6.0%7.0%8.0%9.0%

10.0%11.0%12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%


Por

tfol

io R

etur

n% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%50% 3.1% 9.0%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%


100% stocks

100% bonds

Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. These compromise the

efficient frontier.

15-83

Definition of Risk When Investors Holdthe Market Portfolio

• Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta (β)of the security.

• Beta measures the responsiveness of a security to movements in the market portfolio.

)()(

2,

M

Mii R

RRCovσ

β =

15-84

Estimating β with regression

Secu

rity

Ret

urns

Secu

rity

Ret

urns

Return on Return on market %market %

RRii = = αα ii + + ββiiRRmm + + eeii

Slope = Slope = ββiiCharacte

ristic L

ine

Characterist

ic Line

15-85

Estimates of β for Selected Stocks

0.49Oracle, Inc.

0.20Homestake Mining

0.55Green Mountain Power

1.05Microsoft

0.90Kimberly-Clark Corp.

1.00Du Pont

1.65Travelers, Inc.

2.35Borland International

1.55Bank of America

BetaStock

15-86

The Formula for Beta

)()(

2,

M

Mii R

RRCovσ

β =

Clearly, your estimate of beta will depend upon your choice of a proxy for the market portfolio.

15-87

10.9 Relationship between Riskand Expected Return (CAPM)

Expected Return on the Market:

Expected return on an individual security:

PremiumRisk Market += FM RR

)(β FMiFi RRRR −×+=

Market Risk Premium

This applies to individual securities held within well-diversified portfolios.

15-88

Expected Return on an Individual Security

• This formula is called the Capital Asset Pricing Model (CAPM)

)(β FMiFi RRRR −´+=

• Assume βi = 0, then the expected return is RF.• Assume βi = 1, then Mi RR =

Expected return on a security

= Risk-free rate + Beta of the

security ×Market risk

premium

15-89

Relationship Between Risk & Expected Return

Expe

cted

retu

rn

β


FR

1.0

MR

15-90

Relationship Between Risk & Expected Return

Expe

cted

re

turn

β

%3=FR

%3

1.5

%5.13

5.1β =i %10=MR%5.13%)3%10(5.1%3 =−×+=iR

15-91

10.10 Summary and Conclusions• This chapter sets forth the principles of modern portfolio theory.• The expected return and variance on a portfolio of two securities

A and B are given by

• By varying wA, one can trace out the efficient set of portfolios. We graphed the efficient set for the two-asset case as a curve, pointing out that the degree of curvature reflects the diversification effect: the lower the correlation between the two securities, the greater the diversification.

• The same general shape holds in a world of many assets.

ABAABB2

BB2

AA2P )ρσ)(wσ2(w)σ(w)σ(wσ ++=

)()()( BBAAP rEwrEwrE +=

15-92

10.10 Summary and Conclusions

• The efficient set of risky assets can be combined with riskless borrowing and lending. In this case, a rational investor will always choose to hold the portfolio of risky securities represented by the market portfolio.

retu

rn

σP

efficient frontier

rf

M

CML

Then with borrowing or lending, the investor selects a point along the CML.

15-93

10.10 Summary and Conclusions

• The contribution of a security to the risk of a well-diversified portfolio is proportional to the covariance of the security's return with the market’s return. This contribution is called the beta.

• The CAPM states that the expected return on a security is positively related to the security’s beta:

)()(

2,

M

Mii R

RRCovσ

β =


15-94

Ross / Corporate Finance 7E / CAPM12.1 The Cost of Equity Capital

Invest in project

Firm withexcess cash

Shareholder’s Terminal

Value

Pay cash dividend

Shareholder invests in financial

asset

Because stockholders can reinvest the dividend in risky financial assets, the expected return on a capital-budgeting project should be at least as great as the expected return on a financial asset of comparable risk.

A firm with excess cash can either pay a dividend or make a capital investment

15-95

The Cost of Equity

• From the firm’s perspective, the expected return is the Cost of Equity Capital:

• To estimate a firm’s cost of equity capital, we need to know three things:

)( FMiFi RRβRR −+=

1. The risk-free rate, RF

FM RR −2. The market risk premium,

2,

)(),(

M

Mi

M

Mii

σ

σ

RVarRRCov

β ==3. The company beta,

15-96

Example

• Suppose the stock of Stansfield Enterprises, a publisher of PowerPoint presentations, has a beta of 2.5. The firm is 100-percent equity financed.

• Assume a risk-free rate of 5-percent and a market risk premium of 10-percent.

• What is the appropriate discount rate for an expansion of this firm?

)( FMiF RRβRR −+=

%105.2%5 ´+=R%30=R

15-97

Example (continued)Suppose Stansfield Enterprises is evaluating the following non-mutually exclusive projects. Each costs $100 and lasts one year.

Project Project β Project’s Estimated Cash Flows Next Year

IRR NPV at 30%

A 2.5 $150 50% $15.38

B 2.5 $130 30% $0

C 2.5 $110 10% -$15.38

15-98

Using the SML to Estimate the Risk-Adjusted Discount Rate for Projects

An all-equity firm should accept a project whose IRR exceeds the cost of equity capital and reject projects whose IRRs fall short of the cost of capital.

Proj

ect

IRR

Firm’s risk (beta)

SML

5%

Good project

Bad project

30%

2.5

A

B

C

15-99

12.2 Estimation of Beta: Measuring Market Risk

Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the S&P Composite, is used to represent the market.

Beta - Sensitivity of a stock’s return to the return on the market portfolio.

15-100

12.2 Estimation of Beta• Theoretically, the calculation of beta is straightforward:

• Problems1. Betas may vary over time.2. The sample size may be inadequate.3. Betas are influenced by changing financial leverage and business risk.

• Solutions¤ Problems 1 and 2 (above) can be moderated by more sophisticated

statistical techniques.¤ Problem 3 can be lessened by adjusting for changes in business and

financial risk.¤ Look at average beta estimates of comparable firms in the industry.

2,

)(),(

M

Mi

M

Mi

σσ

RVarRRCovβ ==

15-101

Stability of Beta

• Most analysts argue that betas are generally stable for firms remaining in the same industry.

• That’s not to say that a firm’s beta can’t change.¤ Changes in product line¤ Changes in technology¤ Deregulation¤ Changes in financial leverage

15-102

Using an Industry Beta

• It is frequently argued that one can better estimate a firm’s beta by involving the whole industry.

• If you believe that the operations of the firm are similar to the operations of the rest of the industry, you should use the industry beta.

• If you believe that the operations of the firm are fundamentally different from the operations of the rest of the industry, you should use the firm’s beta.

• Don’t forget about adjustments for financial leverage.

15-103

Modern Financial Markets:Prices, Yields, and Risk Analysis

Blackwell, Griffiths and Winters

Chapter 15

Stock Portfolio Formation andRisk Management

15-104

Learning Objectives

1. Portfolio formation and correlation2. Measuring portfolio risk3. Incremental value-at-risk4. Portfolio risk management strategies using

derivative securities

15-105

The Basics of Portfolio Formation

A portfolio is a group of assets. The relative importance (weight) of an asset in a portfolio is based on the asset’s contribution to the valueof the portfolio.

We will focus on forming a stock portfolio.

15-106

The Basics of Portfolio Formation (cont.)

ExampleLet’s assume we have £15,000 invested in

our portfolio and the investment is divided among three stocks:FLYBY £5,000 33%UO £6,000 40%GDAY £4,000 27%

15-107

The Basics of Portfolio Formation (cont.)Example (cont.)

Investment ExpectedReturn

(Annual)

StandardDeviation(Monthly)

Beta

FLYBY 11% 10% 1.05

UO 9% 5% 0.95

GDAY 12% 7% 1.11

CorrelationCoefficients

FLYBY UO GDAY

FLYBY 1.0 0.8 0.5

UO 0.8 1.0 0.2

GDAY 0.5 0.2 1.0

15-108


E[R] = w1*r1 + w2 * r2 + w3*r3 = 0.33(11%) + 0.40(9%) + 0.27(12%) = 10.47%

E[β] = w1*β1 + w2 *β2 + w3*β3 = 0.33(1.05) + 0.40(0.95) + 0.27(1.11)= 1.03

So, what is the likely return on the portfolio? The answer isthe weighted average of the individual returns.

Also, what is the Beta (β) of the portfolio? The answer is theweighted average of the individual Betas.

15-109


The third calculation we would like to do for our portfolio is its standard deviation. However, the standard deviation of the portfolio is notthe weighted average of the individual standard deviations because of the difference between diversifiable and non-diversifiable risk.

15-110

The Basics of Portfolio Formation (cont.)Correlation

The degree of correlation is a measure of the extent to which returns on two assets move together.

If both move up and down together, they are positively correlated and ρij > 0.

15-111

The Basics of Portfolio Formation (cont.)Correlation (cont.)

Less Than Perfect Positive Correlation

TimeR

etur

n

Stock AStock B

Perfect Positive Correlation

Time

Ret

urn

Stock A

Stock B

15-112


If one moves up when the other moves down, they are negatively correlated and ρij < 0.

Perfect Negative Correlation

Time

Ret

urn

Stock A

Stock B

Less Than Perfect NegativeCorrelation

Time

Ret

urn

Stock A

Stock B

15-113


If the two assets are completely independent, then they are uncorrelated and ρij = 0.

Zero Correlation

Time

Ret

urn

Stock A

Stock B

15-114

Measuring Portfolio Risk

We now want to measure portfolio risk and since a portfolio has more than one asset we have to consider the correlation of the assets in the portfolio.

The standard deviation of a portfolio includes the correlation between the assets in the portfolio and thus provides a measure of portfolio risk.

15-115

Measuring Portfolio Risk (cont.)

The formula for portfolio standard deviation is:

( )[ ] 5.022 2∑ ∑∑+= jiijjiiip www σσρσσ

15-116


Now, let’s return to our portfolio and calculate its standarddeviation.

GDAY

UO

FLYBY

GDAYUOFLYBYINVESTMENT

The shaded area (on the diagonal) represent the weighted totalrisk of the of the individual securities in the portfolio,which in the formula is Σwi

2σi2

15-117


The off-diagonal items represent the correlations between the different assets in the portfolio and the formula is:

2(ΣΣwiwjρijσiσj)

The formula starts by multiplying by 2 because the items above the diagonal are the same as the items below the diagonal.

15-118


(0.27)2(.07)2GDAY

(.4)(.27)(.2)(.05)(.07)(0.4)2(0.05)2UO

(.33)(.27)(.5)(.1)(.07)(.33)(.4)(.8)(.1)(.05)(0.33)(0.33)22(.1)(.1)22FLYBY

GDAYUOFLYBYINVESTMENT

Using our portfolio, the cells of the figure are as follows:

15-119


( )[ ]

6.06%or 0606.

]003678[.

)]000076.000312.000528(.2000357.0004.001089[.

))]07)(.05)(.2)(.27)(.4(.)07)(.1)(.5)(.27)(.33(.)05)(.1)(.8)(.4)(.33((.2

)07(.)27(.)05(.)4(.)1(.)33[(.

2

5.0

5.0

5.0

222222

5.022

=

=

+++++=

++

+++=

+= ∑ ∑∑p

jiijjiiip www

σ

σσρσσ

Now, the calculation for the portfolio standard deviation is:

15-120


The financial definition of risk is uncertainty and standard deviation provides a measure of uncertainty. However, individuals often think of risk in terms of losses and focus on the absolute dollar value of their losses.

The focus on dollar losses as a concept of risk has led to the development of an alternative measure of risk called Value-at-Risk.

15-121

Value-at-Risk

To measure value-at-risk, we change our focus from portfolio percentages to dollar value invested. With this change we can measure portfolio standard deviation in terms of dollar value invested. That is, we replace the portfolio percentage weights with the dollar values invested. The calculation for our portfolio is as follows:

15-122

Value-at-Risk (cont.)

( )[ ]

912.14

]000,832[

)]800,16000,70000,120(2400,78000,90000,250[

))]07)(.05)(.2)(.4000)(6000()07)(.1)(.5)(.4000)(5000()05)(.1)(.8)(.6000)(5000((2

)07(.)4000()05(.)6000()1(.)5000[(

2

5.0

5.0

5.0

222222

5.022

=

=

+++++=

++

+++=

+= ∑ ∑∑p

jiijjiiip www

σ

σσρσσ

Continuing with our portfolio, the portfolio standard deviationbased on value invested is.

15-123


Having calculated the portfolio standard deviation in terms of value invested, we have completed the first step in determining Value-at-Risk.

Value-at-Risk provides the expected maximum loss over a target horizon with a given level of confidence.

15-124


To calculate Value-at-Risk, we need two more pieces of information:

1. Length of holding period, which is chosen to match the amount of time required to liquidate the portfolio in an orderly manner.

2. Confidence interval, which is a function of the amount of risk aversion of the investor. Higher confidence intervals imply higher value-at-risk figures.

15-125


Both measures are somewhat arbitrary and are chosen to fit the situation and the investors.

For a stock portfolio, a common horizon is one month and we will choose a confidence level of 5% for our calculation of value-at-risk.

15-126


Recall from statistics that the point estimate from a confidence level is as follows:

Point estimate = +/- (confidence level critical value) * (standard deviation)

Since we are looking at value losses, we focus only on the left tail of the distribution.

15-127


Assuming that the changes in the value of our portfolio arenormally distributed then the critical value for theconfidence interval can been seen from Figure 15-6.

-3 -2 -1 0 1 2 3

N(d)

1.00

0.50

0.05

d = Standard Normal Variable

1.65 σ

c = 5%confidencelevel

15-128


Value-at-Risk calculation:Portfolio standard deviation in value is

£912.14 and critical value for 5% lower tail confidence interval is 1.65.

So, the value-at-risk is £1,505.03 = (£912.14*1.65).

This means that we are 95% confident that our maximum monthly loss is £1,505.03.

15-129

Incremental Value-at-Risk

Value-at-Risk provides a calculation of portfolio risk. However, we may want to know which security provides the most risk (or threat to maintaining value) in our portfolio.

We can address this question by calculating incremental value-at-risk.

15-130

Incremental Value-at-Risk (cont.)

Incremental value-at-risk is a two step calculation with the steps as follows:

1. Calculate the individual stock value variance in the portfolio followed by

2. Calculating the individual stock contribution to the value-at-risk for the portfolio.

15-131


Step 1 of incremental value-at-risk

=

=

=

£41.300.0007£6,000+0.0035£5,000+0.0049£4,000GDAY

£37.800.0007£4,000+0.004£5,000+0.0025£6,000UO

£88.000.0035£4,000+0.004£6,000+0.01£5,000FLYBY

CovariancePosition+CovariancePosition+VariancePosition

Stock

Recall that covariance = ρijσiσj so the covariance between GDAY and FLYBYis (0.5)(.1)(.07) = 0.0035

15-132


Step 2 of incremental value-at-risk

=

=

=

£1505.04Total(rounded)

£298.94£4,000*£1505.03*832000÷41.30GDAY

£410.27£6,000*£1505.03*832000÷37.80UO

£795.93£5,000*£1505.03*832000÷88.00FLYBY

Stock Wealth Position

*Portfolio Value-at-

Risk

*Portfolio Variance

÷Stock (Variance/

Covariance)

Stock

15-133


Points from incremental value-at-risk.1. FLYBY puts the most value-at-risk even

though at is not the largest investment in the portfolio.

2. UO has the lowest value variance (from Step 1) but is not the least risky portion of our portfolio because of the large value investment in UO.

15-134

Three Important Points about Value-at-Risk

1. The value-at-risk calculation is only an estimate. It is not a guarantee.

2. The value-at-risk calculation assumes that stock returns follow a certain distribution (e.g., normal).

3. To this point we have not borrowed to buy stock. Borrowing to buy stocks is referred to as trading on margin.

15-135

Value-at-Risk with Margin

Our calculations assumed that we owned the stocks in our portfolio. If an investors uses margin to buy stock, then the investor does not own the entire investment because a portion of the cash flows are obligated to the lender.

Buying on margin is a classic example of leverage; we know that leverage increases risk.

15-136

Value-at-Risk with Margin (cont.)

Let’s assume that we borrow £15,000 at a rate of 7% to increase the positions in each of the three stock in our portfolio by £5,000.

£21,000£9,000£15,000£15,000Net Equity Value

£36,000£24,000£30,000£15,000Total Equity Value

£15,000£15,000£15,000£ 0Margin Loan

£10,800£7,200£9,000£4,000GDAY

£13,200£8,800£11,000£6,000UO

£12,000£8,000£10,000£5,000FLYBY

Market Rises 20%

Market Falls 20%

Margin PositionOriginal position

Security

15-137


The impact of margin on portfolio return is the following

(7,950-15,000)/15,000 = -47%

£7,950£16,050£24,000Market falls by 20%

(13,950-15,000)/15,000 = -7%

£13,950£16,050£30,000Market Unchanged

(19,950-15,000)/15,000 =33%

£19,950£16,050£36,000Market Rises by 20%

Investor ReturnNet Gain (Loss)

Repayment of Principal and Interest

Ending Portfolio Value

15-138


Points about investor returns with margin.1. When there is no change in the market, the

investor loses because of the interest owed on the borrowing.

2. Leverage magnifies both gains and losses.

15-139


Now, let’s take the leverage effect of margin and apply it to our value-at-risk calculation for the portfolio.

Re-calculating the money standard deviation with margin, the standard deviation becomes £3,351, which means at a confidence interval of 95% the value-at-risk of our portfolio with £15,000 of margin borrowing is £5,529.15, which is a 3.67 time increase over the original value-at-risk amount.

15-140

Global Portfolios and Value-at-Risk

Let’s use market indices from 13 stock markets around the world and look at value-at-risk for an equal weighted portfolio invested in the 13 market indices.

Returns, standard deviations and correlations for the indices using data from January 1998 through December 1999 are:

15-141

Global Portfolios and Value-at-Risk (cont.)

Country E[R] standard deviationUSA 0.0039 0.0255Egypt 0.0053 0.0154France 0.0079 0.0283Germany 0.0045 0.0369Hong Kong 0.0044 0.0480Israel 0.0067 0.0313Japan 0.0021 0.0303Korea 0.0094 0.0641Mexico 0.0030 0.0494So Africa 0.0030 0.0404Taiwan 0.0003 0.0417Canada 0.0023 0.0252UK 0.0022 0.0240

15-142


S&P Egpt Fr Ger HK Isr Jap Kor Mex So Af Tai Cda UKS&P 1.00 -0.09 0.79 0.68 0.47 0.38 0.35 0.19 0.57 -0.12 0.23 0.73 0.69Egypt 1.00 -0.19 -0.09 -0.02 0.15 0.00 0.06 0.08 -0.07 -0.04 0.02 -0.11France 1.00 0.83 0.45 0.24 0.29 0.26 0.33 0.13 0.16 0.72 0.71Germany 1.00 0.43 0.38 0.24 0.20 0.41 -0.07 0.25 0.67 0.71Hong 1.00 0.09 0.29 0.33 0.40 -0.19 0.23 0.41 0.47Israel 1.00 0.29 0.20 0.36 0.08 0.26 0.35 0.32Japan 1.00 0.28 0.39 -0.06 0.19 0.37 0.31Korea 1.00 0.17 -0.12 0.01 0.26 0.32Mexico 1.00 -0.11 0.24 0.40 0.38So Africa 1.00 -0.06 -0.10 -0.13Taiwan 1.00 0.17 0.11Canada 1.00 0.67UK 1.00

Correlations

15-143


If a $1,000,000 investment is divided equally among the 13 indices, the incremental value-at-risk of each index is:

inc riskUSA 331.21Egypt 121.38France 408.44Germany 692.12Hong Kong 1172.71Israel 499.31Japan 467.54Korea 2091.72Mexico 1240.27So Africa 828.91Taiwan 885.61Canada 323.90UK 293.88

15-144


So, the data suggest that the high risk indices in our equal weighted portfolio are:

Korea, Mexico, and Hong Kong.

15-145


Repeating the portfolio calculations for the first 15 weeks of 2000.

Country E[R] standard deviationUSA -0.0053 0.0445Egypt 0.0038 0.0186France 0.0012 0.0406Germany 0.0024 0.0323Hong Kong -0.0033 0.0489Israel 0.0019 0.0364Japan 0.0051 0.0116Korea -0.0166 0.0343Mexico -0.0081 0.0701So Africa -0.0100 0.0380Taiwan 0.0069 0.0440Canada 0.0028 0.0481UK -0.0038 0.0284

15-146


Country inc riskUSA 682.48Egypt 118.55France 567.50Germany 359.87Hong Kong 823.29Israel 456.73Japan 46.10Korea 405.30Mexico 1691.40So Africa 498.00Taiwan 667.96Canada 798.54UK 278.63

15-147


The new data suggest that high risk indices in our equally weighted portfolio are:

Mexico, Hong Kong, and Canada.

15-148


What do we learn from these calculations about incremental value-at-risk?

Our estimates of VaR are only as good as the data. That is, we assume that historic data is the best predictor of the future, so our calculations are only as good as data is as a predictor.

15-149

Portfolio Risk Management Strategies

Now that we can quantify portfolio risk, both in percentages and dollars, we need to turn our attention to how to manage this risk exposure.

We discuss two techniques:1. Protective puts and2. Protective collars.

15-150

Portfolio Risk Management Strategies (cont.)Protective Puts

Protective puts are often referred to as portfolio insurance. That is, you pay for the put option to protect against ‘bad events’ and if ‘good events’ occur the insurance (put) expires unused.

15-151

Portfolio Risk Management Strategies (cont.)Protective Puts (cont.)

The idea behind a protective put is the investor has a portfolio (or an asset) that is valuable, that the investor does not want to sell, and that the investor is concerned about a near-term decrease in value.

So, the investor buys put options on the portfolio to protect the current value.

15-152


Recalling the vector notation for options in Chapter 10, applying a protective put to a portfolio is:

Long Portfolio + Buy a Put = Protected Position

│ 1│ │0│ │1││-1│ + │1│= │0│

15-153


So, what we see from the vector notation is that the portfolio with a protective put (buy the put so we have the right to sell the portfolio at the strike price) gains when the portfolio gains, but has no losses when the portfolio declines in value.

15-154


Index Level

Profit

Loss

UnprotectedPortfolio

Portfolio withProtective Put

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Portfolio Risk Management Strategies (cont.)Protective Puts (cont.) Example

Example inputsPortfolio = $4,000,000 and is roughly similar to the NASDAQ Composite Index (NMCI), NMCI = 1580,Put option = $37 at a strike index level of 1580 and the size of one contract is index level * 100.

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Example cont.Position and cost of protective put1.$4,000,000/(1580*100) = 25.3 contracts. We

cannot buy a fractional contract so we buy 25 put options.

2.The cost of the position is $37 * 100 * 25 = $92,500.

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Example cont.What happens on the put expiration date?• If the index increases in value, we let the put

expire; the cost of our portfolio insurance, $92,500, is lost.

• If the index declines, we exercise the put to recover the losses on the portfolio. Of course, we still pay the $92,500 for the insurance but this time we use the insurance to cover losses.

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The protective put limits any losses on our portfolio. However, the cost of the insurance (protective put) is high.

A method to limit the cost of the portfolio insurance is to apply a protective collar to the portfolio instead of a protective put.

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Portfolio Risk Management Strategies (cont.)Protective Collar

A protective collar is designed to limit the downside losses on a portfolio at reduced premium costs.

A protective collar is the combination of a buying a put with selling a call on the portfolio with the put and call at different strike prices.

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Portfolio Risk Management Strategies (cont.)Protective Collar (cont.)

The put and call options in the collar are chosen so that they have similar premiums.

The idea behind a collar is that you buy loss protection for the portfolio (buy the put) by selling some portion of the potential gains of the portfolio (sell the call).

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Index Level

Profit

Loss

UnprotectedPortfolio

Protective put from the collar to limit losses on the portfolio

Call limits gains on the portfolio but was sold to pay for the protective put

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Points about the protective collar1. The collar, as drawn, is designed to limit

risk on the portfolio around the current index level.

2. The put strike index level is different (lower) than the call strike index level.

3. Between the two strike levels the portfolio value is NOT hedged and therefore changes.

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stanford2008.pageout week post… · 15-2 Levich Chap 15 Scan Read Pages 554-597 Equity Portfolios Luenberger Solnik Eun Chap 8, 9, 12, 13 Chap Pages Pages 317-84, 385-432, 523-627