tuck bridge finance module 5

8/4/2019 Tuck Bridge Finance Module 5

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1 Investment decisions

We move on to consider optimal investment problems .

Decision: where to invest our wealth.

Our optimality criterion will be to achieve the maximumreturn for a given level of risk .

Consider only one-period models: you invest today, next

period you consume the payoffs from the investment (i.e.ignore the investment/consumption decision).

Finance, Bridge Program 2005 3


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Investment opportunity setBy asset I mean any type of investment:

Stocks (domestic or international).

Bonds (domestic, foreign, corporate, government).

Real estate. Derivatives (options, futures, swaps).

Human capital (college education, Bridge, MBA,. . . ).

Funds (mutual funds and hedge funds).

Indexes (equal versus value weighted).

Many others: venture capital funds, private equity funds,wine, CDs, paintings, . . . .



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The investment decision

The portfolio choice is summarized by set of weights w i , wherew i denotes the proportion of your wealth invested in asset i .

Convenient to call the portfolio decision a vector

w =

w1

w2...

wn

where n denotes the total number of assets available forinvestment.

The weights must add up to 1: ni =1 w i = 1 .

Example: 20% in LNUX, 30% in MSFT, 50% in risk-free bonds.



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ShortingShorting, w i < 0, is in principle allowed.

To short one share of an asset simply means that you get itscurrent price today, and promise to give back the shares in the

future (i.e. to give the other investor the price of the shares ata future date plus any dividends).

Consider an investor with wealth of $200.

Consider investing $300 in PEP and shorting $100 of KO (youshort KO, get $100, and use these and your wealth to buy $300worth of PEP).

The weights are wKO = 50% , wP EP = 150% .



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Academic Shorting Example

Hypothetical Rates of Return: KO =

10% , P E P = +15% .

= Payoff: R P = 50% (10%) + 150% (+15%) = +27 .5% .$100 KO shares borrowed became a liability of $90;

$300 PEP shares invested became an asset of $345;

= The net portfolio gain is $55 on assets of $200;= +27 .5% rate of return.

Hypothetical Rates of Return: KO = +100% , P E P = 0% .

= Payoff: R P = 50% (+100%) + 150% (0%) = 50.%.$100 KO shares borrowed became a liability of $200;

$300 PEP shares invested became an asset of $300;

= The net portfolio loss is $100 on assets of $200;=

50% rate of return.



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The storylineWe will consider nding an optimal portfolio mix w (i.e. it isoptimal to invest 5% in KO, 30% in LNUX, -10% in MSFT,25% in IBM, 15% in long-term risk-free bonds, . . . ).

For a given portfolio w , the returns on the portfolio will be

R P = w1 R 1 + + wn R n =n

i =1w i R i ;

where R i is the actual return on asset i .

Optimality criterion: maximize expected return for a given levelof risk (volatility).

Therefore we now turn to the measurement of risk. Inparticular: (1) how to measure expected returns and risk of individual securities (Statistics), (2) how to measure return andrisk of portfolios knowing return/risk of assets (Finance)?



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2 Statistics and asset returns

Notation. Let Y be an arbitrary random variable whichcan take values Y 1 , . . . , Y k with probabilities p1 , . . . , p k .Then we dene

E f (Y ) =

k

i =1 pi f (Y i ) .

We will measure several things about the returns onassets. Let R denote the return of an asset. Then we

dene:

The expected return E [R ]. Measures the centraltendency of R . Sometimes we use symbol todenote expected returns.



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The correlation between two assets R i and R j isdened as

cor( R i , R j ) =cov( R i , R j )

SD( R i )SD( R j ).

Also use symbol i,j . Note i,j [1, 1]. The beta of asset i with asset m is

i,m =cov( R i , R m )

SD( R m )2=

i m

i,m .

If m is the market we typical abbreviate notation byletting i = i,m .



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Remarks

We normalize variance by taking var so our risk measure ismeaningful (measured in %, not %2 ).

We normalize the covariance in two ways:1. by dividing through by standard deviations to get

correlation;

2. by dividing by variance of one asset (typically themarket) to get beta.

Further note:

We will measure risk by the standard deviation of an asset.

We also measure covariation between assets, not just theindividual risk of the assets.



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Interpretation of standard deviationSay an asset has an annual expected return of 12% and astandard deviation of 15% (close to the S&P 500).

The probability that the returns of this asset are

in [3% , 27%] is about 68.2%; in [18% , 42%] is about 95.4%;

in [33% , 57%] is about 99.7% .Note: these are the 1 SD, 2 SD and 3 SD intervals around the

mean for this asset.Note: always convert variances into SDs (by taking squaredroot), variances are not meaningful numbers.



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Interpretation of beta and correlationSay an asset has a beta of 2 with respect to the S&P 500.

This means that, on average, as the S&P 500 returns move byx %, this asset will more by 2x %.

The correlation between two assets is a number between 1and 1. If it is positive the two assets, on average, movetogether. The higher the correlation the more two assetscomove.

One can interpret 2 as the percent of the variation of one assetthat can be explained by the movements of another asset.



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Assignment 4 - part 1Given historical monthly return data on S&P500 Index return,30-year bonds, and 90-day T-bills.

How come 30-year bonds and T-bills have negative returns?Interest rate movements make the price of the bonds uctuate.They are only risk-free if the investment horizon would beexactly equal to maturity of bonds (and they paid no coupons).

How can we estimate expected returns and standard deviations?

Simply use =AVERAGE() and =STDEV() formulas.



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Summary statisticsIn monthly terms:

S&P500 30-year bonds 90-day T-bills

Means 1.05% 0.46% 0.40%SDs 4.20% 2.56% 0.29%

Minimum -21.60% -7.73% -0.12%

Maximum 16.97% 13.31% 2.13%

Higher return is associated with higher risk .



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Annualizing expected returns and standard deviations

S&P500 30-year bonds 90-day T-bills

Mean 12.61% 5.53% 4.82%SD 14.55% 8.88% 1.01%

E [r AnnualReturns ] = 12 E [r MonthlyReturns ](super-geek slide takes care of compounding)

SD(AnnualReturns) = 12 SD(MonthlyReturns)What is the relationship between daily returns standarddeviation and annualized returns standard deviation?

SD(AnnualReturns) = 250 SD(DailyReturns)(approximately 250 trading days in a year)



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Remarks on estimationConsider two assets with the same standard deviation and

expected return. Given a nite amount of data, we will getdifferent returns for each assets, and therefore differentestimates for their standard deviation and expected return.These estimates obviously have risk!

This risk is usually measured by standard errors .

Some facts on estimation:

Standard errors of means are large and unavoidable.

Standard errors of standard deviations can be large, butthey also can be made arbitrarily small by looking athigh-frequency data (e.g. daily or even intradaily).

Standard errors for covariances (and correlations and betas)are typically also large, although they can also be madearbitrarily small by looking at high-frequency data.



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General n asset case

How can we compute the expected return and standarddeviation of the returns of a portfolio w ?

Expected return formula:

E R P = EN

i =1

w i

R i =

N

i =1

w i

E R i

Variance formula:

Var R P =N

j =1

N

k =1

w j

wk

Cov( R j , R k )

In order to nd standard deviation: SD( R P ) = Var R P .



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My approach to computations (aside)Put all the expected returns in a list, or vector , which I will simply call (but recall

that is a list of n different numbers). Literally, dene:

=

E [ R 1 ]

E [ R 2 ]

.

.

.

E [ R n ]

We organize the variance-covariance information in a n n table, in which ( i, j )element we have ij . Namely we have the following matrix , which well call :

=

11 12 . . . 1 n

21 22 . . . 2 n

. . . . . . . . . . . .

n 1 n 2 . . . nn

Note: diagonal elements are variances and off-diagonal elements are covariances.



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Spreadsheet tips (v.2 ctd)Let cells A1:A3 have the weights w1 , . . . , w N , cells B1:B3 have the

information on each asset expected returns, and C1:E3 have thevariance-covariance matrix.

Then in Excel you can compute the expected return of theportfolio as

= MMULT(TRANSPOSE (A1 : A3 ) , B1 : B3 )

and the variance of the portfolio as

= MMULT(MMULT(TRANSPOSE (A1 : A3 ) , C1 : E3 ) , A1 : A3 )

Note : in order for these formulas to work you need to hit , i.e. hold down the control and shift keysand then hit enter. Dont hit rst, if you do by mistakeyou need to edit the cell again, using F2 , and then hit ).



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Computing expected returns and risk of

portfoliosStock Weights E [r ] SD [r ] covariance

KO 0.2 0.1841 0.289 0.027424

IBM 0.8 0.1479 0.361

The expected return for portfolio A can be computed as

E [R A ] = (0 .2)0 .1841 + (0 .8)0 .1479 = 15 .51%

and its standard deviation

SD( R A ) = (0 .2)2 (0 .289) 2 + (0 .8)2 (0 .361) 2 + 2(0 .2)(0 .8)0 .027424= 30 .9%



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Computing expected returns and risk of

portfoliosStock Weights E [r ] SD [r ] covariance

IBM 0.5 0.1479 0.361 0.01829

EK 0.5 0.0972 0.245

The expected return for portfolio B can be computed as

E [R B ] = (0 .5)0 .1479 + (0 .5)0 .0972 = 12 .26%

and its standard deviation

SD( R B ) = (0 .5)2 (0 .361) 2 + (0 .5)2 (0 .245) 2 + 2(0 .5)(0 .5)0 .01829= 23 .8%



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Recap

Inputs of investment decision problem: means, variancesand covariances.

Estimating (with the aid of a computer) means, variancesand covariances.

Calculating the expected returns and variances of portfolios(with computer, and by hand for case of 2 assets).

Next two classes

Apply these concepts to decide what is the optimalinvestment we ought to make.

Thinking about risk in a portfolio context: the CAPM.


tuck bridge finance module 5

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