Tuck Bridge Finance Module 4

8/4/2019 Tuck Bridge Finance Module 4

1/23

CLASS 5

DIVERSIFICATION AND

OPTIMAL PORTFOLIOS

Bridge Program 2005

Finance module

Finance, Bridge Program 2005 1


2/23

Contents

1 Combining risk-free asset with one-risky asset 4

2 Combining two risky assets 9

3 Optimal portfolios 14

3.1 Two risky assets case . . . . . . . . . . . . . . . . 14

3.2 The general case . . . . . . . . . . . . . . . . . . . 20



3/23

Recap

Estimating expected returns and

variance-covariance matrix.

Finding expected returns and standard

deviations of the returns of given portfolios.

In this class we will use the techniques from class 4to determine optimal portfolios.



4/23

1 Combining risk-free asset with one-risky

assetWe will follow problem 1.

Risky asset with expected return 0.2 and standard

deviation 0.3. Risk-free asset yielding 0.08.

Invest w in risky.

The expected return of the portfolio is:

E[RP] = 0.2w + 0.08(1 w) = 0.08 + 0.12w.

And its variance and standard deviation:

Var[RP] = w2(0.30)2 p = 0.3|w|.



5/23

Risk-return tradeoff

We can further summarize the risk-return trade-off by

noting that w = p/0.3, and

E[RP] = 0.08 + 0.12w

= 0.08 +0.12

0.3p = 0.08 + 0.4p

In general, given a risky asset with expected return E [Ri]and standard deviation SD(Ri) this can be written as

E[RP] = Rf +(E [Ri] Rf)

SD(Ri)p

i.e. the expected return is a linear function of the

standard deviation of the portfolio, with slope

(E(Ri)Rf)

SD(Ri).



6/23

Interpreting the weights

For w < 0 we are short in the risky asset (i.e. we are

selling it).

For w < 1 we are lending (i.e. buying the risk-free asset).

For w > 1 we are borrowing (i.e. selling the risk-free

asset).



7/23

The mean-variance efficient frontier

w 0 are the mean-variance efficient portfolios.



8/23

Questions

If you would like to tolerate a risk of 20%, what portfolioshould you invest in?

Equate the standard deviation of the portfolio to 20%:

w0.3 = 0.2;

so that w = 2/3 or 67% in the risky asset.

If you would like to earn a return of 10%?

Equate the expected return of the portfolio to 10%:

w0.2 + (1 w)0.08 = 0.10

so that w = 1/6 16.7% in the risky asset.



9/23

2 Combining two risky assets

Data

Stock Weights E[r] SD[r] covariance

A w 0.20 0.3 0.5(0.3)(0.4)

B 1-w 0.25 0.4

The expected return is then:

E[RP] = 0.20w + (1 w)0.25 = 0.25 0.05w.

For the variance calculation we need some more work:

Var(RP) = w2(0.30)2+(1w)2(0.40)2+2w(1w)(0.30)(0.40)0.5;

So that:

p =

w2(0.30)2 + (1 w)2(0.40)2 + 2w(1w)(0.30)(0.40)0.5



10/23

Sample portfolios

Weight w E[RP]

p

0.0 0.250 0.400

0.1 0.245 0.376

0.2 0.240 0.354

0.3 0.235 0.334

0.4 0.230 0.317

0.5 0.225 0.304

0.6 0.220 0.295

0.7 0.215 0.289

0.8 0.210 0.288

0.9 0.205 0.292

1.0 0.200 0.300

Variance starts going up at w 0.77, and expected return goes

down: the mean-variance efficient portfolios are those with

w 0.77.



11/23

The efficient frontier



12/23

Relationship between E [RP] and SD(RP)

One can actually be more explicit about the relationship

between P and E[RP] simply by solving from the expected

return equation for w:

E [RP] = 0.25 0.05w w = 5 20E[RP] ;

and then using this expression in the variance formula:

2P = (5 20E[RP])2(0.30)2 + (20E [RP] 4)

2(0.40)2

+2(5 20E[RP])(20E [RP] 4)(0.30)(0.40)0.5

Note that this can be expressed (by solving a quadratic) with

E [RP] on the left hand side (which makes plots easier).

Note: write x for E [RP] and y = 2P when thinking about the

above equation.



13/23

Questions

If you would like to tolerate a risk of 30%, what portfolio should

you invest in?

Equating the standard deviation to 0.30, we have that weshould invest w 54% in asset A.


Equating the expected return of the portfolio to 0.30 we have

that w = 100%, short asset A and invest twice your wealth inasset B.



14/23

3 Optimal portfolios

3.1 Two risky assets case

We can invest in three assets: risk-free, asset A and asset B.

First only A and B in isolation. Then we are allowed to invest

arbitrary amounts in each asset.First-remark: final conclusion is always going to be of the form

invest part of your wealth in risk-free and the rest in a portfolio

that has weights w in asset A and 1 w in asset B.

Graphically, move along the tangency line depending on your

risk-aversion.

How much you put in the risk-free asset versus in the optimal

fund (w, 1 w) of the two risky assets will depend on the

investors risk aversion.



15/23

Investing in the risky assets in isolation

Consider investing in stock A. From problem 1, we have that

the efficient frontier (plot of combinations of risk-free and assetA) is given by

E [RP] = Rf +(E [RA] Rf)

Var (RA)SD(RP) (1)

= 0.10 +

(0.20 0.10)

0.3 SD(RP) = 0.10 +

1

3 SD(RP) (2)

Similar for asset B

E [RP] = 0.10 +3

8SD(RP) (3)

Since 3/8 > 1/3 we see that for any portfolio of the risk-free andasset A there exists a portfolio of the risk-free and asset B

which does better.

Therefore if we could only invest in these assets in isolation we

would prefer asset B.



16/23

Sharpe ratios

Given an asset with returns Ri, the quantity

si =E [Ri] Rf

SD(Ri)

is called the Sharpe ratio of asset i.

Note that if asset i is a portfolio, we can use the same definition

and talk about the Sharpe ratio of a given portfolio.

The Sharpe ratio is the slope of the line in the previous

equations: therefore if we had to choose between A and B in

isolation the problem boils down to looking at which of the twoassets has a higher Sharpe ratio.

With Rf = 4%: sA = 0.533 and sB = 0.525, so asset A is actually

preferred under this scenario.



17/23

Optimal portfolios

The interesting question is what happens if we can invest

arbitrary amounts in each asset.Consider a given portfolio characterized by w. We can easily

compute the Sharpe ratio of this portfolio.

Finding the optimal portfolio boils down to finding the portfolio

with the highest Sharpe ratio (a graphical argument comes in

handy).

Literally it solves

maxw

E [RP] Rf

SD(RP)

Since we have explicit expressions for E [RP] and SD(RP) theabove is a well-defined problem (analytically challenging, but in

principle straightforward).

In the assignment one can check that the optimal portfolio of

risky assets is w 48.3% and 1 w 51.7%.



18/23

A handy graph



19/23

Questions

If you would like to tolerate a risk of 20%, what portfolio shouldyou invest in?

x(0.306) = 0.2 65.3% in the optimal risky portfolio. In other

words: put about 34.7% in the risk-free asset, 31.5% in asset A

and 33.8% in asset B.


x0.226 + (1 x)0.1 = 0.25, or x 1.19 in the risky portfolio. In

other words: put about 57.5% in asset A, 61.6% in asset B, and

borrow an amount equivalent to 19.2%

Very important remark: the optimal portfolio of risky assets is

independent of the investors preferences for risk (different

risk-aversion implies different mixes between the riskless asset

and this optimal fund of risky assets).



20/23

3.2 The general case

The same type of arguments yield the conclusion that the

optimal portfolio of risky assets must maximize the Sharpe ratio

(the slope of the line that joins an portfolio to the risk-free

asset).

The book actually gives a precise analytical treatment of this

problem in the Appendix of chapter 16.

Easy to compute it in a spreadsheet numerically: just create a

cell with the Sharpe ratio of an arbitrary portfolio of risky assetsand then maximize this value.



21/23

Some fun facts

1. The optimal portfolio is proportional to

1( rf),

where is the variance-covariance matrix of the risky

assets returns, and rf is a vector with typical elementE [Ri] rf (the excess expected return of an asset).

2. When all assets are uncorrelated, then the optimal portfolio

is proportional to a vector with typical element

(E [Ri] rf)

Var (Ri) =

SRi

SD(Ri) .

You invest in each asset proportionally ot its Sharpe-ratio

divided by the assets standard deviation.



22/23

One fancy example

Historical data on three international indexes:

E [r] SD[r]

S&P500 0.13 0.12

Nikkei 0.15 0.18

Footsie 0.18 0.15

Variance-covariance matrix:

S&P500 Nikkei Footsie

S&P500 0.0144 0.00432 0.0144

Nikkei 0.00432 0.0324 0.0162

Footsie 0.0144 0.0162 0.0225

Optimal portfolio when Rf = 5% (check!!)

w = (0.0685, 0.0342, 1.0342)



23/23

Recap

The optimal portfolio is given by a combination of the risk-free

asset and a fund of the other (risky) assets.

The optimal fund is obtained by maximizing the Sharpe ratio of

the portfolio of risky assets.

Upcoming

Thinking about risk in a portfolio context.

How to determine discount rates for assets as a function of their

risk.


Documents

Tuck Bridge Finance Module 4