Case 5 - Solution / expected answers

Note: Figures of intermediate results in this solution may be rounded for simplicity of notation only. Final results are the only ones to be rounded according to the rules of the game.

Part 1:

Question 1: The correct answers in Table1:

Table 1(CAUTION: variance values are presented as multiples of 10-4. We kept only two digits after the point)

Stock A Stock B

Average return (%) 3.24% 3.47%

Variance

22.9

64.6

Standard deviation (%) 4.78% 8.03%

Annualized average return (%) 38.88% 41.62%

Annualized variance

274.5

774.6

Annualized standard deviation (%) 16.56% 27.83%

Explanations:

By definition, a return is the rate of increase of a capital between the beginning and end of a specified period. There are two ways to calculate it, the discrete method and the continuous one.

The discrete return (in case of non-dividend distribution) is equal to the final price minus the initial price divided by the initial price: = !!!!!!! . In case of a dividend, this latter should be added in the numerator. The rate of return can be calculated every day or at different periods of time (week, month, year) but a discrete return can also be calculated over the whole period.

The natural logarithmic of the ratio(ST/So) ln !!!! , or also (ln! ln!) is used for continuous returns calculation. This is the convention we adopted here. Using a continuous compounding return assumes that P t = P t-1 exp (rt) where rt is the rate of return during the period (t-1, t), and Pt the price of the asset at period t. Suppose that r1, r2,r12 are the returns for 12 periods ; then the price of the stock at the end of the 12 periods will be:

P12= P0 exp (r1+r2.r12)

We prove that the continuous return for the whole year is the arithmetic sum of the continuous returns calculated on the twelve months: ln !! = ln !!!! + ln !!!!!!+ ln !! + ln !! = ! + !!! +!+!

Average and mean will be used indifferently in the following. So, the average return E(X) or, by definition also, its mean value denoted often , is mathematically the expected value of the variable X. The average return is calculated using the Excel worksheet function AVERAGE.

To measure risk, finance often relies on standard deviation, noted . In portfolio theory, the risk corresponds to the standard error of the stocks return, and not the stocks price. The standard deviation is the square root of the variance. The variance is computed using the Excel worksheet function VAR.P which is used to calculate the population variance (whereas VAR is

used in sample calculation); we used the population variance and covariance because the data do not represent a small sample, it may be considered to be an entire population. its mathematical formula is: = !!!! !!!!! ! = ! ().

Statistically the standard deviation measures the average dispersion to mean (or central value). Standard deviation has the great advantage when compared to variance to be expressed in the same unit as the one in which the data are measured. Mathematically, for data X, we have thus:

= ! ! We used the Excel worksheet function STEDV or equally SQRT (VAR).

Question 2: The correct answers are in Table 2:

(CAUTION: variance values are presented as multiples of 10-4. We kept only two digits after the point)

Table 2

Covariance (A,B) 19.1

Correlation (A,B) 49.58%

Explanations:

The covariance measures the degree to which variables change together. When variables tend to move on the same direction the variance is positive. When it is negative, the variables move in opposite direction. If we take two variables Y and Y we have the mathematical formula:

, = By definition, the covariance of a variable with itself is equal to its variance. To calculate the covariance between the two

stock returns, we constructed the two columns of the differences between the returns and their means, we calculated their products and then the average (or mean) of the products. We can also use the the Excel worksheet function COVAR between those two columns. The correlation coefficient measures the normalized covariance, i.e. the covariance divided by the product of the standard errors : , = (,)! !

Always comprised between -1 and 1, the correlation coefficient measures the degree of linear co-movements between two variables. Another way to look at the correlation coefficient between two stocks A and B is to regress the returns of stock B on those of stock A and use the Excel worksheet function TREND: the correlation coefficient is the square root of the regression R given by the software.

Question 3: The correct answers are in Table 3.

Table 3 (CAUTION: variance values are presented as multiples of 10-4 . We kept only two digits after the point; standard deviation is expressed in %))

Portfolio P1

Average return 3.35%

Variance

31.4

Standard dev. 5.60%

Explanations:

The mean or average return of a portfolio is simply the weighted sum of the assets composing the portfolio. So if the proportion of A and B are respectively 50%, the average return of the portfolio will be 0.5*E(XA) +0.5* E(XB) with E(XA), E(XB) the expected returns of the asset A and B respectively. As for the risk , if the reference proportion for asset A is denoted a , the variance (i.e. the square of ) of the portfolio is given by the equation: ! = !! + 1 !! + 2 1 ,

Or equally, using now the correlation coefficient between A and B: ! = ()+ 1 !!()+ 2 1 ()() To compute it, we built the column of the weighted sum of the average returns and then we applied the Excel worksheet function VAR P to it.

Question 4: The correct answers are in Table 4.

Table 4 (CAUTION: variance values are presented as multiples of 10-4 . We kept only two digits after the point; standard deviation is expressed in %)

Weight of stock A in the portfolio Av. return Standard dev.

0.0% 3.47% 8.034%

7.5% 3.45% 7.616%

15.0% 3.43% 7.212%

22.5% 3.42% 6.825%

30.0% 3.40% 6.457%

37.5% 3.38% 6.113%

45.0% 3.37% 5.796%

52.5% 3.35% 5.511%

60.0% 3.33% 5.264%

67.5% 3.31% 5.060%

75.0% 3.30% 4.904%

82.5% 3.28% 4.801%

90.0% 3.26% 4.755%

97.5% 3.25% 4.766%

105.0% 3.23% 4.836%

112.5% 3.21% 4.960%

120.0% 3.19% 5.136%

127.5% 3.18% 5.358%

135.0% 3.16% 5.621%

142.5% 3.14% 5.919%

150.0% 3.13% 6.247%

Explanations:

Using previous formulas for returns and risks and the proportion invested in asset A as input parameter, we build the Table 4 for the portfolio performance statistics, assuming that borrowing and short selling an asset are possible. It calculates the risk return performance as a function of the proportion of stock A in the portfolio. So, the proportion of A can be higher than 100%.

Question 5: The correct answers are in Table 5.

(CAUTION: variance values are presented as multiples of 10-4 . We kept only two digits after the point; standard deviation is expressed in %)

Table 5

Weight of each stock in C D E F Portfolio P2 20% 30% 40% 10% Portfolio P3 20% 10% 10% 60%

Portfolio P2 Return 9.10% Variance 1.216 Portfolio P3 Return 12.00% Variance .02 Covariance between P2 and P3 .07

Explanations:

The return of a portfolio B is computed as follows: ! = !, where M represents the column-vector of return for the listed stocks (so its transpose M is a row-vector), and X is the column-vector of the computed proportions. In Excel, this

computation is easily done with the function SUMPRODUCT. The individual elements of the vectors need to be in the same order for the answer to be correct.

The variance of a portfolio is: (!) = !, where X represents the column vector of stock proportions and S is the variance-covariance matrix. S is a 4x4 matrix and X is a column-vector containing 4 elements, so the result is a scalar: !,! !,! !,! = (!) . In Excel, this computation can more easily be executed with nested MMULT functions.

The Table 5 is a generalization of a previous one (Table 3). The portfolios P2 and P3 have been randomly constructed now with four assets C, D, E, F. The two variances and the covariance between the portfolios P2 and P3 have been calculated with the functions VAR. P and COVAR.P

The mathematics of this generalization is following, the matrix notation simplifying the writing of the portfolio problem. If the proportion of asset i in the portfolio is denoted by it is convenient to write the portfolio proportions as a column vector:

=

N

.

.1

We write the transposed vector of as :

[ ]NT ..1=

The expected return of the portfolio whose proportions are given by is the weighted average of the expected returns of the individual assets.

=N

iip rErE )()(

We write E(r) and E(r) T the column and row (respectively) vector of asset returns. We write the expected portfolio return in matrix notation as:

=== TTN

iip rErErErE )()()()(

The variance of the portfolio is given by:

= +== +=

+=+=N

i

N

ijijji

N

iii

N

i

N

ijjiji

N

iip rrCovrVarrVar1 11 1

2),(2)()(

We may then write:

= ijjiprVar )(

The most practical representation of the portfolio variance (especially when the market is large) uses matrix notation. We call the matrix that has !" in the ith row and the jth column the variance- covariance matrix:

=

NNN

N

S

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

...

1

2221

11211

The portfolio variance is given by :

= SrVar Tp )(

If now we denote [ ]N ..11 = the proportions of portfolio 1 and [ ]N ..12 = the proportions of portfolio 2 we can evaluate that the covariance of the two portfolios is:

Tp SrCovCov 21)()2,1( ==

Question 6:

The correct answers are in Table 6.

( CAUTION: variance values are presented as multiples of 10-4 . We kept only two digits after the point; standard deviation is expressed in %)

Table 6

Weight of P2 in the portfolio Av. Return Standard dev. -80.0% 14.32% 72.88% -65.0% 13.89% 67.22% -50.0% 13.45% 61.72% -35.0% 13.02% 56.40% -20.0% 12.58% 51.33% -5.0% 12.15% 46.59% 10.0% 11.71% 42.28% 25.0% 11.28% 38.57% 40.0% 10.84% 35.63% 55.0% 10.41% 33.66% 70.0% 9.97% 32.84% 85.0% 9.54% 33.26%

100.0% 9.10% 34.87% 115.0% 8.67% 37.51% 130.0% 8.23% 41.00% 145.0% 7.80% 45.13% 160.0% 7.36% 49.74%

Explanations:

The Table 6 is also similar to Table 4. It is constructed assuming that a portfolio is made out of the two portfolios P2 and P3 described in the Table 5. We again admitted that borrowing and short selling were possible so the proportions invested in one portfolio can be higher than 100%. We have now to test if a linear combination of portfolios P2 and P3 can give an efficient frontier.

In literary terms, the definition of the efficient frontier is a set of optimal portfolios that offers the highest expected return for a defined level of risk or, conversely, the lowest risk for a given level of expected return. Portfolios that lie below the efficient frontier are sub-optimal, because they do not provide enough return for the level of risk. Portfolios that cluster to the right of the efficient frontier are also sub-optimal, because they have a higher level of risk for the defined rate of return. The efficient frontier is curved because there is a diminishing marginal return to risk. Each unit of risk added to a portfolio gains a smaller and smaller amount of return.

Mathematically, an efficient portfolio is the portfolio of risky assets that gives the lowest variance of return of all portfolios having the same expected returns. Alternatively an efficient portfolio has the highest expected return of all portfolios having the same variance. For a given return m, an efficient portfolio p is one that solves )( pijji rVarxxMin = subject

to )( pii rEmrx == and =1ix .

The efficient frontier is the set of all efficient portfolios. Black 1976 proved that the efficient frontier is the locus of all convex combinations of any two efficient portfolios. Therefore , if }{ NN xxxx ,,.... 11 = and }{ NN yyyy ,,.... 11 = are efficient portfolios and if a is a constant then the portfolio z defined by :

+

+

+

=+=

NN yaax

yaaxyaax

yaaxz

)1(...

)1()1(

)1( 2211

is also efficient : thus we can find the whole efficient frontier if we can find any two efficient portfolios. With this theorem once we have found two efficient portfolios x and y we know that any other efficient portfolio is a convex combination of x and y. If we denote the mean and the variance of x and y by }{ 2),( xxrE and }{ 2),( yyrE and if yaaxz )1( += then :

)()1()()( yxz rEaraErE +=

!! = !!! + 1 !!! + 2 1 , =!!! + 1 !!! + 2 1 !

Question 7: The correct answers are in Table 7.

(CAUTION: variance values are presented as multiples of 10-4 . We kept only two digits after the point; standard deviation is expressed in %)

Table 7

The portfolios resulting from the combination of P2 and P3 are lying on the efficient frontier False Because A single stock has a better risk/return profile than the combination of P2 and P3 True The risk-return curve of the combination of P2 and P3 is an hyperbola False The combination of two diversified portfolio always makes an efficient frontier False

Explanations:

To answer the question of the efficient frontier that is supposed to be drawn from the two portfolios P2 and P3 and mathematically described above, it is necessary to draw the efficient frontier obtained when combining the two portfolios P2 and P3 to obtain a global portfolio.

This is done by plotting all the points of this global portfolio (risk, returns in blue for the bottom branch and green for the upper branch of the hyperbole which is precisely the frontier) simulating the proportions invested respectively in the two initial portfolios. But if we plot the four basic assets C, D, E and F we can see that one asset D (in red) displays a higher return for the same quantity of risk than the upper limit (in green). It dominates the efficient frontier.

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

Stock A

Stock B

Stock C

Stock D

Risk

Return

Part 2

Question 1: The correct answers in Table 1:

Table 2 (CAUTION: We kept only two digits after the point)

Beta Stock G 1.48 Stock H 1.08 Stock I 1.31 Stock J 1.30 Stock K 0.26 Stock L 0.49 Universe City Stock Index 1.00

Explanations:

The equity beta, also referred to as the levered beta, can be estimated by taking a ratio of the the historical covariance between the share price return and the index return (used as a proxy for the whole market) and the variance of the market return. ! = (! ,!)(!)

The beta is a measure of the volatility, or systematic risk, of a security or a portfolio in comparison to the market as a whole. By definition, the market itself has a beta of 1.

Beta can be calculated using regression analysis or using the mathematical definition above. Beta is described as the tendency of a security's returns to respond to swings in the market. A beta of 1 indicates that the security's price will move

with the market. A beta of less than 1 means that the security will be less volatile than the market. A beta of greater than 1 indicates that the security's price will be more volatile than the market. For example, if a stock's beta is 1.2, it's theoretically 20% more volatile than the market. To compute the betas, we used the Excel COVAR function and the VAR. P function on the matrix of the returns. The mathematical rationale behind the result is the following one, referred to as the Single Index Model. The model s basic assumption is that the returns of each asset !can be linearly regressed on some market index !: ! = ! + !! + !

Given this assumption we easily prove with the help of previous equations: (!) = ! + !(!) !" = !!!! The Single Index Model leads to great simplifications in the calculation of the variance-covariance matrix. Regressing the

returns of each asset on an index like the Universe City Stock index, the betas can be calculated either with function SLOPE applied on the column of returns to the column Index or with the function COVAR / VAR.P applied with the same syntax. The table above has been calculated using both methods to check that they give the same results.

Question 2: The correct answers in Table 2:

Table 2

True / False The portfolio P4 is lying on the efficient frontier True The portfolio P5 is lying on the efficient frontier True The portfolio P6 is lying on the efficient frontier False

Explanations:

The rationale of the calculation is to check if portfolios are actually efficient. We begin with some calculations and well known result due to F.Black 1972. Let c be a constant and R-c denote the following column vector, which is the column of excesses of expected returns related to c.

=

NN crE

crEcrE

cR

)(...

)()(

2

1

S being the variance-covariance matrix, let the vector z solve the system

SzcR = Then this solution produces a portfolio x on the envelope of the set of portfolio means and standard deviations generated by

any portfolio whose proportions sum to 1 (called feasible portfolios) in the following way:

{ }cRSz = 1

and }{ NN xxxx ,,.... 11 = with =

jj

ii z

zx

All envelope portfolios and in particular all efficient portfolios are of this form and are the result of the procedure described

above: if x is any envelope portfolio then there exists a constant c and a vector z such as SzcR = and

=

jj

ii z

zx . Any

two envelope portfolios }{ NN xxxx ,,.... 11 = and }{ NN yyyy ,,.... 11 = are enough to establish the whole envelope which is simply a convex combination of x and y.

This means that given any constant a, the portfolio

+

+

+

=+

NN yaax

yaaxyaax

yaax

)1(...

)1()1(

)1( 2211

is on the envelope of the efficient

frontier. If y is any envelope portfolio then for any portfolio x, we have the relationship:

(!) = + ! ! where ! = !"#(!,!)!!!

Besides, c is the expected return of a portfolio z whose covariance with y is 0: = (!) and , = 0. If y is on the envelope, the regression of any and all portfolios x on y gives a linear relationship. If the market portfolio M is efficient (a heavy assumption we will test hereafter) the previous result is also true for the market portfolio. That is the Security Market Line with (!) substitute for c: (!) = (!)+ ![ ! (!)]

Where ! = !"#(!,!)!!! And , = 0

Note that here, the role of the risk free asset is played by a portfolio with a zero beta with respect to the particular envelope portfolio y. When there is a risk free asset, the result above specializes to the security market line of the classical Capital Asset Pricing Model : ! = ! + ! ! !

If all investors choose their portfolios only on the basis of portfolio mean and standard deviation, then M is a portfolio composed of all risky assets in the economy, each asset taken in proportion of its value. The only real test of the CAPM is whether the true market portfolio is mean variance efficient. If M is an efficient portfolio solution of SzrR f = and

=

=

1jj

ii z

zM , all (convex) combinations of portfolio M and the risk-free asset will be called the Capital Market line. A

portfolio located on that line will display the following risk/return relation: ! = ! + (1 ) ! ! = (1 )! M is referred to as the Market portfolio, the only portfolio of risky assets included in any optimal portfolio. It includes all

the risky assets with each asset weighted in proportion to its market value. To find M when ! is known, implies simply to solve the above system given that the constant = !. When the risk- free rate changes, we get a different market portfolio. So we create first two efficient portfolios by implementing the procedure just described above (determination of two portfolios generating the convex frontier which is plotted thereafter). A convex combination of those two portfolios, named P4 and P5, will generate the whole efficient frontier. The frontier is plotted. The portfolio P6 is clearly inefficient since it is outside the upper part of the hyperbole.

Question 3: The correct answers in Table3:

Table 3

G H I J K L G 0.20595 H 0.03752 0.07904 I 0.10775 0.03548 0.08674 J 0.04926 0.10284 0.04429 0.44353 K 0.02083 0.00891 0.01944 0.01925 0.00831 L 0.00586 0.04058 0.01484 0.02736 -0.00148 0.03921

Explanations:

The clearest and quickest method for calculating the variance covariance matrix as it appears above is an on- screen method which uses the excess return matrix. Suppose that we have N risky assets and that for each asset we have return data for M periods. Then the excess return matrix A will look like:

==

rrrr

rrrrrr

returnsexcessofmatrixA

NMM

N

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

......

1

12

111

The transpose of the matrix is:

=

rrrrrrrr

ANMN

MT

......

......

1

111

Multiplying TA times A and dividing through by the number of periods M gives the variance covariance matrix S:

= !" = ! . That is what has been done here. We first calculated the mean return for each asset , by using the already mentioned Excel

function AVERAGE on each column of the data, then subtracting each assets mean return from each of the periodic return gives the excess return matrix. Using the array function TRANSPOSE gives the transposed matrix To enter the formula TRANSPOSE, mark the area of the matrix, type TRANSPOSE on this area and finish with [Ctrl] +[Shift] +[Enter]. To calculate the variance-covariance matrix it is necessary to multiply TA times A . We used again the array function MMULT(A_TRANSPOSE, A)/N. In the same way, type TRANSPOSE on the concerned area and finish with [Ctrl] +[Shift] +[Enter].

Question 4: The correct answers in Table 4:

Table 4

Weight Portfolio P4 (%) -1.45 Portfolio P5 (%) 2.45 Return Return of the portfolio P7 (%) 22.78%

Explanations:

The efficient frontier method and the simulations constructed at the previous step help us to construct a portfolio whose risk equals the market risk. Indeed we have to find the respective proportions of portfolio 4 and portfolio 5 to build a portfolio P7 which will display a risk of 0.1896 which is precisely the global market risk. We can use an Excel GOAL SEEK to manipulate a specific cell so that one of the portfolios sigma will be equal to the standard deviation of the Universe City Stock Index. This specific cell will show the difference between the various portfolios proportions. It is also possible to find it numerically by

decreasing little by little the initial proportion of P4. We took -2.00 (i.e. -200%) as an initial value for that proportion. The table shows that, for a risk (s)of 0.1896, the expected return of P7 is 22.78%.

Question 5: The correct answers in Table 5:

Table 5

Sharpe ratio Portfolio P4 1.54 Portfolio P5 1.23 Portfolio P6 0.09 Portfolio P7 0.93

Explanations:

The Sharpe ratio is a measure of stock or fund performance, it measures the reward per unit of risk. By definition, it is the ratio of an assets excess return to its volatility. It is also known as the reward-to-variability ratio. The Sharpe ratio can be computed either as an ex-ante or ex-post. Mathematically, the Sharpe ratio, in its ex-post expression (based on realized returns) is as follows:

= !(!!)!!!!!

where E(rP )is the return on an asset or portfolio P, rf is the return on the benchmark asset, and P is the standard deviation of rP.

We take the efficient portfolios P4 and P5 as reference mutual funds portfolios. The risk free rate is 5%. We select the portfolios according to their performance. Other measures of performance are available, the most used being:

Treynor ratio

The Treynor ratio is a measure of stock performance. It measures the performance of an asset i compared to a risk free asset (typically Treasury bills) per unit of assumed market risk. Mathematically, the Treynor ratio is as follows: = (!) !! Where ri is the return on asset I (or a portfolio), rf is the return on the risk free asset, and i is the beta of asset (resp.the portfolio) i.

The Treynor ratio can also be computed for a portfolio; the higher is the Treynor ratio, the better is the performance of the fund. Treynor ratios are commonly used to rank fund managers. The ranking given by Treynor ratios is usually the same as the ranking obtained via Jensens alpha because both of them rely on systematic risk. Theoretically, the Treynor ratio can be negative, but in this case it is very difficult to interpret. A negative Treynor ratio may imply that the fund manager has outperformed the risk free rate while reducing systematic risk (negative Beta) which is a favorable situation. The opposite situation with a positive beta but having underperformed the risk free return is very unfavorable.

Jensens alpha

Jensens alpha is a measure of a securitys excess return with respect to the expected return given by the Capital Asset Pricing Model. Investors are looking for assets or portfolios with positives alphas, as it signals positive abnormal return. An asset with a positive alpha has a higher return than the risk adjusted return estimated by the CAPM. Computations of Jensens alpha are based on realized returns. For a stock or portfolio i, we thus have: ! = ! ! + ! ! !

where ri is the return on asset i, rf is the return on the risk free asset, bi is the beta of asset i, and rM is the market return.

Question 6: The correct answers in Table 6:

Table 6

Portfolio Given your previous answers, which portfolio is the nearest of the super-efficient portfolio? [P4/P5/P6/P7]

Explanations:

According to the classical criteria, P4 is the nearest of the super -efficient portfolio since, being efficient, it also displays the highest performances measures.