Corporate finance and Financial institutions - appunti completi

Investment Analysis

A BB200.000 200.000Overall investment Overall investmentExpected return of the portfolio- 30.000 170.000R = X R + X R = ⋅ 0.03 + ⋅ 0.05p A A B B 200.000 200.000Can be used for : historical return (average realized return) and future return‣- The expected return is weighted average of returns or a linear combination of returns (can be used with historical values or future ones)Expected return in the future or in the past depending on what I use is the weighted average or the linear combination‣ 2 2 2 2Variance of a Portfolio : [sigma A, B is the covariance of A and B]- X σ + 2X X σ + X σA B A,B B BA AFrom this if I use the square root I can find the Standard Deviation‣ I have to know the standard deviation of the stocks VAR or sigma, then multiply the square of each weight fo the variance‣- Expected return = linear combination- Variance = square of the weigh multiplied by the variance of A + square of the weigh multiplied by the variance of B +

covariance (involvesboth) involves the multiplication of the two weights

Example : Invest 60% in A and 40% in B- (it lies in between 17.5 and 5.5)

R = 0.6 ⋅ 17.5% + 0.4 ⋅ 5.5 % = 12.7 %

p2 2 4σ = 0.6 ⋅ 0.0668 + 2 ⋅ 0.6 ⋅ 0.4 ⋅ (− 0.0049) + 4 ⋅ 0.0132 = 0.023851p 2 (lies between the two standard deviation 15 and 11 of the two stocks)

σ = σ = 0.023851 = 0.1544 = 15.44 %p p- Diversification of risk :Standard deviation of portfolio is less compared to the simple weighted average of individual‣ standard deviations

Simplest way in which you can explain risk of return• 3 - 25 {chapter10}- We have past returns : use formula to find the past (apply backward)

EXERCISE : Weights of Coca Cola and Apple- Amount invested

Weight of each stock Formula = If you sum the two value you obtain 1;

Overall value of the portfolio

578.54 565.10

Amount invested Amount invested= = = 37.946 %= = = 62.054 % WeightWeightApple CocaCola 913.27932.31 Overall

standard deviation will follow this rule [Not correct]

Weighted standard deviation : multiply each standard deviation for the weight of each stock = 62.054 · 3.114 + 37.946 · 0.849 =• 2.254

Error of using the weighted standard deviation :The weighted standard deviation overestimates the standard deviation , the calculated one gives a smaller error

If I simplify, I over estimate the risk (which is lower) 2.254 > 1.948• →Actual ways to compute the standard deviation :

• [direct]
• [indirect]

2 2 2 2 0.5Time series of a portfolio returns = 1.997% = 1.948(X σ + 2X X σ + X σ )A B A,B B BA A-- standard deviation of daily return of portfolio in a standard deviation : the square root of the- -period variance

EFFECT OF THE DIVERSIFICATION-- Diversification : the property of the risk (standard deviation of the portfolio) of two stocks or more such as the risk is less the simplecombination of the risks of the stock in the portfolioPutting

stocks is less then the weight average of the standard deviation4 - 25 {chapter10}

While the expected return is the weighted average return, if I increase each single expected return I have a proportional return in the‣ portfolio

This is not true for the standard deviation which increases less than proportionally compared with the increase of each standard‣ deviation

- When does diversification does not work?ρ = 1

When- This happens when the correlation between to stocks is equal to one‣ σ ρ ⋅ σ ⋅ σA,B A,B A Bρ = = ρ = 1 σ = σ ⋅ σwhen , thenA,B A,B A B• σ ⋅ σ σ ⋅ σA B A B

- Compute now the variance of a portfolio of A and B, the formula would be :2 2 2 2VA R(ρ) = X σ + 2X X ⋅ σ σ ⋅ σ + X σ σ σ ⋅ σ, instead of writeA B A,B A B A,B A BB BA A 2(X σ + X σ )

Can be simplified as this can never be equal to

zero since →A A B Bσ σ X X X + X = 1 and are positive and and are both positive or however there is the condition :• A B A B A Bρ = X σ + X σ Since we are in this condition the standard deviation of a portfolio is :- P A A B Bρ = 1 We find that when correlation we are in a condition in which we have a perfect balance‣ A,B The standard deviation of a portfolio, in this case is the weighted average• THE EFFICIENT SET FOR TWO ASSETS-- Standard Deviation of Selected Securities- Correlation 1 Straight line→Increasing the Standard deviation : the relationship between standard deviation of the portfolio and expected return of the portfolio is a‣ straight line Both standard deviation and expected return follow the rule of the linear return (same mathematical rule)• I can increase the risk and accordingly the return in a proportional way (no diversification)• ρ = − 1 Condition where , how much is the variance of the

portfolio?- 2 2 2 2VA R(ρ) = X σ − 2X X ⋅ σ ⋅ σ + X σ σ = − 1 σ ⋅ σ, negative sign since‣ A B A B A,B A BB BA A 2(X σ − X σ )This is the square of the binomial : could be equal to zero, which could happen even tough‣ A A B Bσ σ X Xand are positive and and are both positive• A B A B- It is interesting to notice that this expression can to go zero under some conditionsX σ − X σ = 0‣ A A B BX X = 1 − X X σ − (1 − X )σ = 0Instead of I can write the result is• B B A A A A Bσ σB AX = X =X σ − σ − X σ = 0 X ⋅ (σ + σ ) = σ and→ →- A A B A B A A B B A Bσ + σ σ + σA B A B5 - 25 {chapter10}Example :σ = 10 % = 0.10 σ = 12 % = 0.12Assume andA B Apply the formula and find that :σ σ0.12 0.10AB X = = = 0.4545 =

45.45 %X = = = 0.5454 = 54.54 % BA σ + σ 0.12 + 0.10σ + σ 0.12 + 0.10 A BA B-

In this way I am able to find what are the weights for A and B, which can guarantee that in this special case with correlation equal to - 1 :( )ρ = − 1

The risk of the portfolio is equal to 0 (standard deviation = 0)

‣ My two stocks are risky, but in this way the overall risk is equal to zero

•- Space which represents the relationship between standard deviation of portfolio and expected return of portfolio and when we have twostock : Lies inside this perimeter (triangle), which is given by two extreme cases :ρ = 1 and

‣ ρ = − 1 : two segments, starting from the point that represent each stock (points with zero risk)

‣ ρIn practice is nor one nor minus one, it is between those two extremes

- In more realistic situations, it will follow one of the convex curves

‣ The more convex it is the higher the correlation (closer to the diagonal)ρ = 1

growth companies - the correlation are higher than value

Are in the same sector: Knowledge sector (online activities, intangible assets - most important aspect)

Share common properties: important result

• Correlation: powerful tool to address how much we can diversify
• 6 - 25 {chapter10}

Correlation = 1 NO diversification