This dissertation focuses on the presentation of the basic aspects of modern portfolio theory, considering utility theory as fundamental as a theoretical support for the results obtained.
This document is divided into five parts. The first of them refers to the basic concepts present in portfolio investment without delving into mathematical aspects. Furthermore, the justification for studying these topics is given by presenting relevant aspects of real life in the markets. The second part deals with the theory of utility in which expected utility is shown as the means of choosing between random alternatives. The third part is devoted to the analysis of risk, return and correlation as key components in an investment portfolio. The fourth part investigates the nature of the efficient frontier through the two-fund and one-fund theorems. In the fifth part, the CAPM for the entire economy is derived and the application of beta in determining the cost of capital is shown.
To clarify some of the concepts in this presentation, an appendix on normal distribution and market definitions was designed.
basic-aspects-of-modern-portfolio-theory1 INTRODUCTION TO KEY CONCEPTS
The investment
Investment refers to the use of resources in the production of satisfiers in order to obtain potential profits in the future. The investment can be made in two facets in the national economy or abroad:
1. Direct. It is carried out in tangible assets such as machinery and intangible assets such as education. This investment is generally long-term given the low liquidity it presents.
2. Indirect or portfolio investment. Refers to the purchase of financial instruments such as shares. Generally short term given the existence of secondary markets that provide liquidity to financial assets.
In this presentation it will be the second type of investment, however, before entering the study of investment portfolios, the elements of the definition of investment must be broken down:
o Production of satisfiers
o Uncertain earnings
Production of satisfactors in investment
The use of resources to produce goods and / or services that do not satisfy any need is unfeasible because nobody would want to buy them.
The case of a municipality that issues debt for the construction of a bridge shows the need to create wealth for the inhabitants and carriers who need better communication channels. On the other hand, the holders of the municipal bonds are the ones who provide the financial resources.
Uncertain gains
Investments are not safe, even those made in government papers, as they are subject to market, credit and operational risks. The risk of an investment determines the profitability offered by it depending on the opportunity cost. Thus, the higher the risk, the higher the yield.
Once the investment concept is understood, the next step is to analyze the suitability of one investment over another. In direct investment there are techniques for evaluating investment projects, while in portfolio investment there is a stock market analysis and there is the Modern Portfolio Theory that is the topic of this exposition.
Investment portfolio in a first definition.
It is a set of at least two financial instruments in which it has been invested simultaneously.
The financial instruments with which a portfolio can be created are varied and can come from the following markets:
• Money
market • Capital
market
• Derivatives market
• Currency market • Commodities market
Portfolios have their raison d'être in the idea of diversification that facilitates the reduction of risk and sustenance in performance.
Diversification. The following example provides an intuitive idea of what it means to diversify.
The inn
Lemonade is offered to diners in a small inn. On hot days, lemonades increase income, but on cool days people decrease the consumption of cold drinks, so they feel a decrease in sales. If the owner of the inn introduces some coffee in his menu, then, when the days are hot, lemonade could be offered, while on cold days coffee can be offered, thus reducing the possibility of losses.
In this case, product diversification leads to compensation for losses in lemonade through coffee sales on cool days. When the days are hot, the sale of coffee will decrease but the sale of lemonades will increase, so in both cases the possibility of loss is reduced.
Diversification finds its origins in the bullseye theory that was developed by Alfred Cowles in the 1920s. The idea of this theory indicates that it is preferable to buy from everything that is in the stock market to form a diversified portfolio. Cowles concluded that the diversified portfolio is better, on average, than following the best investment strategies of stockbrokers due to commission payments.
Later, Cowles' idea was perfected through the Modern Portfolio Theory initiated by Markowitz. The classic diversification phrase "don't put all your eggs in one basket" due to Tobin should not be forgotten.
The key to diversification lies in the dependence between the instruments that make up a portfolio. Such dependency relationships are estimated with the correlation. The lower the asset correlation, the more diversified the portfolio will be.
Yield and Risk. When choosing between two portfolios, the most important indicators are the risk and the return they present.
Yield shows growth in portfolio value. A distinction must be made between realized performance and expected performance. The first refers to the performance that the portfolio actually had, while the second is an estimate of the future performance of the portfolio.
Risk is often defined as the possibility of loss and can be linked to a declining market, but even in this scenario it is possible to obtain gains through short positions. Therefore, for these notes, risk indicates the dispersion of the returns made with the expected return.
Both return and risk have different estimation methods such as moving averages for performance and GARCH models for volatility. However, this material uses only the average return and the standard deviation as estimates of return and risk, respectively.
Investor investments
In financial markets, there are different types of investors, but a first classification considers two classes: individual and institutional. However, the need to invest and the conditions the investor is in are the factors that determine the types of investments.
Banks
The same financial institution may have different portfolios based on senior management policies. Thus, in a financial institution, you can find a trading portfolio made up of liquid instruments to rebalance (change the composition of the portfolio) frequently and a pension fund composed of longer-term instruments with less liquidity but that offer the possibility of exploit certain regulatory arbitrations. Regulatory arbitration (only applicable to multiple banks) consists of investing in instruments for which regulatory bodies request regulatory capital that is less than economic capital.
Insurance
Insurance institutions must invest reserves because these are the resources with which claims are responded to. Circular S-11.2 of the National Commission of Insurance and Bonds establishes the characteristics of the investments in which an insurance company can participate. This circular indicates investment limits based on the type of assets and the reserve class as shown in Tables 1 and 2.
The current pension system in Mexico
Table 1. Restrictions on investment of reserves by the regulatory authorities.
Type of value Percentage of the portfolio
Securities issued or backed by the federal government Up to 100%
Securities issued or backed by credit institutions Up to 60%
Any investment other than the above Up to 30%
Source: National Commission of Insurance and Bonds
Table 2. Liquidity restrictions on reserve investments.
Reserve Minimum percentage of short-term investment
OPC 100
IBNR 75 Ongoing
Risk 50
Mathematics 30
Forecasting 30
Contingency Special 30
Catastrophic Risks 20
Source: National Commission of Insurance and Bonds
These are just simplified examples of reality in the financial markets. The models shown in this material are only the beginning of the long learning process that must be followed by those who intend to get involved in the financial markets.
2 THEORY OF UTILITY
The decision in the face of uncertain alternatives is modeled through the theory of utility, which is presented in a limited way but without subtracting the key elements for understanding portfolio selection. In the following paragraphs, the axioms of utility theory are shown, as well as the derivation of expected utility as the tool for choosing before random alternatives. Subsequently, the topics of stochastic dominance, risk aversion and the mean-variance criterion are discussed. Illustration 1 indicates the course of this content block.
Illustration 1. Theoretical support of the selection of investment portfolios.
The expected value criterion to value uncertain alternatives.
Until before the 1930s, people were considered to decide on uncertain alternatives based on the criterion of expected value. The invalidity of said criterion is exemplified below:
Suppose an individual has the following alternatives:
1. A lottery ticket that rewards 2000 currency units (CU) with 5% chance and loses CU 2 with 95% chance. The representation of this bet is:
⎧2000 0.05
G = ⎨ E = 2000 * 0.05 + (- 2) * 0.95 = 98.1
⎩− 2 0.95
2. The H investment of CU100 in a bank account that safely pays 1% interest.
E = 101
3. A game M consisting of a fair coin toss that stops at the first appearance of the front and in this case pays 2nd currency units where r is the number of tosses until the game stops. The value of the probability of the r-th throw is 2-r, so the hope of this game is
E
r = 1
Under the criterion of expected value, the third alternative is the correct choice. However, having an infinite expected prize, the cost of participating in such a game is also infinite, so no one would want to participate in it. Game M is better known as the St. Petersburg Paradox.
The inconsistency of this criterion is solved with the idea of the expected utility that in the following paragraphs is constructed from axioms.
Axioms of Utility Theory
Before presenting the axioms of the theory in question, an understanding of the idea of lottery is indispensable. A lottery is a game in which different mutually exclusive prizes with associated odds are obtained and has the following expression:
⎧x1 p1
G (x1, x2,.., xn: p1, p2,.., pn) = ⎪⎪⎨x2 p2
⎪
⎪⎩xn pn
where prize xi has probability pi. This simple lottery expression can be abbreviated by grouping the prizes and probabilities into vectors x = (x1, x2,…, xn) and p = (p1, p2,…, pn) so G (x: p) is simpler notation.
There are also compound lotteries such as
⎧ G1 (x: q) p
G (G1, G2: p) = ⎨ in which each prize is a lottery.
⎩G2 (y: r) 1− p
Examples. Let two simple lotteries be G1 (x: q), G2 (x: r) and let G (G1, G2: p) be a compound lottery. The lotteries are such that x = (2,4,6) q = (0.5,0.3,0.2) and = (6,8) r = (0.6,0.4).
⎧2 0.5
G1 (2,4,6: 0.5,0.3,0.2) = ⎪⎨4 0.3 G 2 (6.8: 0.6,0.4) = ⎧⎨6 0.6 G = ⎨⎧G1 0.5
⎪⎩6 0.2 ⎩8 0.4 ⎩G2 0.5
The G lottery can be reduced to a simple lottery by being understood as a linear combination of lotteries. That is, G = 0.5G1 + 0.5G2, so it takes the following simple form:
⎧2 p = 0.25
⎪4 p = 0.15
G = ⎪⎨
⎪6 p = 0.40
⎪⎩8 p = 0.20
Note that the prize value of 6 is offered in the G1 and G2 lotteries, so the probability of this is 0.5 (0.2) +0.5 (0.6). The odds of the other prizes are calculated analogously.
Axioms
Now that the idea of lottery has been mastered, the five axioms of utility theory can be presented.
Let Γ be the set of lotteries concerning an individual and let the bounded set X be the set of possible, non-negative outcomes for all lotteries.
Axiom 1. Completeness. For all x, y ∈ X the agent can do one of the following:
• prefers ax over y denoted y x
• prefers ay over x denoted x y
• is indifferent between the two (y ≈ x)
Axiom 2. Transitivity It occurs with the following situations for x, y, z ∈ X:
• x y ∧ y z ⇒ x z
• x ≈ y ∧ y ≈ z ⇒ x ≈ z
Axiom 3. Strong independence. Let x, y, z ∈ X and G1, G2 ∈Γ. This axiom indicates that:
x ≈ y ⇒ G1 (x, z: p) ≈ G2 (y, z: p).
Axiom 4. Measurability. Let x, y, z ∈ X and G∈Γ. The axiom indicates that
x yz∨ xy z ⇒∃! p such that y ≈ G (x, z: p).
Axiom 5. Graduation. Let four results be x, y, u, z ∈ X
Assuming that (xyz) ∧ (xuz) it is had by axiom 4 that there are lotteries G1, G2 ∈Γ such that y ≈ G1 (x, z: p) yu ≈ G1 (x, z: q).
This axiom indicates the following: If q ≤ p ⇒ uy.
Theory of expected utility
Two more assumptions are required to develop the expected utility theory:
1. Individuals always prefer more wealth
2. For the individual, favorable deviations from the average wealth cannot compensate for unfavorable deviations from the average wealth.
Assumption 1 indicates the logical condition of individuals who always want greater well-being. Assumption 2 details that there is aversion to risk because, however high the prize, the possibility of a large loss keeps individuals from uncertain events. With these assumptions and the five written axioms, the development of the theory is viable.
Utility function.
It is a scalar function that is defined in the set of results X such that it represents the degrees of preference for the different results that actually represent levels of wealth. In mathematical form the utility function takes the following form:
U: X → ℜ
x → U (x)
The functional value U (x) is irrelevant since what matters is the preservation of the order (X, ) to the order of the real numbers, so increasing transformations such as powers or related transformations V (x) = aU (x) + b with a> 0. To exemplify this situation, consider three alternatives: pears, apples and oranges. In addition there is an individual with the following preferences on fruits:
Apple orange pear
The utility function of the individual is
U (apple) = 12
U (orange) = 16 U (pear) = 20
Note that we now have real numbers that can be compared and it is clear that
U (apple) <U (orange) <U (pear) = 20
The values that a utility function takes are irrelevant because the important thing is that they preserve the order of preferences through the order of the real numbers. In this way the affine transformation 2 * U (x) +3 is equivalent to the function U (x) since it preserves the initial order.
Theorem. For all x, y ∈ X the utility function must respect the order of preferences as follows:
U (x)> U (y) ⇒ x y
U (x)
Demonstration
Since X is a bounded set, the element xI = inf (X) is called hell x and is the worst result; the maximum xP = sup (X) is known as paradise x and is the best result.
For all x, y ∈ X we have xP xxI ∨ xP x xI and xP yxI ∨ xP y xI
Based on axiom 4, there are the equivalences
x≈G1 (xI, xP: p (x)) and y≈G2 (xI, xP: q (y)).
If U (x) = p (x) and U (y) = q (y), then by axiom 5 we have:
o U (x)> U (y) ⇒ x and U (x)
Theorem of the expected utility. The utility function is used to compare random alternatives through the expected utility.
Demonstration
Let x, y, z ∈ X. Starting from the previous equivalences x≈G1 (xI, xP: p (x)) and y≈G2 (xI, xP: q (y)), a compound lottery is constructed such that z≈G (G1, G2: r) as shows.
⎧ ⎧xP p (x)
⎪ x ≈⎨ r (z) z ≈⎪⎨ ⎩xI 1- p (x)
⎪y ≈⎧⎨xP q (y) 1- r (z)
⎪⎩ ⎩xI 1- q (and)
Then z≈G (xP, xI: r (z) p (x) + (1-r (z)) q (x)) and it is remembered that U (x) = p (x) and
U (y) = q (y) so U (z) = r (z) U (x) + (1-r (z)) U (y) which is understood as the
expected profit.
More generally, the expected utility of future wealth is E = ∑U (xi) pi.
Solution to the St. Petersburg Paradox
The expected utility theorem solves the St. Petersburg paradox by finding a finite value.
E
r = 1
Features of the utility function
Individuals' preference for greater wealth based on assumption 1 implies an increasing utility function. This condition is equivalent to that the derivative of a utility function, known as marginal utility, is positive U (x) /> 0.
Assumption 2 means that the individual is risk averse so that the marginal utility is decreasing, that is, U (x) // <0, and this condition is equivalent to a concave utility function.
Examples. The utility function U (x) = x is increasing and concave since U / (x) = 1> 0 and U // (x) = - 1 x− <0.
2 x 4
Illustration 2 Characteristics of the root utility function with decreasing derivative.
However, the quadratic utility function U (x) = ax2 + bx + c can be concave and increasing depending on the parameters a, b, and c.
Assuming that the function is increasing, it must be considered that as the level of well-being progresses, a point of inflection is reached in which the first derivative changes sign so that the utility function is increasing and concave only in the interval
⎡ ⎢⎣0, - 2ba⎤⎥⎦ while for values greater than - 2ba the individual prefers
less and less wealth.
Utility functions provide the mathematical tool for decision-making in the face of random alternatives such as the returns on stocks in a portfolio. In the topic covered in this document, a rational investor always opts for the portfolio with the highest expected profit.
Profit and returns distributed as normal.
Up to this point the idea of a utility function was presented as the representation of an individual's preferences. Such a function was assumed to be increasing U (x) /> 0 and concave U (x) // <0.
In addition, examples of utility functions such as the root function and the quadratic have been given, but the selection of portfolios should not be restricted to a family of utility functions, therefore the following assumption is required:
Course. The utility function can be approximated by a Taylor polynomial.
If x0 is a point in the domain of a utility function U (x) then
UU k (x0) (x - x0) k
k = 0 k!
Let w be a random variable with hope µ <∞ and variance σ2 <∞ such that it represents the future benefit of an investment.
U k (µ) k
If U (w −µ) is done then to determine the utility
k = 0 k!
expected E = ∑k∞ = 0 U kk (! µ) E all the
central moments of the random variable w must be known. This situation is avoided when the utility function is quadratic since the derivatives of order greater than or equal to three cancel. Unfortunately, this function cannot be assigned to all investors, so it is preferable to suppose that w ~ N (µ, σ) because all the moments of this random variable are obtained from the first two as shown in the appendix..
Under the assumption of normality for w, no further assumptions are required for the utility function, requesting only that it be approximated by a Taylor polynomial as well as being concave and increasing.
Risk aversion
The concavity of a utility function is a symptom of the investor's risk aversion, but more information on the amount of risk that an investor is willing to tolerate can be obtained through the following measures:
• Arrow-Pratt coefficient A (x)
• Risk aversion R (x)
Previous derivation of such measures, the concept of true equivalent must be known.
Equivalent true.
The true equivalent of an uncertain level of wealth is a certain amount such that the utility of the second is equal to the expected utility of the first.
In mathematical terms the value of C is a true equivalent of the level of wealth x when U (C) = E or explicitly C = U −1 (E).
To exemplify, consider an investor with a utility function
U (x) = −e − x, a current wealth of 10 and a new wealth x = 10 + G such that
⎧ 1
G = ⎪⎨− 5 with p = 12
⎪ 5 with p =
⎩ 2 So
E¨ = - 1 = −0.003369 so the true equivalent 2
is C = -ln (- (- 0.003369)) = 5.6931 and U (C) = 0.003369.
Therefore, the investor is indifferent between 5.69 certain monetary units and the new level of wealth. The difference between the current level of well-being and the true equivalent 10-5.6931 = 4.3069 is understood as an insurance premium that the investor would pay for not facing the G lottery.
This difference is a measure of absolute risk aversion and its development is as follows:
Arrow-Pratt absolute risk aversion coefficient.
Consider an investor with a utility function U (x) such that x is the initial wealth level and a final wealth level x + ε where ε is a random variable with variance σε2 that represents a fair game so E = 0.
With these data, it is desired to calculate the premium Π that the investor would pay for not facing the uncertainty of the final wealth level.
Let C be the true equivalent of x + ε, that is to say that U (C) = E. In order to find an analytical expression for the prime Π a second order Taylor approximation is made around the level of x for U (x + ε).
U (x + ε) = U (x) + U / (x) (x + ε− x) + U // (x) (x + ε− x) 2
Take the hope of this approximation remembering that x is a given value
E = U (x) + U / (x) E + U // (x) E = U (x) + U // (x) σε2
If we remember that the premium is the difference between the current level of wealth and the true equivalent, we have the following expression:
Π = x −C ⇒ C = x −Π⇒U (C) = U (x −Π)
Performing a first order Taylor approximation around x gives:
U (x −Π) = U (x) + U / (x) (x −Π - x)
Since C is a true equivalent, then U (x −Π) = E, so when equating the approximations we have:
/// 2/1 2 //
U (x) + U (x) (- Π) = U (x) + U (x) σε ⇒ − ΠU (x) = σεU (x) ⇒
2
//
1 2 U (x) Π = - σε /
2 U (x)
This prime Π is known as the Arrow-Pratt prime and since
1σε2 is constant, the definition of the aversion coefficient at 2
U // (x) Arrow-Pratt risk A (x) = - / is made.
U (x)
To analyze an individual's risk aversion, the derivative of the coefficient is taken. If the derivative is positive, then the individual is willing to allocate more resources to risky investments. When the derivative is negative, then there is risk aversion, which means that less and less resources will be allocated to risky investments and, if the derivative is null, the same number of monetary units is maintained in risky investments.
Risk aversion coefficient
Risk aversion indicates the percentage of wealth that would be sacrificed for not participating in a lottery.
As in the previous case, a positive derivative indicates that the individual increases the percentage of wealth destined for risky investments. If the derivative is negative, then there is risk aversion, a lower percentage of wealth will be allocated to risky investments and, if the derivative is null, the same percentage of monetary units is maintained in risky investments. In an analogous way to the Arrow-Pratt coefficient, the coefficient of
xU // (x) is obtained relative to risk aversion R (x) = - /.
U (x)
Example: Analyze an individual with a utility function U (x) = x. For the definition of the coefficients, the first two derivatives with respect to wealth are required.
U / (x) = 1> 0 and U // (x) = - 1 x− <0. A (x) = −U // (x) = 1 ⇒ A / (x) = - 12 <0
2 x 4 U (x) 2x 2x xU // (x) 1 /
R (x) = - = ⇒ R (x) = 0 U (x) 2
It is observed that the derivative of the absolute risk aversion coefficient is negative, so the individual will invest more resources in risky assets. The relative aversion to risk is constant, so the individual will always invest the same percentage in risky assets. Figure 3 shows the behavior of both coefficients.
RISK AVERSION
Figure 3. Risk aversion coefficients of the square root utility function.
Stochastic dominance
If the objective is to choose between different portfolios based on the risk and performance indicators, the definition of stochastic dominance must be made to establish the decision criteria. For this section A and B are two different assets, RA and RB are the returns and have distribution functions FRA (x) and FRB (x) respectively.
First-order stochastic dominance. Asset A dominates asset B in this sense when FRA (x) ≤ FRB (x).
To understand this definition, a few mathematical operations are required as shown:
FRA (x) ≤ FRB (x) ⇔ −FRB (x) ≤ −FRA (x) ⇔1− FRB (x) ≤1− FRA (x) ⇔ P {RA ≥ x} ≥ P {RB ≥ x}
In other words, the probability of obtaining a higher return with asset A is greater than with asset B.
Second order stochastic dominance. Asset A dominates in
this sense tt asset B when FRB (x) dx.
This definition assumes risk aversion on the part of the investor and means that asset A will be preferred because it accumulates less probability in the left tail, which is the least unfavorable regardless of the renunciation of a better return.
To land these stochastic dominance ideas, the distributions of three normal random variables with different parameters are shown below.
Normal distribution Mean Standard deviation
F1 0.1 0.17
F2 0.2 0.17
F3 0.21 0.3
Table 3. Normal distributions and stochastic dominance.
Illustration 4. Stochastic dominance.
Figure 4 shows that F2 dominates F1 in the first order, while F3 is dominated by F2 in the second order, since it accumulates less probability in the left tail despite having a lower mean than F3 and this shows risk aversion.
Stochastic dominance and utility function.
First order stochastic dominance with expected utility. Asset A is said to dominate asset B in this sense when
E ≥ E and U /> 0.
Second order stochastic dominance with expected utility. Asset A dominates asset B in this sense when E ≥ E and U // <0.
Stochastic dominance with expected utility. If we consider that
U /> 0 and U // <0, asset A dominates asset B when E ≥ E.
This latter definition of stochastic dominance and the assumption of returns with normal distribution leads to dominance criteria known as mean-variance.
Mean and variance criteria for stochastic dominance.
Let RA ~ N (µA, σA), RB ~ N (µB, σB), Y ~ N (µ, σ) with U /> 0 and U // <0 and let y0 be the initial wealth level. Then the following dominance criteria are valid.
First-order stochastic dominance. Asset A dominates asset B when µA ≥ µB and σA = σB.
Demonstration.
Y = σZ + µ with Z ~ N (0.1)
The future wealth level is y0 (1 + σZ + µ) with expected profit
E.
Taking the partial derivative of this expectation with respect to the location parameter µ, it is observed that it is positive, so the expected utility is increasing with respect to the mean of normal returns and the new definition of stochastic dominance of first order is maintained.
e
Edz
2π
∂E / e dz> 0 since U /> 0.
2π
With this result we have the following rule:
Given a level of risk, choose the asset or portfolio with the highest return.
Second order stochastic dominance. Asset A dominates asset B when σA ≤σB and µA = µB. The proof of this statement follows the same trend as the previous one, but the concavity of the utility function is used. Demonstration.
∂E e
= ∫U (y (1 + σz + µ)) zy dz + ∫U (y (1 + σz + µ)) zy dz
∂σ −∞ 0 0 2π 0 0 0 2π
∞ / e ∞ / e
dzdz 2π2π
Since U // <0 and U is an increasing function, we have that the partial derivative of the expected utility with respect to the standard deviation is negative, so the lower volatility is affected to a lesser degree
to the utility.
Then risk aversion U // <0 means the following rule:
Given a level of performance choose the lowest risk asset.
To show these ideas we have the following list of assets that are identified based on risk and performance.
ASSET YIELD RISK
A 30% 17%
B 30% 53%
C 30% 19%
D 15% 12%
E -2% 12%
F 18% 12%
Table 4. Examples of stochastic dominance.
First-order stochastic dominance.
To apply this criterion, a risk level must be established. For assets D, E and F the risk level is 12%, so they are ordered below.
ASSETS RISK PERFORMANCE
F 18% 12%
D 15% 12%
E -2% 12%
Table 5. First order stochastic dominance.
Asset F dominates assets D and E in this regard.
Second order stochastic dominance.
ASSETS YIELD RISK
A 30% 17%
C 30% 19%
B 30% 53%
Table 6. Second order stochastic dominance.
In this case, asset A dominates assets C and B by having less volatility given a level of return. In illustration II are the six assets. At this point the question arises about the preference between assets A and F. To answer this question, the expected profit is required. If the expected profit of asset A is greater than the expected profit of asset F, then the asset chosen is A. If not, choose F.
Illustration 5. Stochastic dominance with mean and standard deviation.
3 PERFORMANCE, RISK AND CORRELATION
Performance. As justified, the returns on assets are distributed in a normal way, so it is now time to determine them based on share prices, assuming that there is no dividend payment.
Let St be the price of an asset on day t. So the return on an
asset on that day is Rt = ln⎛⎜⎜⎝ SSt − t1 ⎞⎟⎟⎠.
Performing a first order Taylor approximation around the previous price, we obtain another definition for the yield, which is the one that considers the percentage variation.
ln⎜⎜⎛⎝ SSt − t1 ⎞⎟⎟⎠ ≈ ln⎛⎜⎝⎜ SStt −− 11 ⎟⎞⎟⎠ + S1t − 1 SStt −− 11 (St - St − 1) ⇒ Rt ≈ St S − t − S1t −1
However, from a theoretical point of view, the use of this approximation leads to positive probabilities for negative prices, since Rt is normally distributed when
St - St − 1 1⇔ St - St − 1 <−St − 1 ⇔ St <0 starting from St-1> 0.
Rt → −∞ ⇒ <-
St − 1
With the use of logarithmic returns this theoretical detail is saved because when ln⎛⎜⎜⎝ SSt − t1 ⎞⎟⎟⎠ → −∞ ⇒ SSt − t1 → 0 ⇒ St → 0 ⇒ St> 0, so
there are never negative prices as they are lower than zero.
Another advantage of logarithmic returns is that they can be added by facilitating the presentation of the annualized average return. The yield for n periods is given by
ln⎜⎜⎝⎛ SSt − tn ⎞⎟⎟⎠ = ln⎛⎜⎜⎝ SSt − t1 SStt −− 12 SSt − t − nn + 1 ⎟⎞⎟⎠ = ∑kn = - 10 Rt − k
Generally, it is considered that a year has 250 days for the stock market, so when having the estimate of the average daily return E, it is simply multiplied by this number of days to obtain the annualized average return.
From the parametric statistics it is had that the maximum estimator
T
∑R i
plausible of the average yield is µ = i = 1 for a sample of
T
size T.
It is clear that the performance of an asset cannot be less than –1 and that it does not have higher levels, but the assumption of normality is still viable since it is difficult for an asset to change too much in a short period of time.
Standard deviation
The standard deviation indicates the dispersion around the average return of the observations and serves as an estimator of the risk that the investment in an asset represents. The maximum plausible estimator of
the standard deviation is but the estimator to be
used in the following is that it is unbiased.
There are numerous methods to estimate the standard deviation and among them are the GARCH models that perceive changes in the conditional variance over time but the unconditional variance remains constant. In other words, the stochastic process followed by the actions is not stationary locally, but it is asymptotic.
To annualize volatility, the square root rule must be considered and to explain this idea, suppose that there are T observations of the performance of an asset which are considered independent due to the efficient market assumption.
If R1, R2,…, RT are the independent observations with variance σ2, then the aggregate of these variables is the yield for a
T period of T days, so the variance of ∑Rt which is the sum of the
t = 1 variances of returns individually given independence.
Var⎛⎜∑T Rt ⎞⎟ = ∑T Var (Rt) = Tσ2 ⇒ det.est⎜⎛∑T Rt ⎞⎟ = Tσ
⎝ t = 1 ⎠ t = 1 ⎝ t = 1 ⎠
In other words, the volatility for a period T is the square root of that period by the daily volatility. To annualize daily volatility, it must be multiplied by the root of 250, which is the number of days the market is active.
Short sales
For entrepreneurs, the rule of buying low and selling high is common and necessary for the viability of a firm. For a portfolio investor, in addition to this rule, the following can be fulfilled: sell high and buy low, which comes from the possibility of short sales.
For a better explanation of this concept, the meanings of long position and short position must be understood.
Long position. A long position in an asset is assumed when you bet on its price to increase. In other words, an increase in the value of the property benefits the owner. In this sense, the owner buys cheap with the hope of selling expensive.
As an example you have a long position in the future. If on the delivery date the cash price of the underlying is greater than the delivery price then the buyer will have obtained a profit due to the increase in the price of the underlying.
Short position. The short position implies the possibility of making profit in a declining market. In other words, the owner of the short position benefits if the price of the asset falls and the example is the sale of a future.
Short sale. A particular case of short position is short selling. This idea can be explained from the following steps:
• Loan an asset with the promise of delivering it after a period of time T.
• At the time of receiving the asset, it is sold for an amount S0.
• After the term, the asset must be purchased at a ST price and delivered to the original owner.
As appreciated, short selling means the sale of an asset that is not owned and this operation provides profit when the price of the asset decreases. In other words, it will have been won when S0> ST and the realized profit is S0 - ST.
Short sales involve high risks because the gains are limited since the price can only decrease to zero while the loss can be unlimited when the price tends to infinity.
Note that the cash flow of this operation is always negative since it is –S0 at the beginning and -ST at the end. Curiously, the rate of return is negative when you have profits, since in this case
T ST - S0 0 but since the initial investment is –S0, you have
S0> S ⇒ <
S0
positive profit - S0 ST - S0 = S0 - ST> 0.
S0
should be noted that in practice short sales require guarantees procured by the high risk they represent. In addition, if in the period of time in which the action is taken on loan there is a dividend payment, it must be paid to the owner. Figure 5 shows the payment of a short sale.
Short sale example.
Suppose that an economic agent owns 1,000 shares of issuer A, which are currently trading at 25 currency units. Short selling is exemplified as follows:
• An investor requests these shares from the agent with the promise of delivering them in seven days.
• The investor sells those shares at the current price of 25, obtaining 25,000 monetary units that may or may not invest in other instruments in which it would be long.
• After seven days the investor buys 1000 shares of Issuer A at a price of CU24 and returns them to the agent obtaining a profit of CU1,000
4 PORTFOLIO OF INVESTMENTS
From the long and short position definitions, a new portfolio definition can be enunciated:
Purse. It is a set of financial instruments in which you have a position.
The following assumptions are valid from now on:
1. The number of stations is finite.
2. The total number of shares of the issuers is constant.
3. There are no mergers or bankruptcies.
4. The negotiation is continuous.
5. There are no transaction costs, taxes, or stock divisibility issues.
6. There is no dividend payment.
Initially, consider N instruments S1, S2,…, SN with returns
R1, R2,…, RN Let wi be the
percentage per unit that is assigned to the asset Si, so it is clear that ∑wi = 1.
i = 1
Finding the optimal portfolio selection means finding a combination of weights or weights such that they minimize risk given a level of return. To do this you must first determine the performance and risk of the portfolio. The value of wi is also known as the weight or weight of the asset Si.
Yield of a portfolio. The return on a portfolio, denoted by RP, is the weighted average of the returns on assets.
RP = w1R1 + w2 R2 +… + wN RN
The expected return on the portfolio is the weighted average of the expected returns on the assets.
E = w1E + w2 E +… + wN E
Portfolio risk. Risk is estimated from the standard deviation which is the square root of the variance. The variance of a portfolio contains the concept of a matrix of covariances of asset returns that has the following form:
⎡σ12
⎢
Σ = ⎢σ21
⎢
⎢
⎢⎣σn1 σ12 σ22
σn2 σ1n ⎤
⎥
σ2n ⎥
⎥
σn2 ⎥⎥⎦
where σi2 is the variance of the returns of the i-th asset and σij is the covariance between the assets i, j with i ≠ j.
Based on the performance of the portfolio RP = w1R1 + w2 R2 +… + wN RN, the variance denoted by σP2 is obtained.
N
wiwjσij
i = 1 i ≠ j
The variance of a portfolio can be shown in matrix form and for
⎡ w1 ⎤
⎢ ⎥
it is defined to the vector W = ⎢w2 ⎥ that contains all the weights of the
active
⎢ ⎥ ⎢ ⎥
⎣wN ⎦
Then the variance is expressed as the following quadratic form:
σP2 = W / ΣW
Volatility is simply σP = W / ΣW and likewise follows the root of time rule for a period of T days σP = TW / ΣW.
Examples for return and risk of a portfolio. To illustrate these ideas, two assets S1 and S2 are considered with the following data:
E = 0.15
E = 0.12 σ1 = 0.21⇒σ12 = 0.0441 σ2 = 0.17 ⇒σ22 = 0.0289 σ12 = 0.01785
w1 = 0.3
w2 = 0.7
So the portfolio return is 12.9%
E = w1E + w2E = 0.3 * 0.15 + 0.7 * 0.12 = 0.129
and the volatility of the portfolio is 16.08%
σP2 = w12σ12 + w22σ22 + 2w1w2σ12 = (0.3) 2 (0.0441) + (0.7) 2 (0.0289) +2 (0.3) (0.7) (0.01785) = 0.025627 σP = 0.1608
Now suppose that you have an initial amount of CU1,000,000 and the S1 asset is sold short, obtaining an additional CU300,000 after the operation, so you now have CU1,300,000 that is invested in S2. So the weight of asset S1 is w1 = −300000 = −0.3 which is negative since
1000000
this instrument was borrowed and can be seen as a liability.
The weight of the S2 asset is w2 = 1,300,000 = 1.3 since the
1,000,000
initial amount plus the amount obtained from the short sale have been deposited.
It is clear that w1 + w2 = −0.3 + 1.3 = 1 and it is concluded that a short sale implies a negative weight for that asset.
With these new weights or weights, the yield is 11.1% E = w1E + w2E = −0.3 * 0.15 + 1.3 * 0.12 = 0.111 and the standard deviation is 19.71%.
σP2 = w12σ12 + w22σ22 + 2w1w2σ12 = (- 0.3) 2 (0.0441) + (1.3) 2 (0.0289) + 2 (−0.3) (1.3) (0.01785) = 0.3888 σP = 0.1971
Up to this point, little attention has been devoted to covariance between market assets as it has only been mentioned as part of a formula and not as a key factor for good diversification. Covariance and correlation measure asset dependency and form the basis for diversification, so a somewhat detailed study of such dependency measures is required.
Covariance. With i, j∈ {1,2,…, N} let Si and Sj be the prices of two assets with returns Ri and Rj. The covariance between the assets is defined as σij = E) (Rj - E)] and has the characteristics of an internal product but two properties of interest are indicated:
1. σii = σi2
2. σij = σji so the matrix Σ is symmetric.
The sign of the covariance and its nullity provide information on the dependence of the assets Si and Sj as indicated:
• σij> 0 It means that, on average, when one asset produces a greater or less than average return, the other will tend to the same pattern. In other words, Si accompanies Sj when the latter appreciates or depreciates.
• σij <0 It means that, on average, when one asset produces a yield lower or higher than its average value, the other will tend to reverse in each case.
• σij = 0 In this case, a clear link on the assets cannot be established.
T
∑ (Rit - E) (Rjt - E)
The estimator of the covariance is σˆij = t = 1 for T
T
observations where Rit is the return of asset i on day t.
Correlation. The covariance depends on the magnitude of the random variable, so a standardized measure is preferable. Such a measure of dependency is found in the correlation defined as follows:
σij
ρij = in addition −1≤ ρij ≤1 σiσj
To exemplify the importance of correlation, consider the portfolio of two assets S1 and S2 with the following data:
E = 0.12 E = 0.15
σ1 = 0.17 ⇒σ12 = 0.0289
σ2 = 0.21⇒σ22 = 0.0441
The variance of the portfolio is σP2 = w12σ12 + w22σ22 + 2w1w2σ12 and from the equality σ12 = σ1σ2ρ12 a new formula is obtained for the variance of the portfolio.
σP2 = w12σ12 + w22σ22 + 2w1w2σ1σ2ρ12
If the weights of the assets and the correlation are varied, the following graph is obtained:
Illustration 6. By decreasing the correlation, better returns are achieved for a risk level.
It is observed that as the correlation decreases, it is possible to find better returns for a risk level. So it is preferable that the portfolios have negatively correlated assets.
It has been mentioned that covariance has the properties of an inner product and this makes correlation a measure of linear dependence. The geometric interpretation of the correlation can be seen with the following transformations in the formula
σP2 = w12σ12 + w22σ22 + 2w1w2σ1σ2ρ12.
a = σP b = w1σ1 c = w2σ2
So using the law of cosines for the triangle a, b, c we have the equality a2 = b2 + c2 - 2bccos (θ) from which it turns out that cos (θ) = −ρ12 and θ is the angle between the sides b and c.
The economic interpretation is that side a is the volatility of the portfolio and that this side grows with increasing correlation. When the correlation is null, we have a2 = b2 + c2, which is the Pythagorean theorem, and the volatility of the portfolio can be seen as the hypotenuse of the triangle.
It should be clarified that the correlation is a linear dependency measure, so it has limitations when the assets have a non-linear relationship, as suggested by the following example:
Let V1 ~ U (-1,1) and V2 = 1 − V12
It can be proved that E = 0 and E = 0 so that σV1V2 = 0 and we have random variables with null covariance but that are related in a non-linear way because V12 + V22 = 1.
The variance of a portfolio as a function of weights
By enrolling the definitions of stochastic dominance, it was derived the need to achieve, for a portfolio, the lowest risk given an expected level of return. Then you have an optimization problem in which the target variable is the variance of the portfolio and you can have numerous restrictions such as the prohibition of short sales. The first step in solving this optimization problem is the study of variance as a function of asset weights.
N
The variance of the portfolio is ij that in the form
i = 1 i ≠ j
matrix is σP2 = W / ΣW where W is the vector of weights and Σ is the matrix of variances and covariances.
Under the assumption that the inputs of Σ are constant, then the variance is a function of the weights of the assets
σP2 = f (w1, w2,…, wN).
From this point on, a further level of abstraction is desirable by placing the mean returns and asset weights into vectors as shown below along with an auxiliary vector.
⎡ E ⎤ ⎡w1 ⎤ ⎡1⎤
⎢E w 1
R = ⎢ ⎥ W = ⎢ ⎥ I = ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣E⎦ ⎣wN ⎦ ⎣1⎦
Based on the different strategic or legal requirements, there are different optimization problems. Below are some of them.
minσP2 = W / ΣW
sa
W / R = EW / I = 1
This problem seeks to minimize the risk subject to a given level of performance and the restriction that the weights or weights add up to the unit.
minσP2 = W / ΣW
sa
W / R = EW / I = 1
wi ≥ 0 ∀i
In this problem, there are other N restrictions when short sales are prohibited since, as is known, a short sale implies a negative weight. It should be noted that Markowitz made portfolios with short sales illegitimate.
A more general problem is the one that forces the incorporation of the weights at intervals defined by the authorities, as is the case in Mexico with the SIEFORE portfolios.
minσP2 = W / ΣW
sa
W / R = EW / I = 1
δi ≤ wi ≤γi ∀i
The first problem can be solved by using the Lagrange multipliers of the differential calculus while the following problems belong to the area of nonlinear programming.
In both cases, the study of the characteristics of the objective function and of the set that generate the restrictions is essential to build the efficient frontier.
Efficient portfolio.
When one of the optimization problems described above is solved, an efficient portfolio has been determined. In other words, for a given level of performance, the portfolio with the lowest risk has been obtained.
Efficient frontier. When any optimization problem for portfolios is solved for all possible levels of expected return, then the points generated form the efficient frontier as long as they have economic significance.
For simplicity to understand further topics, the solution to the problem in which short sales are allowed is given. To do this, a brief review of the convexity analysis is shown and these ideas are applied to the present financial problem.
Convex analysis
Convexity is a common feature in optimization problems. The construction of the efficient frontier without risk-free assets involves the solution of two optimization problems, starting from the solutions of such problems and by virtue of the theorem of two funds, which is presented later, any efficient portfolio is obtained. The following definitions and results constitute the mathematical justification for the construction of the efficient frontier.
Convex set. The set E ⊆ ℜN is convex if given x, y∈E ⇒αx + (1 − α) y∈E with α∈.
Intuitively, a convex set is one in which given two points of it, the segment that joins them is a subset of it.
The point αx + (1 − α) y∈E is known as the convex combination and can be generalized as α1 × 1 + α2 × 2 +,.., + αnxn ∈ E for n elements x1, x2,.., xn ∈ E and
n for n scalars α1, α2,.., αn ≥ 0 such that ∑αi = 1.
i = 1
As examples of this type of sets we have in triangles, the real line and in general ℜN, but so is (ℜ + ∪ {0}) N.
Convex function. A function f: →n → ℜ defined in a convex set E is convex if for x, y∈E and α∈ we have the following inequality:
f (αx + (1 − α) y) ≤αf (x) + (1 − α) f (y)
In the case of doubly differentiable functions in the entire set E, the following theorem represents an alternative definition of a convex function that, for the objectives pursued in these notes, will be adequate.
Theorem. Let be a function f: RN → R doubly differentiable defined on a convex set. This function is convex if and only if it has a defined semi-positive H (x) Hessian matrix.
N
Example. The portfolio variance function ij
i = 1 i ≠ j is convex.
Test
∂σ 2
j ij wjσij
∂w ≠ ij = 1
ij
∂2σ2
P = 2σij
∂wj∂wi
This means that the first derivative of the quadratic form σP2 = W / ΣW is 2ΣW and the second derivative is 2Σ which is a non-singular matrix.
So the Hessian of the portfolio variance is a positive matrix defined by being twice the covariance matrix so it can be stated that the variance of a portfolio is a strictly function
convex.
⎡σ12
⎢
H = 2 * ⎢σ21
⎢
⎢
⎢⎣σn1 σ12 σ22
σn2 σ1n ⎤
⎥
σ2n ⎥
⎥
⎥
σn2 ⎥⎦
Theorem. (Uniqueness). Given the following optimization problem:
min f (x)
sa x ∈ E
Where f: ℜn → ℜ is a strictly convex function and the set E is convex then the optimization problem has at most a minimizer.
Demonstration.
Suppose that a, b∈E are two different solutions, that is, f (a) = f (b) ≤ f (x) ∀x ∈E since f is convex so f (αa + (1- α) b) <αf (a) + (1- α) f (b) = f (a) = f (b)! for α∈ (0.1).
The contradiction lies in the fact that the point αa + (1- α) b∈ E, so there would be a value less than the minimum.
From this moment on, the objective is to create the efficient frontier and for this, the Lagrange multipliers are used, since it is already known that the variance of a portfolio has unique minimums.
Efficient frontier
As it has been proven, the variance of a portfolio is a strictly convex function, so minimizing it will not find technical details with the solutions found as long as the constraints form a convex set.
minσP2 = W / ΣW
sa
W / R = EW / I = 1
It is convenient to point out that minimizing σP2 is equivalent to minimizing
1σP2, so to solve this last case, two
2 scalars λ1 and λ2 are considered in the Lagrange function.
L (w1,.., wN, λ1, λ2) = W / ΣW + λ1 (E −W / R) + λ2 (1 − W / I)
The derivative of the quadratic form is 2ΣW, so when deriving the function L with respect to its arguments and equaling zero, we have:
ΣW = λ1R + λ2 I
E = W / R 1 = W / I
The first of these last three equations shows the general form of the efficient frontier and the important two-fund theorem.
ΣW = λ1R + λ2 I ⇒W = λ1Σ − 1R + λ2Σ − 1I
Interesting relationships are obtained from the solution of the optimization problem and for convenience the following terms are defined:
A = R / Σ − 1I
B = R / Σ − 1R
C = I / Σ − 1I D = BC - A2
If the solution is multiplied by the left by the transposed yield vector and by the vector I / then we have
R / W = λ1R / Σ − 1R + λ2 R / Σ − 1I I / W = λ1I / Σ − 1R + λ2 I / Σ − 1I
In reality, what was obtained is a system of equations whose solutions lead to the geometric interpretation of the efficient frontier.
Bλ1 + Aλ2 = E where λ1 = CE - A and λ2 = B - AE.
Aλ1 + Cλ2 = 1 DD
Multiplying from the left with the vector transposed of weights to equality ΣW = λ1R + λ2 I, we obtain an identity for the variance of the portfolio.
W / ΣW = λ1W / R + λ2W / I
CE 2 - AE B - AE
σP2 = λ1E + λ2 = PP + P
DD
CE 2 2AE B
σP2 = P - P +
DDD
This last equality corresponds to a parabola in the half-variance plane. The minimum of this function is obtained through the derivative of the variance with respect to the mean performance.
g A
2 E =
dσP = 2 CE - A = 0 ⇒ C
dE D σPg2 = 1
C
where the superscript g indicates that it is the global minimum variance portfolio.
The same result is obtained considering the efficient frontier as a hyperbola in the standard deviation-yield plane.
2 CE 2 2AED DB CD ⎜⎜⎛E 2 - 2 CA E + CA22 ⎟⎟⎞⎠ + C1 ⇔
σP = - + ⇔
D ⎝
⎛⎜E - A⎞⎟
2
σP ⎝ C ⎠
- = 1
1 D
CC 2
Global minimum variance portfolio
The portfolio that is located at the vertices of the parabola and the hyperbola is the least risky combination of assets regardless of the desired return.
The performance, variance and standard deviation of this portfolio are:
E g
σPg2
σPg
The vector of weights in this portfolio will be denoted as W gy and to determine this we must find the Lagrange multipliers corresponding to E g and they are:
1 Σ − 1I λ1g = 0 λ2g = ⇒W g = from the solution W = λ1Σ − 1R + λ2Σ − 1I.
DC
Two-fund theorem. The weight vectors of two efficient portfolios can be set in such a way that any efficient portfolio is generated from those two initial portfolios. This means that the efficient frontier can be created from two funds.
W = αW d + (1 − α) W g
Demonstration.
Efficient portfolio weights take the form
W = λ1Σ − 1R + λ2Σ − 1I
−1 −1
Doing W d = Σ R and W g = Σ I we have that W = λ1 AW d + λ2CW g in the
AC
that as already observed λ1 A + λ2C = 1.
Making α = λ1 A ⇒ 1 − α = λ2C gives the desired result, so any weight vector of an efficient portfolio is
a linear combination of two other efficient portfolios. Use of the investment portfolio technique
In order to demonstrate the technique, a portfolio of three assets has been designed, and the techniques developed in the preceding paragraphs are applied.
Suppose an economy with three risky assets whose returns and covariance matrix are presented below:
E = 0.14 ⎡ 0.23 0.02 −0.10⎤ ⎡ 9.71
E = 0.11 Σ = ⎢⎢ 0.02 0.15 0.10 ⎥⎥ ⇒Σ − 1 = ⎢⎢− 8.39
E = 0.13 ⎢⎣ − 0.10 0.10 0.17 ⎥⎦ ⎢⎣10.64
- 8.39 10.64 ⎤
⎥
18.22 −15.65
⎥
−15.65 21.35 ⎥⎦
The first step is to determine the constants A, B, C and D and then determine the vectors W d and W g.
A = ⎢⎢− 8.39
⎢⎣10.64 - 8.39 10.64 ⎤⎡1⎤
⎥⎢ ⎥
18.22 −15.65 1 = 3.1584
⎥⎢ ⎥ −15.65 21.35 ⎥⎦⎢⎣1⎥⎦
B = ⎢⎢− 8.39
⎢⎣10.64 - 8.39 10.64 ⎤⎡0.14⎤
⎥⎢ ⎥
18.22 −15.65 0.11 = 0.4829
⎥⎢ ⎥
−15.65 21.35 ⎥⎦⎢⎣0.13⎥⎦
C = ⎢⎢− 8.39 18.22
⎢⎣10.64 −15.65 10.64 ⎤⎡1⎤
⎥⎢ ⎥
−15.65 1 = 22.4796
⎥⎢ ⎥
21.35 ⎥⎦⎢⎣1⎥⎦
D = BC - A2 = 0.2053
⎡ 9.71 - 8.39 10.64 ⎤⎡0.14⎤
⎢ ⎥⎢ ⎥ - 8.39 18.22 −15.65 0.11
⎢ ⎥⎢ ⎥ ⎡ 0.5761 ⎤
d ⎢⎣10.64 - 15.65 21.35 ⎥⎦⎢⎣0.13⎥⎦ ⎢ ⎥
W = = −0.3816
A ⎢ ⎥ ⎢⎣ 0.8055 ⎥⎦
⎡ 9.71 - 8.39
⎢
- 8.39 18.22
⎢
g ⎢⎣10.64 −15.65
W =
C
Observations:
10.64 ⎤⎡1⎤
⎥⎢ ⎥
−15.65 1
⎥⎢ ⎥ ⎡ 0.5321 ⎤
21.35 ⎥⎦⎢⎣1⎥⎦ ⎢ ⎥
= −0.2591
⎢ ⎥
⎢⎣ 0.7270 ⎥⎦
• w1g + w2g + w3g = 0.5321−0.2591 + 0.7270 = 1.
• Asset 2 is sold short and the proceeds from this operation are sent to assets 1 and 3.
The minimum return and the minimum variance independent of the expected return of the portfolio is:
σPg2 = ⎢⎢ 0.02
⎢⎣ − 0.10 0.02
0.15
0.10 −0.10⎤⎡ 0.5321 ⎤
⎥⎢ ⎥
0.10 −0.2591 = 0.0445
⎥⎢ ⎥
0.17 ⎥⎦⎢⎣ 0.7270 ⎥⎦
The pair (σPg, RPg) = (0.2110, 0.1405) is the first point of the efficient frontier.
By the two-fund theorem, the efficient frontier is constructed from the following linear combination as shown in illustration 7:
⎡ 0.5761 ⎤ ⎡ 0.5321 ⎤
Wα = α⎢ − 0.3816⎥ + (1 − α) ⎢ − 0.2591⎥ α∈ℜ
⎢ ⎥ ⎢ ⎥
⎢⎣ 0.8055 ⎥⎦ ⎢⎣ 0.7270 ⎥⎦
Illustration 7. The efficient frontier is a hyperbola in the volatility-yield plane.
Expected profit and efficient portfolios
In order to know which efficient frontier portfolio should be considered when investing, the indifference curves of the expected profit are used.
The point of tangency between some indifference curve and the efficient frontier will be that which the individual will stochastically dominate the others and will be the individual's best choice.
This is how the expected utility allows decision-making between efficient investment portfolios and the first part of this material is justified. The same process is found when building the Capital Market Line in the following paragraphs.
The assumption of Gaussian returns allows individuals to have different utility functions and therefore different points of tangency with the efficient frontier. Once again, this situation is of great importance in equilibrium models as will be appreciated with the CAPM.
Illustration 8. The portfolio is chosen tangent to some indifference curve of the expected profit.
Inclusion of risk-free assets
Up to this point, only risky assets have been treated as shares, but a risk-free asset such as the T-bill, a bank account or the Cetes de México may be included.
The risk-free asset is denoted by S0, so there are now N + 1 instruments. This risk-free asset offers a known RL return.
With the inclusion of this asset, it is of interest to know if there are alterations in the efficient frontier, since now you can create a portfolio with a portfolio of risky assets and a risk-free asset.
The answer to this concern is obtained by solving a new optimization problem. The additional assumption is made that you can borrow and borrow at the risk-free rate.
E - RL maxTan (θ) = σP
To determine the efficient frontier with risky assets and with the risk-free rate, we must maximize the tangent of the angle formed by the line that joins the risk-free rate and any portfolio of risky assets.
N
∑wi (E - RL)
Tan (θ) = i = 1 this expression is derived with respect to wi
NN
∑∑wi wjσij
i = = 1 j 1
and then equals zero.
NN
∑wi (E - RL) ∑wjσij
i = 1 j = 1
= 2 = 0 ⇒
∂wi σP
NN
∑wi (E - RL) ∑wjσij
i = 1 2 j = 1 = E - RL
σP
N
∑wi (E - RL) N ξ = i = 1 2 ⇒ ∑ξwjσij = E - RL ∀i
σP j = 1
If vj = ξwj is done then the system can be expressed that is easily solved as observed.
⎡σ12 σ12
⎢ 2
⎢σ21 σ2
⎢
⎢
⎢⎣σN1 σN2
σ1N ⎤⎡v1 ⎤ ⎡ E - RL ⎤
⎥⎢ ⎥ ⎢ ⎥ σ2N ⎥⎢v2 ⎥ = ⎢ E - RL ⎥
⎥⎢ ⎥ ⎢ ⎥ 2 ⎥⎢ ⎥ ⎢ ⎥ σN ⎥⎦⎣vN ⎦ ⎣ E - RL ⎦
However, the values obtained from the solution of this system cannot be considered as weights or weights, so they must be normalized in order to obtain the weights of the WM market portfolio with inputs wiM = Nvi ⇒ N.
∑vi i = 1
i = 1
From these percentages, the volatility σM and the average return of the market E are determined, and then the capital market line (LMC) is constructed together with the risk-free rate. The slope of the LMC is E - RL and the equation in the dot and
σM
slope form is E = RL + E - RL σP. σM
Bottom theorem
All portfolio in the capital market line is built from a linear combination between the market portfolio and the risk-free asset.
Demonstration.
This result is obtained by solving the optimization problem by means of Lagrange multipliers.
minσP2 = W / ΣW
s ~.a. ~ where W ~ = ⎡w0 ⎤ R ~ = ⎡⎢RL ⎥⎤ ~ I = ⎡⎢1⎤⎥
⎢ ⎥
W / R = E ⎣W ⎦ ⎣ R ⎦ ⎣I⎦
W ~ / ~ I = 1
Based on these vectors, the following equalities are obtained:
ΣW = λ1R + λ2 I −λ2 −λ1RL = 0
Applying mathematical tricks we obtain that each vector of the CML is of the form
⎡1⎤ ⎡ 0 ⎤
⎢ ⎥ ⎢ ⎥
W ~ = α⎢0⎥ + (1 − α) ⎢w1M ⎥ α∈ℜ
⎢⎥ ⎢ ⎥
⎢ ⎥ ⎢ M ⎥
⎣0⎦ ⎣wN ⎦
Figure 9 shows the combination of the 10% risk-free rate and the efficient frontier portfolios of the three-asset economy described above. The result is the Capital Market Line (LMC).
Assuming the same data for the economy of three assets, a risk-free rate of 10% is added and the CML is determined as observed.
⎡ 0.23 0.02 −0.10⎤⎡v1 ⎤
⎢ ⎥⎢ ⎥
0.02 0.15 0.10 v2 ⎥
⎢ ⎥⎢
⎢⎣ − 0.10 0.10 0.17 ⎥⎦⎢⎣v3 ⎥⎦
⎡0.14 −0.10⎤ ⎡v1 ⎤ ⎡ 0.6237 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = 0.11−0.10 ⇒ v2 ⎥ = ⎢ − 0.6230⎥
⎢ ⎥ ⎢
⎢⎣0.13 −0.10⎥⎦ ⎢⎣v3 ⎥⎦ ⎢⎣ 0.9098 ⎥⎦
As can be seen v1 + v2 + v3 = 0.6237 −0.6230 +0.9098 = 0.9105 ≠ 1, so it is normalized to obtain the weights of the market portfolio and thereby solve the problem.
w1M == 0.6850 w2M == −0.6842 and w3M == 0.9992.
Based on these weights, the market performance
E = 0.14 * 0.6850 +0.11 * (- 0.6842) + 0.13 * 0.9992 = 0.1505 is obtained together with the volatility σM = 0.2356. The capital market line has the following equation RP = 0.10 + 0. σP.
Illustration 9. The risk-free asset gives rise to the Capital Market Line.
5 CAPITAL ASSET VALUATION MODEL (CAPM)
The capital asset valuation model (CAPM onwards) seeks to explain the performance of an asset in terms of market risk considering, among other assumptions, that investors in the economy make up their portfolios according to modern portfolio theory and that have homogeneous expectations.
Limitations of diversification.
Diversification is very useful to reduce the risk of an investment portfolio. However, this risk treatment mechanism is limited as can be seen when constructing the following portfolio: Suppose a set of N risky assets such that in pairs they have the average covariance covm that is assumed to be positive, the variance of each asset is the same for all and the weight of the ith asset is 1 = wi. So the variance of this portfolio is
N
constant at the limit.
2 N ⎛ 1 ⎞ 2 2 1 1 ⎛ 1 ⎞ 2 N 2 2 σ 2 ⎛ 1 ⎞
σ N = ∑i = 1 ⎜⎝ N ⎟⎠ σ + 2∑i ≠ j NN cov m = ⎝⎜ N ⎟⎠ ∑ i = 1 σ + N 2 ∑i ≠ j cov m = N + ⎜⎝1 - N ⎠⎟ cov m
σ 2 ⎛ 1 ⎞
+ ⎜1 - ⎟ cov m = cov m
N ⎝ N ⎠
This means that the greater the number of assets the variance of the portfolio decreases but the diversification is limited so that there is always risk regardless of the number of assets in a portfolio of risky assets. This observation gives rise to the following definitions of risk:
Diversifiable risk.
It is one that is potentially eliminated by diversification and comes from the particular characteristics of a station. It is important to note that the market portfolio has the maximum possible diversification, so the remaining risk gives rise to systematic risk.
Systematic risk.
It is the one that diversification cannot eliminate because it derives from factors that affect the entire economy, such as political changes.
Illustration 10. Diversifiable risk and systematic risk.
Then the total or specific risk of the instrument is equal to the aggregate of the diversifiable risk plus the systematic risk.
Total Risk = Diversifiable Risk + Systematic Risk.
In an economy in which investors use diversification to shape their portfolios, financial assets only have to pay a differential for systematic risk, since diversification has been pushed to the limit.
CAPM links the systematic risk premium of a financial asset to the premium of the market portfolio through a linear relationship. A generalization to this model is found in the Arbitrage Pricing Theory. In the following paragraphs, two derivations of the CAPM are shown in addition to the treatment of inflation, taxes, consumption and the Single Index Model (MIU) as an alternative for the construction of the efficient frontier.
CAPM assumptions
• Investors decide based on the half-variance criterion with normally distributed returns.
• Investors have the same time horizon.
• Investors have homogeneous expectations about asset returns, which means they see the same efficient frontier.
• The market is efficient.
• There is a risk-free instrument at the rate of which investors can lend and borrow unlimited amounts. • The market is perfect
Some of these assumptions can be weakened in order to obtain extensions of the CAPM, but among all of them, the one that refers to the homogeneity of the expectations of the returns on assets is fundamental because it enables the efficiency of the market portfolio.
Derivation of CAPM
Consider M participants in the capital market. Let Xi be the initial wealth of the i-th investor i = 1,2,…, M.
The economic equilibrium is reached when the supply and the demand of some satisfactor are equal. CAPM is an equilibrium model because it considers this situation. In this model, demand is the weighted sum of all the portfolios belonging to the M investors, while supply is seen in the market portfolio.
Demand
Let Wi be the weight vector of the portfolio of the i-th investor
M, then W ~ D = X1 ∑i = 1 X iW ~ i is the weight vector of
total demand M with X = ∑ X i.
i = 1
Based on a fund theorem, we have that the vector of
M ⎡1⎤ M ⎡ 0 ⎤
total demand is W ~ D = i = 1
X 0
⎢ ⎥ + i = 1
⎢⎥
⎢ ⎥
⎣ 0⎦
X w
⎢ 1 ⎥.
⎢ ⎥
⎢ M ⎥
⎣wN ⎦
MMM
∑ X iαi ⎢ ⎥ ∑ (1 − αi) X i ⎢ M ⎥
∑ X iαi ∑ (1 − αi) X i ∑ X iαi
Also i = 1 + i = 1 = 1 therefore, when αD = i = 1, we have
XXX
that the vector of total demand weights belongs to the CML.
For the WD vector, the following equalities are used to obtain the value of the first Lagrange multiplier.
ΣW D = λ1R + λ2 I
−λ2 −λ1RL = 0
ΣW D = λ1 (R - RL I)
WD / ΣW D = λ1 (E - RL)
ΣW DR - RL I
=
D / D
W ΣW E - RL
Offer
The total supply is given by the WM market portfolio.
The equilibrium
The equilibrium is obtained when WM = WD, so that the CAPM for the entire economy is obtained from equality.
ΣW MR - RI
WD = WM ⇒ M / M = L
W ΣW E - RL ⎡β1 ⎤ ⎡RL ⎤ ⎡ E ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ β RE
⇔ ⎢ 2 ⎥ (E - RL) + ⎢ L ⎥ = ⎢ 2 ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣βN ⎦ ⎣RL ⎦ ⎣E⎦
The i-th input of the vector ΣWD is the covariance between the yield Ri and the market yield RM and βi = cov (Ri2, RM).
σM
The beta and applications
The beta of an asset is a systematic measure of risk and helps to show the sensitivity to market risk of a stock.
If the beta of an asset is greater than unity, then the return on that asset, on average, will show an increase or decrease more than proportional to the market portfolio.
When the beta of the asset is less than unity then the return on the asset will accompany it in a way that is less than proportional to the performance of the market portfolio.
In the event that the asset has unit beta, then the return on the asset will move, on average, in the same proportion as the market portfolio.
Estimating the beta requires the performance of the market portfolio. The latter cannot be determined exactly, but there are proxy variables that allow simulating it. Said proxy variables are the stock indices such as the S&P 500 of the United States and in the Mexican case there is the Price and Quotation Index of the Mexican Stock Exchange that includes a group of around 35 shares, ratified or replaced each year, with weights that vary in real time.
Once an approximate market portfolio is obtained, the beta can be determined from its definition or through a linear regression in which it is considered that the performance of an asset depends linearly on the performance of the market portfolio..
CAPM finds real-life applications to determine a firm's cost of capital. The WACC (Weighted Average Cost of Capital) is the weighted average of the cost of equity and the capital cost of financial debt.
from
WACC = Kd + Ke d + ed + e where
Kd is the capital cost of the financial debt Ke is the cost of own capital
d is the market value of the financial debt
e is the market value of the firm's equity
In particular, the beta serves to estimate the cost of own capital Ke, which in the Mexican case takes the following form:
Ke = RL + β (E-RL) * RVA + Rsm + RP
Where
RL is the rate that 30-year treasury bills pay β is determined relative to the S&P 500 index
E is the average return on the S&P 500
RVA is an adjustment for an investment outside the United States environment
Rsm is a premium to consider Due to the size of the
RP firm, it is the country risk of the Mexican Eurobonds
The CAPM is a model that has extensions and criticisms of the very restrictive hypotheses that it uses, since, as has been seen in the cost of own capital, adjustments must be made to the theoretical CAPM. However, the model is still in force.
APPENDIX
The normal distribution
The random variable X is said to follow a normal distribution with localization and scale parameters µ and σ respectively if the density function has the following form
(µ) 2
n
- ∞ <µ <∞ σ> 0
When X has a normal distribution with their respective parameters it is denoted as X ~ N (µ, σ).
If the transformation Z = X −µ is carried out, it is obtained that Z ~ N (0.1) and Z σ is
known as standard normal. If we have Z, the transform X = σZ + µ leads to the original normal X.
For convenience, the variable Z is then used to derive results on any normal variable X.
Theorem. Let Z ~ N (0,1). So all the moments of this variable are finite.
PD E <∞ ∀n∈ N
Demonstration.
z22
E - z -ne− 2 dz ze dz
z2
If the change of variable y = is made, the following expression
2 is obtained
in which Γ denotes the range function.
nn
E = 22 ∞∫ and n2−1e − ydy = 2π2 Γ⎜⎛⎝ n2 + 1⎞⎟⎠ <∞
π 0
Result 1. If n is odd then E = 0.
2
E ze dz = 0
z2
This is because f (z) = zne− 2 is an odd function.
Result 2. If n is even then E = 1⋅3⋅5⋅… ⋅ (n −1)
n 2
E = −∫∞zeye − ydy = 2π2 Γ⎛⎜⎝n2 + 1⎞⎟⎠
2π
By induction on k ∈ N such that n = 2k it is proved that
2 k Γ⎛⎜ k + 1 ⎞⎟ = 1 ⋅ 3 ⋅ 5 ⋅… ⋅ (2k - 1) π ⎝ 2 ⎠
Once results have been obtained for the standard normal, it is possible to find other results for any normal.
Let mn = E and X ~ N (µ, σ).
If we remember that Z = X −µ then mn = E = E⎡⎢ (X −nµ) n ⎥⎤. σ ⎣ σ ⎦
The values of m3 and m4 are of interest because they lead to the values of bias and kurtosis of any normal.
Particular case n = 3
For the result 1 m3 = 0 = E⎡⎢ (X −3µ) 3 ⎤⎥ ⇒ k3 = E = 0 and therefore
⎣ σ ⎦ σ
has another result:
Result 3. The bias k3 of any normal random variable is zero.
Particular case n = 4
By the result 2 m4 = 3⋅1 = E⎡⎢ (X −4µ) 4 ⎤⎥ ⇒ k4 = E = 3 this leads
⎣ σ ⎦ σ
to another important result in the study of financial time series.
Result 4. The kurtosis of any normal random variable is equal to three.
From equality X = σZ + µ we have that X n = (σZ + µ) ny based on Newton's binomial
(σZ + µ) n = ∑j = n0 C njσ n− j Z n− jµ j where C nj = (n −n! j)! j!
Then we have the following result:
Result 5. The nth moment of a normal random variable is a function of the values of the mean µ and the standard deviation σ. In other words, any moment greater than the second of any normal random variable depends solely on the first two moments.
PD E = f (µ, σ)
Proof
n
If the expression n− j Z n− jµj
j = 0 is taken as hope,
then the linearity of the expression gives the desired result.
nn
E (n− jmn− jµ) = f (µ, σ)
j = 0 j = 0
This fifth result is critical when combining the utility and yield function ideas that are normally distributed.
MARKETS
PERFECT MARKET
A capital market is perfect when the following conditions are true:
• The market is friction-free; that is, there are no transaction costs or taxes, all assets are perfectly divisible and liquid, and there are no legal restrictions.
• There is perfect competition in the merchandise and stock markets.
• Information is received by all individuals and is free.
• Individuals are rational and seek to maximize their expected utility.
EFFICIENT MARKET
An efficient capital market allows the transfer of assets with a small loss of wealth, which is why it is integrated into the concept of efficiency in the Pareto sense. A market is of this type when the prices of the financial assets in commercialized in it reflect all the available information and therefore are fair prices.
There are three forms of efficiency:
1. Weak form of efficiency. In this circumstance, no individual can make extraordinary profits by following investment strategies based on historical price information. In other words, prices discount past information.
2. Semi-strong form of efficiency. In this form of efficiency, no investor obtains extraordinary returns through rules generated from publicly available information, so prices are said to discount that public information.
3. Strong form of efficiency. In this type of efficiency, no individual can earn returns above the market for any information. So the prices reflect all the information.
REFERENCES
- Copeland & Weston. (1988). Financial Theory and Corporate Policy. Addison Wesley Elton, Edwin J., Gruber Martin J. (1995). Modern Portfolio Theory and Investment Analysis. John Wiley & Sons. Heyman, Timothy. (1998). Investment in Globalization. IMEF, Milenio, IMCP, ITAM and BMV.
