Example. Suppose n = 2 and we wish to invest 200 in the portfolio d = (J, ) to do this

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1 o Wk ^ Stochastic Modelling in Finance - Mean-Variance Analysis Three basic assumptions (i) Agents only care about the mean and variance of returns. (ii) Agents have homogeneous beliefs (that is they can all calculate the same expected returns and variance of returns). (iii) Markets are frictionless so any number of units of an asset can be bought or sold at any time at the market price. Assets Consider a one-step market model with n-assets. The price of asset j at the initial time (t = 0), denoted Sj(0), is a constant. The price of asset j at the end of the period (t = 1), denoted S?(l), is a random variable. The return per unit invested in asset j is j which is a random variable. The expected returns on asset j are E[rj] and the variance of Tj will be denoted a? and can be calculated using a* = The covariance between assets i and j is denoted a^j and can be calculated using f Pu (observe that a^ = erf). The correlation between asset i and asset j is denned as p(i, j) = COV(ri, r^jgio-j. This formula is handy as sometimes the matrix of correlation coefficients is provided and sometime the matrix of covariances. Portfolios A portfolio is a collection of assets. We describe a portfolio via a vector of portfolio weights $ = ($1, #2?...»$n).The portfolio weight of asset j is the proportion of the value of the portfolio held in asset j, i.e. n cash invested in asset j 3 ' total cash invested in portfolio Example. Suppose n = 2 and we wish to invest 200 in the portfolio d = (J, ) to do this we invest 50 in asset 1 and 150 in asset 2.

2 -2- A portfolio is feasible if the agent is able to purchase the portfolio. By definition all feasible portfolios have 5^=1 ^» = 1- ^n some circumstances it may be desirable to select 'dj < 0 for some 1 < j < n, which is commonly called 'short-selling'. In some markets short-selling is prohibited in which case all portfolio weights in the feasible portfolio satisfy 'dj > 0. The returns on the portfolio fl will be denoted z=l which (like r,-) is a random variable. The expected returns on the portfolio $ are The variance of a portfolio # is denoted o% and can be calculated using o% := VAR(i^) = i=l i=\ j=l i=l j=l Example. Suppose n = 2 and a&e any portfolio $ = (#i,#2)- ^e variance of this portfolio is u% = V^ie^in + tf2r2) = ^i<7? + t9 «t + 2?9i^2cri(72Pi,2L Mean Standard deviation diagram - One risky and one risk free asset Suppose we have a market of only two assets. The first asset is risk-free so has zero variance and a constant return 77 and the second asset (asset A) has returns denoted with standard deviation a a- These two assets are illustrated in Figure 1. The feasible portfolios can be written as da = (a, 1 - a) for a G R. The return on the portfolio?9q is denoted Ra- The mean and standard deviation of Ra are v E[Ra) = otrf + (1 - ai)e[ra] ; aa = l - a\aa. For a < 1 we have a = 1 - Sa- substituting this into the equation for the expected returns gives So for a < 1 the feasible portfolio axe on the upper line in Figure 1. The bold line indicates where the assets are both bought in positive quantities and the dotted line indicates the region where the risk free asset has been short sold. Mean Standard deviation diagram - Two risky assets Suppose we have a market of only two assets, denoted A and B, with returns ra,rb and standard deviations aa,ab and correlation coefficient p G [ 1,1]. These two assets are illustrated in Figure 2.

3 -b- A rf, -> Figure 1: Mean-standard deviation diagram for 1 risky and 1 risk free asset. The feasible portfolios can be written as da = (a, 1 - a) and suppose that short-selling is not allowed so that a [0,1]. The return on the portfolio #Q is denoted Ra and E[Ra] = ae[ra] + (1 - a)e[rb] and the variance of Ra is VAR(JRQ) = a2a\ + (1 - a)2a2b + 2a(l - a)o-acrb/?, Suppose that we would like to construct a portfolio with a given expected return r G [?M, E[rB\], to do this set r = a{e[ra] - E[rB]) solving for a gives a*(r)= E[rA) - E[rB]' such that E[Ra*(r)] = r. The standard deviation of Ra*(r) depends on the correlation between the two assets Case 1: p 1 In this case we may write = {a*{r)va + (1 - a*(r))<rb)2.

4 -H- E[rB\ p=-l -> Figure 2: Mean-standard deviation diagram for 2 risky assets. It is an accident that the minimum variance portfolio in the case p = 0 has a return in [i?[r,4], [rs]], this need not always be the case. Inserting the expression for a* (r) into the standard deviation and rearranging gives and this line is illustrated in Figure 2. Case 2: p = 1 In this case we may write Let a : ob/(oa + vb) then it is easy to check that VAR(.Rq) = 0 and E[ta} < Ra<E[rB]. Define a risk-free asset C with re = Ra- Repeating the analysis in the previous paragraph we can show that By combining fund C and asset A we can obtain all portfolios on the line, E[rA] - rc r{cr) = rc-\ a. &A By combining fund C and asset B we can obtain all portfolios on the line, E[rB]-rc r(a) = rc-\ = a.

5 -s- Case 3: p G (0,1) The variance of the portfolio is VAR{Ra) = a2a\ + (1 - a)2a% 2a(1 - (x)aatrbp which is a quadratic function of a as illustrated. Minimum variance portfolio Typically when \p\ < 1 it is not possible construct a risk-free portfolio from 2 risky assets, however, we might be interested in the lowest variance portfolio that can be constructed. In Figure 3 the minimum variance portfolio is labeled Z (and has mean returns rmin and variance crmin). Figure 3: Minimum variance portfolio. Our goal is to find a e (0,1) such that the portfolio d = (a, 1 - a) has E[R \ = E[rm[n] and VAR(i?^) = <rmin. Observe that the expected returns of a feasible portfolio can be expressed as a function of a r(a) := E[R#a} = ae[ra] + (1 - a)e[rb\. Similarly, the variance of a feasible portfolio can also be expressed as a function of a a2{a) = VAR{R#a) = cx2a2a + (1 - afa\ + 2a(l - a)aaabp. At the point Z we have S^ = 0 which we may expand as da(a) dcr(a) da dr(a) da dr(a)'

6 -Gwhere -^gp = E[rA] E[rB] is a (non-zero) constant. Next using the chain rule -r<j2{a) = 2a(a)crf(a) da from which we deduce that 4-cr2(a) = 0 <=> -t-v{oi) = 0. However, hence i d 1. o-2(a) = 2a(<j\ + cr%- ^a^bp) - 2(aB - abaap) a = + o-% - 2<jAcrBp' Dominated portfolios and the efficient frontier A portfolio Z dominates the portfolio Y' \iuz = cry and E[RZ) > E[Ry], i.e. if you get more return for the same risk by investing in portfolio Z. The feasible set A is the set of all points on the mean standard deviation diagram which can be attained by combining the assets 1,2,...,n into a feasible portfolio. The boundary of the feasible set is called the efficient frontier. Figure 4 illustrates some feasible portfolios from combining assets A and B as well as B and C. The red line illustrates the efficient frontier. Theorem (Two fund separation). If funds X and Y are both on the efficient frontier all points on the feasible combinations can be expressed as a linear combination of these 2 funds. J Capital market line and efficient portfolios Next we reintroduce a risk free asset with return 77. By combining any fund Z (point) in A and the risk-free asset we may obtain all portfolios on the line, E[Rz]-rf yz(a) = rf + J-cr. However, for some a the portfolio on this line might be dominated by another point on the efficient frontier i.e. (2/z(cr)i c) <(max{r G R (r, a) G A}} 6*). As illustrated in Figure 5 there is a unique fund T for which the line

7 B Figure 4: Mean-standard deviation diagram for more that 2 assets. The efficient frontier is f */\ck/ lom strictly dominates the efficient frontier, i.e. for all a > 0 foz(cr), a) > nax{r R (r, o-) e A}; and (when A is convex) this inequality holds with equality on for a = <jt. The fund T is called the tangency portfolio and the line a i-)- yr(^) is called the Capital Markets Line (CML). Portfolios on the CML are referred to as efficient portfolios as they have a higher mean return than all other feasible portfolios with the same variance. The slope of the CML is referred to as the Sharpe Ratio of the tangency portfolio. All efficient portfolios have the same Sharpe ratio. Capital asset pricing model Recall that each agent is a mean-variance optimiser so will hold (according to his/her level of risk aversion) a combination of the tangency portfolio and the risk free asset. Suppose that the total number of asset i available is iv» the total value of asset i at time zero is JV^O) and the total value of all assets is Y%=\ NiSi(fy- The market portfolio is the portfolio with portfolio weights WACO) NnSn{0) The market portfolio coincides with the tangency portfolio because in total assets in the market must equal the total assets bought and sold by all of the agents.

8 -8- Figure 5: The tangency portfolio and capital markets line. The risk premium on the market portfolio is E[RT] Tf. The Capital Asset Pricing Model is a model for risk premium of a specific asset (or portfolio) The term pi(e[rr\ 77) is the part of the risk of asset i which is correlated with the market risk premium. This term is often called the systematic risk. Secondly c^ is the firm specific section of the risk. The pi of asset i is denned as _COV(ri,flT) Pi o and is the sensitivity of the excess asset returns to the expected market returns. Usually, we identify oti and # using regression analysis (not covered in this course). Efficient markets Hypothesis The efficient markets hypothesis states that: Adjusting for risk, it is not possible to outperform the market portfolio. In mathematical terms for any feasible portfolio Z E[RT] ~rf o~x E[RZ] - rf &z This assumption is naturally built into the mean-variance analysis outlined above as all feasible portfolios lie below the CML. To test the efficient markets hypothesis we can examine the a of various portfolios

9 -3- When cx.i < 0 asset i earns less return than a point on the CML so is dominated by an efficient portfolio. When oti = 0 asset i is on the efficient CML. When a* > 0 asset i outperforms the market portfolio. In this final case the Sharpe ratio of asset i exceeds the Sharpe ratio of the market portfolio, i.e. E[n] -rf ^ E[RT] - rf di ax which contradicts the efficient markets hypothesis. Multi-factor models It is common to think that the market risk premium is not the only thing that influences the risk premium of asset i. The risk premium can also be modeled as where (Zj : 1 < j < n) are various risk factors. Broadly these are usually either (i) Macroeconomic factors (exchange rates, growth forecasts etc) or (ii) firm specific factors (which industry, country the firm is located or size of firm etc). We will not focus in detail on multi-factor models in this course as in general asset prices will be considered to be exogenous. crj , R M. V S.