Lecture 5 Theory of Finance 1

Lecture 5 Theory of Finance 1 Simon Hubbert s.hubbert@bbk.ac.uk January 24, 2007 1 Introduction In the previous lecture we derived the famous Capital Asset Pricing Model (CAPM) for expected asset returns, given by: µ i = r 0 + β i (µ m r 0 ) i = 1, 2,..., n (1) where β i = cov(r i, r m ) σm. 2 From this we were able to propose a model for asset returns: (2) r i = r 0 + β i (r m r 0 ) + ɛ i, for i = 1,..., n. where E[ɛ i ] = 0, i = 1,..., n; E[ɛ i r m ] = 0, i = 1,..., n; (3) E[ɛ i ɛ j ] = 0 i j. In this lecture we explore some of the more useful applications of the CAPM and the model for asset returns (3) upon which it is based. In order to move the theory along we remark that the chronology of our development so far was as follows: firstly, derive the CAPM, a model for expected rates of return; then, deduce the model for the actual rates of return (3) which is consistent with CAPM. The overwhelming appeal here is that we have made natural step by step progress by building upon our mean-variance analysis, so everything is consistent. Another approach is to work in reverse, that is: 1

firstly, propose a model for the rates of return, similar in spirit to (3); then, hope to deduce a sensible CAPM-like model. The problem with the second step here, is that we do not have the comfort of the mean variance theory. In this lecture we will begin to investigate how this can be overcome with the aim of proposing different, hopefully more accurate models for asset returns. 2 Uses of CAPM 2.1 Data Reduction So far in this course we have become expert in mean-variance optimization, but what about implementing our solution techniques on a computer. Let us first consider the information we need at hand to perform this task. We need: n means µ 1,..., µ n ; n variances σ 2 1,..., σ 2 n; and (n 2 n)/2 covariances σ ij 1 i < j n. So in total our job requires 1 2 (n2 + 3n) parameters. Even if n is moderately large (n = 1000) this represents a huge data set (over 1/2 million). and CAPM can help to slim down this complexity. We have already discovered µ i = r 0 + β i (µ m r 0 ) σ 2 i = β 2 i σ 2 m + var(ɛ i ). For the covariances we can compute: σ ij = cov(r i, r j ) = cov(r 0 + β i (r m r 0 ) + ɛ i, r j ) = β i cov(r m, r j ) + cov(ɛ i, r j ) = β i σ mj + cov(ɛ i, r 0 + β j (r m r 0 ) + ɛ j ) = β i σ mj +β j cov(r m, ɛ j ) + cov(ɛ i, ɛ j ). }{{}}{{}}{{} =β j σm 2 =0 =0 2

In the above development, we have just substituted in the model for random returns, applied the linearity of covariance and the assumptions of CAPM, to yield σ ij = β i β j σ 2 m. All this means that with CAPM, all the key mean-variance data can be retrieved from r 0, µ m, σm, 2 and the sequences {β i } n i and {var(ɛ i } n i, that is, only 2n + 3 parameters are required. This is a substantial reduction. 2.2 The Market Portfolio As we know, the market portfolio is a portfolio consisting of a weighted sum of every asset in the market, with weights in the proportions that they exist in the market. In some sense this portfolio is only a theoretical concept, it ought to consist of every available risky investment including precious metals, real estate, wine etc.. In practice such a portfolio is impossible to assemble and track and thus approximations are used. In the UK the FTSE-100 is used to approximate the UK market and in the US the S&P-500 is taken. There are many criticisms of approximating in this way. The perfect market portfolio has the following properties: It is efficient; It has unit β. An interesting exercise is to check these properties, often one finds small deviations but sometimes these deviations can be large rendering the approximation useless. It is for this reason that Richard Roll, in his famous 1977 critique of CAPM, states that these proxies cannot provide an accurate representation of the entire market. An alternative may be to enforce these conditions by solving a constrained optimization problem. To set this up we suppose our universe of risky assets is represented by the set of n return rates {r 1,..., r n } and we suppose that we know the betas of each asset which we store in a vector b = (β 1,..., β n ) T R n. Given any vector w = (w 1,..., w n ) of feasible weights the associated portfolio beta is given by w T b = w i β i. For an approximation to the market portfolio we require a feasible, efficient portfolio with unit beta. This can be achieved by solving the following constrained optimization problem: minimize f(w) = 1 2 wt Vw subject to g 1 (w) = w T b 1 = 0 g 2 (w) = w T 1 1 = 0, (4) 3

the solution to this follows exactly the same recipe as the mean variance optimization problem. 2.3 Performance Evaluation The theory of CAPM can be used to evaluate the performance of an investment portfolio, and indeed it is now common practice to evaluate many institutional portfolios using the CAPM framework. We shall present the main ideas by going through a simple example. Suppose we have ABC is a mutual fund; FTSE represents the market; Gilts represent the risk-free asset. A fund manager wants to beat the market with ABC. We have 10 years worth of data (in the table below) and we want to use CAPM theory to evaluate the performance of ABC. Year Return ABC Return FTSE Return Gilts 1 14 12 7.0 2 10 7 7.5 3 19 20 7.7 4-8 -2 7.5 5 23 12 8.5 6 28 23 8.0 7 20 17 7.3 8 14 20 7.0 9-9 5 7.5 10 19 16 8.0 Step 1. We begin our analysis by computing the average rates of return and the standard deviation of the rates as implied by the 10 samples. These quantities are estimates based on the available data. In general, given r i for i = 1,..., n, the average rate of return is ˆµ = 1 n and this serves as an estimate of the true expected mean µ. The average standard deviation is given by sample estimate ˆσ = 1 n r i (r i ˆµ) 2. 4

Applying these identities to the data with n = 10 gives ABC FTSE Gilts Average 13 12 7.6 Standard deviation 12.4 9.4 0.5 Next we calculate an estimate of the β of ABC. For the covariance we use the sampling identity cov(abc, F T SE) = 1 n (r (ABC) i with n = 10. We can then deduce that ˆβ ABC = (F T SE) ˆµ ABC )(r i ˆµ F T SE ) cov(abc, F T SE) ˆσ F T SE = 0.0107 0.094 = 1.20375. We note already that the beta of ABC is greater than 1 and so within CAPM ABC beats the market. Step 2. Our next step is to check how well CAPM explains our sample data. We now write the formula ˆµ ABC r 0 = J + ˆβ ABC (ˆµ F T SE r 0 ). This looks like the familiar CAPM pricing formula, except that we have replaced the expected rates of return by measured average returns, similarly with the beta (this is the best we can do with the data), also we have added an error term J. The J stands for the Jensen index. According to CAPM the value of J should be zero when true expected returns are used. Hence J measures, approximately, how much the performance of ABC has deviated from the theoretical value of zero. A positive value of J presumably implies that the fund performs better than the CAPM prediction (but of course we recognize that approximation errors are introduced by the use of a finite amount of data to estimate the important quantities). We also remark that a large Jensen index casts serious doubt on the validity of CAPM to our data set. For ABC we find that J = 0.00104 > 0, and hence we may conclude that ABC is an excellent fund. But is this really a correct inference? Aside from the difficulties inherent in using short histories of data in this way, the inference that ABC is a good mutual fund is not entirely warranted. The fact that β ABC > 1 and J > 0 is nice and may tell us that ABC is good in 5

terms of its expected return, but it doesn t say that it is, by itself, efficient. It can be argued that the Jensen index tells us nothing about the fund, but instead is a measure of the validity of the CAPM. If the CAPM is valid, then every security (or fund) must satisfy the CAPM formula exactly, since the formula is an identity if the market portfolio is efficient. The CAPM is often applied to (new) financial instruments that are not traded and hence not part of the market portfolio. In this case the Jensen index can be a useful measure. Step 3. In order to measure the efficiency of ABC we must see where it falls relative to the capital market line. Only portfolios on that line are efficient. We do this by considering the Sharpe index, we write: S ABC = ˆµ ABC r 0 ˆσ ABC. Here S ABC stands for the Sharpe index for the fund, it is the slope of the line drawn between the risk-free point and the ABC point in (σ, µ) space. We find that 13 7.6 S ABC = = 0.435. 12.4 This must be compared to the gradient of the capital market line, which is just the Sharpe index of the F T SE. We compute Now since S ABC = ˆµ F T SE r 0 12 7.6 = = 0.468. ˆσ F T SE 9.4 0.435 = S ABC < S F T SE = 0.468 the mutual fund does not sit on the capital market line and so is not efficient, see Figure 1 We may conclude that ABC may be worth holding but, by itself, it is not quite efficient. To achieve efficiency it would be necessary to supplement the fund with other assets in order to eliminate its specific risk. 2.4 CAPM as a pricing formula As the final two letters in its acronym suggest, the CAPM is a pricing model. The standard CAPM formula does not contain prices explicitly, only expected return rates.to see why the CAPM is a pricing model we must go back to the definition of return. Suppose that an asset is purchased today at price P and sold at in the future at price Q. The rate of return on this asset is given by r = Q P P. 6

portfolio mean FTSE ABC specific risk risk free rate portfolio volatility Figure 1: Position of fund ABC If the asset is a risky one, then its future payoff Q is unknown today - that is, Q is a random variable. The expected return rate is given by µ = E[ Q P P ] = E[Q P Putting this in the CAPM formula, we have Solving for P we obtain 1] = E[Q] P 1. µ = E[Q] P 1 = r 0 + β(µ m r 0 ). P = E[Q] 1 + r 0 + β(µ m r 0 ). (5) This pricing formula has a form that very nicely generalizes the familiar discounting formula for risk-free assets. In the risk free case, the payoff Q is known and so its present value is found by discounting as P = Q 1 + r 0. In the random case the appropriate interest rate is r 0 + β(µ m r 0 ), which can be regarded as the risk-adjusted rate for this asset. 2.4.1 The linearity of CAPM pricing We now discuss a very important property of the pricing formula - namely, that it is linear. This means that the price of two assets is the sum of their prices 7

and the price of a multiple of an asset is the same multiple of the price. This is quite startling because the formula does not look linear art all. For example, if P 1 = E[Q 1 ] 1 + r 0 + β 1 (µ m r 0 ) and P 2 = E[Q 2 ] 1 + r 0 + β 2 (µ m r 0 ), where β 1 and β 2 denote the betas of assets 1 and 2 respectively, it does not then seem obvious that P 1 + P 2 = E[Q 1 ] + E[Q 2 ] 1 + r 0 + β 1+2 (µ m r 0 ), where β 1+2 is the beta of the new asset, which is the sum of assets 1 and 2. We can easily take care of this doubt by converting the formula into another form, which appears linear. Suppose that we have an asset with price P and future payoff Q. Here again P is known and Q is uncertain. Using the fact that r = Q P 1, the value of its beta (using the linearity of covariance in one argument) is β = cov(r, r m) σm ( 2 ) cov Q P 1, r m = σm 2 = cov(q, r m) P σm 2 Substituting this into the pricing formula (5) and dividing by P yields 1 = E[Q] P (1 + r 0 ) + cov(q,r m)(µ m r 0 ) σ 2 m Finally, solving for P we obtain the following formula P = 1 [ E[Q] cov(q, r ] m)(µ m r 0 ) 1 + r 0 σm 2. (6) The term in the square brackets is called the certainty equivalent of Q. This value is treated as a certain amount, and then the usual discount factor 1/(1+r 0 ) is applied to it to obtain P. The certainty equivalent form clearly shows that the pricing form is linear because both terms in the brackets depend linearly on Q.. 8

The reason for linearity can be traced back to the principle of no arbitrage: if the price of the sum of two assets were not equal to the sum of the individual prices, it would be possible to make arbitrage profits. For example, if the combination asset were priced lower than the sum of the individual prices, we could buy the combination (at the lower price) and sell the individual pieces (at the higher price), thereby making a profit. By doing this in large quantities we could make arbitrarily large profits. If the reverse situation held - if the combination asset were priced higher than the sum of the two assets - we would buy the assets individually and sell them as a combination package, again making arbitrage profits. Such arbitrage opportunities are ruled out if and only if the pricing of assets is linear. 3 Factor Modeling Based upon the formula of the CAPM for assets µ i = E[r i ] = r 0 + β i (µ m r 0 ) = r 0 + β i (E[r m ] r 0 ) = E[r 0 + β i (r m r 0 )], i = 1,..., n. we have proposed the following model for the evolution of the random return rates: r i = r 0 + β i (r m r 0 ) + ɛ i, i = 1,..., n, where ɛ i denotes the random noise specific to asset i and satisfies the three conditions in (3). The model can be written as r i = r 0 + β i r m β i r 0 + ɛ i = r 0 (1 β i ) + β i r m + ɛ i (7) = α }{{} i + β i r }{{ m } + ɛ }{{} i intercept common factor random noise In brief we see that, through our investigation of mean variance analysis we have naturally arrived at the CAPM and, from this theoretically tight standpoint, we could propose a model for asset returns. The view of this model given by (7) tells us that all random returns are driven by a single factor - the market, represented by the market portfolio. However, each asset has its own specific noise component unrelated to the market and so, also, unrelated to each other. At this point we ask ourselves whether the discovery that the market portfolio is the key driver of our portfolio returns, perhaps not. Suppose that our portfolio is composed of random assets taken from a particular sector of the market, for example commodities, inflation linked or fixed income products then we may want to argue that the market portfolio is not the key driver of our assets. In 9

the examples just given a more likely driver is oil prices, inflation targets or interest rate fluctuations. We may also believe that the market isn t the ONLY driver for our assets. Perhaps there are several source of randomness which we can observe that clearly drive the evolution of the assets we are investing in. To account for all this we decide to generalize CAPM and write r i = a i + b i f + ɛ i, i = 1,..., n, (8) where f is the new driving random variable, we called (8) a single factor model. Alternatively, we can also write r i = a i + b i1 f 1 + b i2 f 2 + + b im f m + ɛ i i = 1,..., n, (9) where here we have m random factors driving the asset returns, we call (9) a multi factor model. Just as with CAPM we assume, for the specific noise components, that Zero mean E[ɛ i ] = 0, i = 1,..., n. Uncorrelated with the driving factors E[ɛ i f] = 0, (or E[ɛ i f j ] = 0, j = 1,..., m) i = 1,..., n. Uncorrelated with each other E[ɛ i ɛ j ] = 0, i j. 3.1 Implications of employing factor models We now want to turn to the implications of directly employing factor modelling to our investment decisions. Consider a feasible portfolio of n risky assets with weights w 1,..., w n. The random return rate is then given by r p = w i r i. If we use a one-factor model for asset returns then this becomes ( n ) r p = w i a i + w i b i f + w i ɛ i } {{ } =a p } {{ } =b p } {{ } ɛ p = a p + b p f + ɛ p. 10

Focussing upon ɛ p we know that this random variable has zero mean, its variance is computed as var(ɛ p ) = E[ɛ 2 p] n = E[ w i ɛ i w j ɛ j ] = = = j=1 j=1 w i w j E[ɛ i ɛ j ] wi 2 E[ɛ 2 i ] wi 2 var(ɛ i ). At this point we recall from the CAPM framework: (10) the quantity var(ɛ p ) represents the specific risk of the portfolio which is unconnected to the risk of the driving factor r m ; the specific risk can be eliminated through diversification, and this is achieved by ensuring the portfolio is mean-variance efficient. The new factor models that we are proposing do not arise from a mean variance framework, in fact only when the market portfolio is chosen as the single driving factor do we have this link. Thus, if our new models are to generalize the CAPM in a consistent way we need to construct arguments to show that the specific risk associated with factor modeling can also be eliminated. Suppose we choose the special weights w i = 1 n i = 1,..., n, and that then, using (10) we can deduce that σ 2 = max{var(ɛ i ) : i = 1, 2,..., n} var(ɛ p ) σ2 n. If the number of assets n is large then we can see that the specific risk var(ɛ p ) is small. Technically, if we want to argue that we can eliminate the specific risk we make the assumption that there are infinitely many assets in the economy, in which case for infinetly many assets : var(ɛ p ) σ2 n 0 as n. 11

4 The Arbitrage Pricing Theorem The factor model framework leads to an alternative approach to asset pricing, called Arbitrage Pricing Theory (APT). This theory does not require the assumption that investors evaluate portfolios on the basis of means and variances; only that when returns are certain, investors prefer greater return to lesser return. In this sense, the theory in much more satisfying than CAPM theory, which relies both on the mean-variance framework and a strong version of equilibrium, which assumes that everyone uses the mean-variance framework. Instead the APT assumes that the universe of assets being considered is infinite and these assets differ from each other in non-trivial ways. The APT uses an arbitrage argument to force the relationship between a i and b i so that a i can be eliminated from the pricing formula. We will prove 3 separate cases of the Arbitrage Pricing Theorem. We deal first deal with the simplistic case where the assets follow a single factor model but with no noise component. This is an over simplification, indeed the assumption of infinitely many assets is not required here since there is no specific risk. 4.1 APT - single factor case with no noise Here we assume that the returns on the assets are modelled as r i = a i + b i f, i = 1,..., n. The uncertainty comes from the driving factor f. The APT says that the intercepts a i (i = 1,..., n) and the factor loadings b i (i = 1,..., n) are related if there are no arbitrage opportunities. Theorem 1. Suppose the economy consist of n assets whose return rates evolve according to a single factor model Then there exist a constant λ such that where r 0 represents the risk free rate. r i = a i + b i f, i = 1,..., n. µ i = r 0 + b i λ i = 1,..., n, Proof. Choose two distinct asset i and j such that and form a portfolio with weights b i b j w i = α and w j = 1 α. 12

The return rate on this portfolio is given by r p = αr i + (1 α)r j = α(a i + b i f) + (1 α)(a j + b j f) = αa i + (1 α)a j + [αb i + (1 α)b j ] f We now retrospectively choose α so as to eliminate the factor-risk, that is we solve αb i + (1 α)b j = 0 this gives then the portfolio return b j α = b j b i r p = a ib j b j b i + = a ib j a j b i b j b i a jb i b i b j (11) has no exposure to f, it is risk-free and thus we set it equal to r 0, to give r 0 = a ib j a j b i b j b i r 0 (b j b i ) = a i b j a j b i a j r 0 = a i r 0. b j b i This relationship must hold for all assets, therefore (a i r 0 )/b i must be a constant c which is independent of i. Hence we obtain a i r 0 b i = c, i = 1, 2,..., n. We can use this information to write a formula for the expected return of asset i µ i = E[r i ] = a i + b i E[f] If we set λ = c + µ f then we get = r 0 + b i c + b i E[f] = r 0 + b i (c + µ f ). µ i = r 0 + λb i, i = 1, 2,... Notice that once the constant λ is known, the expected return of an asset i is then determined entirely by the factor loading b i. 13

5 Advert for next lecture! In the next lecture we will continue with the arbitrage pricing theorem, we will consider the multi-factor case, again with the simplifying assumption of no noise. We will then consider the case where the assets follow a multi-factor model with noise attached, it is in this case where the assumption of infinitely many assets is needed. We will also investigate how the APT is implemented in practice. 14