PS4/Asset Pricing

SML Graph

-x-axis: Expected Return -y-axis: Beta -Upward sloping line -Starts at rf -Mark the Market portfolio on the line at some point (1, rₘ)

Consider the Capital Asset Pricing Model (CAPM). What are the main assumptions of the model? What is the Capital Market Line (CML)? What is the Security Market Line (SML)?

The Capital Asset Pricing (CAPM) model is a theory whereby the equilibrium rates of return on all risky assets are a function of their covariance with the market portfolio. It is a model that is used to determine the expected rate of return on a security based on its risk characteristics. It describes the relationship that holds between systematic risk and expected return for assets, in particular stocks. Systematic risks are market risks that cannot be diversified away, such as interest rates. All the portfolios that optimally combine the risk-free rate of return and the market portfolio of risky assets are represented. The CAPM model tells us that the return on a security is equal to the risk-free rate plus a market risk premium multiplied by the market relative risk factor for the security measured as beta. CAPM Equation: rᵢ = rf + 𝛽ᵢ (rₘ - rf) The main assumption underlying CAPM is that the capital market is perfect and: ➢ There are no transaction costs or taxes involved in buying or selling securities in the capital market ➢ All the information about the securities in the market is freely available to all investors ➢ All investors can borrow or lend any amount of money at a given interest rate ➢ All investments mature in one period ➢ Investors are risk averse and make decisions using the mean variance rule (i.e. investors seek to maximise their expected return for a given amount of risk) The Capital Market Line represents portfolios that optimally combine risk and return. It is a straight line passing through the risk-free rate of return and the expected rate of return on the market portfolio. Risk is measured by the standard deviation on the horizontal axis. The rate of return is on the vertical axis. Under CAPM, investors will choose a position on this capital market line at the equilibrium, by borrowing or lending at the risk-free rate, as this will maximise their return for a given level of risk. It optimally combines risk and return and leads to an efficient portfolio. The slope of the CML is the sharpe ratio of the market portfolio. It can be stated that an investor should buy assets if the sharpe ratio is above the CML, and sell if below the CML. The equation of the CML: rₓ = 𝑟𝑓 + [(rₘ − 𝑟𝑓)/σₘ]𝜎ₓ where 𝑟ₓ is the expected return on an asset/portfolio 𝑟𝑓 is the risk free rate rₘ is the expected market return σₘ is the standard deviation of the market returns 𝜎ₓ is the standard deviation of the asset/portfolio returns CML Graph: -x-axis: Volatility -y-axis: Expected return -Upward sloping line -Line starts at rf -Mark the Market portfolio on the line at some point (σₘ, rₘ) -Mark rₘ - 𝑟𝑓 The Security Market Line (SML) is derived from the CML. While the CML shows the rates of return for a specific portfolio, the SML represents the market's risk and return at a given time, and shows the expected returns of individual assets. And while the measure of risk in the CML is the standard deviation of returns (total risk), the risk measure in the SML is systematic risk, or beta. The SML is a line which measures the relationship between beta (or systematic risk) and a firm's expected rate of return. Securities that are fairly priced will plot on the CML and SML. Securities that plot above the CML or the SML are generating returns that are too high for the given risk and are under-priced. Securities that plot below CML or the SML are generating returns that are too low for the given risk and are over-priced. Beta is a measure of a security's sensitivity to market movements or systematic risk 𝛽ᵢ = Cov(rᵢ, rₘ)/Var(rₘ) This formula represents the covariance of the security relative to the variance of the market. By definition, the 𝛽 of the market is 1. -The SML equation is: rᵢ = 𝑟𝑓 + 𝛽(rₘ − 𝑟𝑓) where rᵢ is the expected return on an asset, 𝑟𝑓 is the risk free rate, rₘ is the expected market return SML Graph -x-axis: Beta -y-axis: Expected Return -Upward sloping line -Starts at rf -Mark the Market portfolio on the line at some point (1, rₘ)

CAPM Assumptions

The main assumption underlying CAPM is that the capital market is perfect and: ➢ There are no transaction costs or taxes involved in buying or selling securities in the capital market ➢ All the information about the securities in the market is freely available to all investors ➢ All investors can borrow or lend any amount of money at a given interest rate ➢ All investments mature in one period ➢ Investors are risk averse and make decisions using the mean variance rule (i.e. investors seek to maximise their expected return for a given amount of risk)

CML Equation

rₓ = 𝑟𝑓 + [(rₘ −𝑟𝑓)/σₘ]𝜎ₓ

Beta

-A measure of a security's sensitivity to market movements or systematic risk -It measures the riskiness of stock i relative to the risk of the market 𝛽 = how risky asset i is/ how risky the stock market is 𝛽ᵢ = Cov(rᵢ, rₘ)/Var(rₘ) -By definition, the 𝛽 of the market is 1. -A share with a beta of 1 tends to move by a similar percentage to the market; one with a beta of 2 tends to move up or down -𝛽 > 1 amplifies the overall movements of the market -0 < 𝛽 < 1 tend to move in the same direction as the market but not so far -E.g. Tesco had a beta of 1.36 which means on average when the market raises/falls by 1%, Tesco's price will rise/fall by 1.36%

APT

-Arbitrage Pricing Theory (APT) is a multi-factor asset pricing model. -It is based on the concept that an asset's returns can be predicted using the linear relationship between the asset's expected return and a number of macroeconomic variables that capture systematic risk. -With APT, all securities (and all portfolios) sit on the Security Market Line (SML). An arbitrage portfolio can be created by selling (or buying) an undervalued asset and buying (or selling) a portfolio of other assets with the same beta and then closing out the position later to realise a profit. So, in APT, any asset or portfolio with the same beta should give the same expected return. Arbitrage opportunities may exist if this does not hold true.

Explain how the CAPM provides a simple rationale for the following portfolio strategy: • Diversify your holdings of risky assets according to the proportions of the market portfolio; • Mix this portfolio with the risk free asset to achieve a desired risk-return combination

-CAPM is a theory whereby the equilibrium rates of return on all risky assets are a function of their covariance with the market portfolio. -If an investor buys a portfolio of assets in proportion to the market weightings (otherwise known as passive investing or indexing) of some broad market index like the S&P500, the MSCI global index; the FTSE 100, an investor will by definition earn a market return (rₘ). -If all the investors capital is invested within this portfolio, full exposure to the market conveys that they will neither under nor outperform the market. -From the CAPM model, rₘ symbolises the return from complete investment in the market portfolio, also having a 𝜷 of 1. -The risk level an investor is willing to take may alter the value of beta. For example, putting all their money into risk free assets and none to the market portfolio, would earn a return of 𝒓𝒇 with a 𝜷 of 0. This shows that the market will generate varying returns dependent upon the percentage of the portfolio invested in the market. -A 𝜷 of ½ implies a 50/50 portfolio, meaning the portfolio return will be equal to the average of the risk-free rate and the market return. -A portfolio with a 𝜷 > 𝟏 could be achieved if the investor was more of a risk taker, they could borrow money to invest in the market and this would therefore give an expected return that is higher than the market return. -This strategy can be used alongside their tolerance to risk to determine where on the SML return they wish to be and the percentage of the portfolio to invest in the market.

2015/16 3. Capital Asset Pricing Model (b) Assume a perfect capital market in which investors are constrained to holding portfolios that consist of a single risky asset and the riskless asset. In equilibrium the following relationship between two risky securities i and j holds: Security i Security j Exp. ret. (%) 18 25 Standard Dev. (%) 8 12 What is the rate of interest in this market?

-CML equation is used -Equate the slopes of the CML and find rf The Capital Asset Pricing (CAPM) model is a theory whereby the equilibrium rates of return on all risky assets are a function of their covariance with the market portfolio. It describes the relationship that holds between systematic risk and expected return for assets, in particular stocks. Systematic risks are market risks that cannot be diversified away, such as interest rates. Under CAPM, investors will choose a position on this capital market line at the equilibrium, by borrowing or lending at the risk-free rate, as this will maximise their return for a given level of risk. Therefore the risk-free rate (rf) is the rate of interest in this market. In equilibrium, the two securities i and j must be on the same CML. The equation of the CML is rₓ = 𝑟𝑓 + [(rₘ −𝑟𝑓)/σₘ]𝜎ₓ where 𝑟ₓ is the expected return on an asset/portfolio 𝑟𝑓 is the risk free rate rₘ is the expected market return σₘ is the standard deviation of the market returns 𝜎ₓ is the standard deviation of the asset/portfolio returns The slope of the Capital Market Line(CML) is the Sharpe Ratio of the market portfolio. In equilibrium: (rₘ −𝑟𝑓)/σₘ = (rₘ − 𝑟𝑓)/σₘ (0.18 - rf)/0.08 = (0.25 - rf)/0.12 Solve for rf rf = 0.04 4%

Draw the decision tree describing the situation that the Your Bank faces if it does not implement the credit check. What is the expected return to each loan that is offered?

-Decision tree has two branches: loan is paid back (0.8) or loan is defaulted (0.2) Payoff: 18% return if loan is paid back so r = 0.18 Payoff: 100% loss if loan is defaulted so r = -1 Expected return on each loan offered: ∑𝑝ᵢ𝑟ᵢ = (0.8 ∙ 0.18) + (0.2 ∙ −1) = −0.056 = −5.6%

Draw the decision tree describing the situation that the Your Bank faces if it implements the credit check. Would you recommend the bank to go ahead and implement the credit check? If so, why? How valuable is the information contained in the credit check to Your Bank?

-First decision tree section: CREDIT CHECK - Favourable (0.8) or Unfavourable (0.2) -Second decision tree section (no probabilities) : LOAN DECISION - Granted or Not granted -Third decision tree section: When the loan is granted, PAYMENT - Loan repaid or Not repaid -Calculate the payoffs of each outcome by taking away the cost of implementing the check from the return figures given in the question -Calculate the expected return on a favourable credit check -Calculate the expected return on an unfavourable credit check -Total these to get the expected return when a credit check is implemented -Compare this figure to the expected return without a credit check to see if it is worth implementing

Non-systematic Risk

-Specific/idiosyncratic/diversifiable risk -Risk that can be eliminated by diversification Four components: -Management risk - the risk that the managers running the firm are incompetent and lead the firm into insolvency. It is quite high in new firms with untried and untested managers. -Business risk - the risk from the asset side of the firm's balance sheet. It is the risk that the firm will not generate sufficient sales of revenue to finance the fixed costs of its operations. -Financial risk - the risk from the liability side of the firm's balance sheet. It is the risk that the firm will not generate enough sales revenue to finance the fixed-charge liabilities on the balance sheet. -Collateral Risk - the risk that investors face if they have poor collateral and claims to the assets of the firm and behind other investors.

2015/16 3. Capital Asset Pricing Model (a) The Betas of four stocks in a perfect capital market are as follows: βA = -1 βB = 0 βC = 1 βD = 2 Assume that the market is in equilibrium, that the returns on the risk-free asset is 6% and that the expected return on the "market portfolio" is 14%. Calculate the expected returns on shares A, B, C and D.

-Sub rf = 6, rm = 0.14 and the different betas into the the CAPM equation -CAPM Equation: rᵢ = rf + 𝛽ᵢ (rₘ - rf) -Define CAPM: The Capital Asset Pricing (CAPM) model is a theory whereby the equilibrium rates of return on all risky assets are a function of their covariance with the market portfolio. It is a model that is used to determine the expected rate of return on a security based on its risk characteristics. It describes the relationship that holds between systematic risk and expected return for assets. Systematic risks are market risks that cannot be diversified away, such as interest rates.

Systematic Risk

-Systematic risk (market/covariant/undiversifiable risk) is the risk that cannot be diversified away. -It is the risk that faces every firm in the economy and is due to market conditions (interest rates/inflation rate/business cycles etc). -Example: A firm's earnings are positively correlated with business cycles, so if there is a recession all firms' earnings are negatively affected

CAPM

-The Capital Asset Pricing (CAPM) model is a theory whereby the equilibrium rates of return on all risky assets are a function of their covariance with the market portfolio. -It is a model that is used to determine the expected rate of return on a security based on its risk characteristics. -It describes the relationship that holds between systematic risk and expected return for assets. -Systematic risks are market risks that cannot be diversified away, such as interest rates. -All the portfolios that optimally combine the risk-free rate of return and the market portfolio of risky assets are represented. -The CAPM model tells us that the return on a security is equal to the risk-free rate plus a market risk premium multiplied by the market relative risk factor for the security measured as beta.

CML

-The Capital Market Line represents portfolios that optimally combine risk and return. -It is a straight line passing through the risk-free rate of return and the expected rate of return on the market portfolio. -Risk is measured by the standard deviation on the horizontal axis. The rate of return is on the vertical axis. -Under CAPM, investors will choose a position on this capital market line at the equilibrium, by borrowing or lending at the risk-free rate, as this will maximise their return for a given level of risk. -It optimally combines risk and return and leads to an efficient portfolio. -The slope of the CML is the sharpe ratio of the market portfolio. -It can be stated that an investor should buy assets if the sharpe ratio is above the CML, and sell if below the CML. -The equation of the CML: rₓ = 𝑟𝑓 + [(rₘ −𝑟𝑓)/σₘ]𝜎ₓ where 𝑟ₓ is the expected return on an asset/portfolio 𝑟𝑓 is the risk free rate rₘ is the expected market return σₘ is the standard deviation of the market returns 𝜎ₓ is the standard deviation of the asset/portfolio returns CML Graph: -x-axis: Volatility -y-axis: Expected return -Upward sloping line -Starts at rf -Mark the Market portfolio on the line at some point (σₘ, rₘ) -Mark rₘ - 𝑟𝑓

SML Equation

-The SML equation is: rᵢ = 𝑟𝑓 + 𝛽(rₘ − 𝑟𝑓) where rᵢ is the expected return on an asset, 𝑟𝑓 is the risk free rate, rₘ is the expected market return

SML

-The Security Market Line (SML) is derived from the CML. -While the CML shows the rates of return for a specific portfolio, the SML represents the market's risk and return at a given time, and shows the expected returns of individual assets. -The measure of risk in the SML is the systematic risk, or beta. -The SML is a line which measures the relationship between beta (or systematic risk) and a firm's expected rate of return. -Securities that are fairly priced will plot on the CML and SML. -Securities that plot above the CML or the SML are generating returns that are too high for the given risk and are under-priced. -Securities that plot below CML or the SML are generating returns that are too low for the given risk and are over-priced. -Beta is a measure of a security's sensitivity to market movements or systematic risk 𝛽ᵢ = Cov(rᵢ, rₘ)/Var(rₘ) This formula represents the covariance of the security relative to the variance of the market. By definition, the 𝛽 of the market is 1. -The SML equation is: rᵢ = 𝑟𝑓 + 𝛽(rₘ − 𝑟𝑓) where rᵢ is the expected return on an asset, 𝑟𝑓 is the risk free rate, rₘ is the expected market return SML Graph -x-axis: Expected Return -y-axis: Beta -Upward sloping line -Starts at rf -Mark the Market portfolio on the line at some point (1, rₘ)

To create a portfolio with zero investment and zero beta

-The basic idea behind the APT model is that investors can create a zero-beta portfolio with zero net investment. -To create an arbitrage portfolio with zero investment and zero beta, two conditions need to be satisfied: 1) For zero investment: ∑𝑥ᵢ = 0, where 𝑥ᵢ is the weight of 𝑖𝑡ℎ security 2) For zero beta: ∑𝑥ᵢ𝛽ᵢ = 0, where 𝑥ᵢ is the weight of 𝑖𝑡ℎ security and 𝛽ᵢ is the 𝛽 of 𝑖𝑡ℎ security

CML Graph

-x-axis: Volatility -y-axis: Expected return -Upward sloping line -Starts at rf -Mark the Market portfolio on the line at some point (σₘ, rₘ) -Mark rₘ - 𝑟𝑓

Investment strategy past paper q?

An investor can reduce risk through investing in a diverse range of investments, where the returns of these investments are not highly correlated. The benefits of diversification only hold true if the securities within a portfolio are not perfectly correlated. An investor can reduce risk through investing in a diverse range of investments, where the returns of these investments are not highly correlated. The benefits of diversification only hold true if the securities within a portfolio are not perfectly correlated. The diversified portfolio will include a mix of asset types and investment vehicles in an attempt to limit exposure to any single asset or risk.The positive performance of some investments will neutralise the negative performance of others. For example, mutual funds are a portfolio of stocks of many different companies. Diversification can eliminate non-systematic/specific/idiosyncratic risk. However, diversification cannot eliminate systematic (market/covariant/undiversifiable risk). This is the risk that faces every firm in the economy and is due to market conditions (interest rates/inflation rate/business cycles etc). For example, a firm's earnings are positively correlated with business cycles, so if there is a recession all firms' earnings are negatively affected Providing the rate of return on these stocks are not strongly correlated, the portfolio will have a lower variance than any of the individual stocks. As the number of stocks in the portfolio increases, the portfolio's variance decreases, portraying the benefits in having a more diversified portfolio.

The following describes the mean returns and betas of stocks A, B and C: Stock A Mean Return (%): 4.6 Stock B Beta: 0.86 Stock B Mean Return (%): 10 Stock B Beta: 0.74 Stock C Mean Return (%): 11.2 Stock C Beta: 0.71 Determine the arbitrage portfolio with zero investment and a zero beta. Is there room for arbitrage profit?

Arbitrage Pricing Theory (APT) is a multi-factor asset pricing model. It is based on the concept that an asset's returns can be predicted using the linear relationship between the asset's expected return and a number of macroeconomic variables that capture systematic risk. Inherent to the arbitrage pricing theory is the belief that mispriced securities can represent short-term, risk-free profit opportunities. APT differs from the more conventional CAPM, which uses only a single factor. With APT, all securities (and all portfolios) sit on the Security Market Line (SML). An arbitrage portfolio can be created by selling (or buying) an undervalued asset and buying (or selling) a portfolio of other assets with the same beta and then closing out the position later to realise a profit. So, in APT, any asset or portfolio with the same beta should give the same expected return. Arbitrage opportunities may exist if this does not hold true. In this question, we can see that the mean returns and betas of stocks A, B and C do not fit the linear relationship as shown in the market equilibrium because as the expected returns increase, the beta decreases, as opposed to increasing as the SML would suggest. Therefore, implying the existence of an arbitrage portfolio. Graph -x-axis: Mean Return -y-axis: Beta -Plot points from the question - use dotted lines and label on the axis -The basic idea behind the APT model is that investors can create a zero-beta portfolio with zero net investment. -To create an arbitrage portfolio with zero investment and zero beta, two conditions need to be satisfied: 1) For zero investment: ∑𝑥ᵢ = 0, where 𝑥ᵢ is the weight of 𝑖𝑡ℎ security 2) For zero beta: ∑𝑥ᵢ𝛽ᵢ = 0, where 𝑥ᵢ is the weight of 𝑖𝑡ℎ security and 𝛽ᵢ is the 𝛽 of 𝑖𝑡ℎ security Therefore, to construct the portfolio, we have to solve 2 equations: 𝑥A + 𝑥𝐵 + 𝑥𝐶 = 0 0.86𝑥𝐴 + 0.74𝑥𝐵 + 0.71𝑥𝐶 = 0 As illustrated in the graph, it is evident that stock A is the most overvalued stock since its expected return is low, compared to its volatility. Therefore, we assume that this is a stock to sell (short). The weighting we give for this portfolio is 𝑥𝐴 = −1 -1 + 𝑥𝐵 + 𝑥𝐶 = 0 𝑥𝐵 + 𝑥𝐶 = 1 𝑥𝐵 = 1 − 𝑥𝐶 -0.86 + 0.74𝑥𝐵 + 0.71𝑥𝐶 = 0 0.74𝑥𝐵 + 0.71𝑥𝐶 = 0.86 0.74(1 − 𝑥𝐶) + 0.71𝑥𝐶 = 0.86 0.74 − 0.74𝑥𝐶 + 0.71𝑥𝐶 = 0.86 −0.03𝑥𝐶 = 0.12 𝑥𝐶 = −4 Sub 𝑥𝐶 = -4 into 𝑥𝐵 = 1 − 𝑥𝐶 to find: 𝑥𝐵 = 1 − (−4) 𝑥𝐵 = 5 The arbitrage portfolio comprises of 1) Buying stock B and 2) Selling stock A (20% of investment amount) and stock C (80% of investment amount) The criteria is satisfied, since: 1. The sum of the weightings is 1 − 0.2 − 0.8 = 0 (zero investment) 2. The beta of portfolio is (1 ∙ 0.74 ) + (−0.2 ∙ 0.86) + (−0.8 ∙ 0.71) = 0 (zero beta) Through the implementation of this strategy, the expected arbitrage profit can be calculated by ∑𝑥ᵢ𝑟ᵢ , whereby 𝑥ᵢ is the weight of security 𝑖 and 𝑟𝑖 is the expected return of the security ᵢ . If in equilibrium, it would give a value of zero. (1 ∙ 0.1) + (−0.2 ∙ 0.046) + (−0.8 ∙ 11.2%) = 0.0012 = 0.12% The above workings demonstrate that even when an arbitrage profit is available, the figure is very small and due to transactions costs not being accounted for in the calculations this potentially further undermines how if this is a significant this profit is .

2017/18

Through the implementation of this strategy, the expected arbitrage profit can be calculated by ∑𝑥ᵢ𝑟ᵢ , whereby 𝑥ᵢ is the weight of security 𝑖 and 𝑟𝑖 is the expected return of the security ᵢ . If in equilibrium, it would give a value of zero. (1 ∙ 0.1) + (−0.2 ∙ 0.046) + (−0.8 ∙ 11.2%) = -0.001492 -0.001492 X 100 = -14.92% The above workings demonstrate that there is no room

CAPM Equation

rᵢ = rf + 𝛽ᵢ (rₘ - rf) Expected returns on an asset = risk free rate of return return + the risk adjustment

Expected Arbitrage Profit

∑𝑥ᵢ𝑟ᵢ , whereby 𝑥ᵢ is the weight of security 𝑖 and 𝑟𝑖 is the expected return of the security ᵢ