SECURITY ANALYSIS AND PORTFOLIO MANAGEMENT 1. The investment environment 1.1. Definition and types of assets An investment can be defined as the current commitment of money or recourses in expectance of future benefits. In an economy we can distinguish two types of assets: • Real assets: They include land, building, machines and knowledge that are used to produce goods and services • Financial assets: They are claim or rights on the income generated by real assets (or government income). Rather than generating extra income, financial assets simply allocate or distribute wealth among investors. There are three types of financial assets o Fixed-income or debt securities: They promise either a fix stream of income or a stream of income determined by a formula. The performance of this types of assets is tied to the financial performance of the issuer § Money Market: It includes short-term, highly makeable and very low risk assets such as short-term Government debt, short-term deposits… § Capital markets: Long-term government and corporate bonds o Equity: It represents the ownership share in a company. Its value is tied to the success of the firm and its real assets. o Derivative securities: These types of financial assets provide payoffs related to other financial assets called underlying assets. They can be used to hedge risk or to indirectly invest in real assets.
1.2. Financial markets and the Economy Financial markets are the responsible for capital allocation and ultimately decide which companies are funded to survive or which one will not succeed. Financial markets are risky due to this uncertainty. Because of that, markets may fail in allocation capital to companies that might be successful but they do not understand their business model. On the other hand, individuals have some income. Financial markets allow individuals to differ or advance consumption by lending them money or borrowing it and give them extra income in the future. In addition, financial markets have played an important role in taking advantage of the economies of scale. Most companies nowadays require a large amount of capital and are owned by a large amount of shareholders and managed by a board of directors. Financial markets provide liquidity to the shareholders who do not longer want to hold this shares thus, increasing the willingness of the investors to provide capital because they know that it will be easy to liquidate. This system has its drawbacks because it creates agency problems.
1.3. The investment process An investment portfolio is simply a collection of different assets. Yet, configuring that portfolio involves different type of decisions: • Asset Allocation: The share of income that should be invested in each type of asset • Security selection: The decision that involves selecting which particular securities to hold within every category Investors may choose different strategies when constructing a portfolio. The top-down strategy starts with asset allocation. Hence, the most conscious decision is the share of money invested in each category. After that, there is a security analysis process that involves the valuation of the particular securities that should be included in the portfolio. On the other hand, the bottom-up strategy starts with security analysis without any particular prerequisite as for allocation. 1.4. Financial markets characteristics The first assumption is that markets are competitive. That is, they consist of a large number of very clever investors. Because of that, there exists the No-free-lunch rule (or no arbitrage rule): Financial markets are competitive enough to rule out any profit opportunity from investing in obviously mispriced securities. Another characteristic is that risk generates return. That is, investor requires more return the higher is the possibility to lose their invested capital. This is because if not, there will not be demand for highrisk assets (assuming investors are risk averse). As there is no way to measure implicit risk, investors focus on volatility, or the percentage change in the price of an asset. An additional highly accepted assumption is that markets are efficient. Efficiency means that any relevant information is immediately incorporated in the price of an asset. Yet, this may contradict the different types of investment-management strategies: • Passive strategies assume that the efficient market hypothesis is true and do not try to improve return using security analysis. • Active strategies: They assume that this hypothesis may not hold and spend recourses in security analysis to observe if any relevant information has not been incorporated and thus, there is a mispricing that could be used to improve return. 1.5. Financial crisis of 2008
2. How securities are trade 2.1. How Firm issue Securities When companies are in need of capital, they may choose to float securities. These newly created securities are market by investment banks to the investors in what is called the primary market. On the other hand, already issued securities are traded in the secondary market. • IPO vs Seasoned raise of capital: An initial Public Offer is done when a private company issues stock to common or institutional investment. On the other hand, a seasoned offer is the sale of new shares of already public companies. Both rise occur in the primary market. Investment banks play an important role when rising capital. When new issues or bonds are offered to the public using investment banks, they are called underwriters. They advise of the appropriateness of the time and price of the issuance. The information of the offer must be submitted to the regulatory body (SEC in the US) in what is called a prospectus. Normally, investment banks purchase the stocks from the company and resell them to the public. The difference is what they earn. Stocks can also be sold to instructional investments in what is called private placement. 2.1.1. IPO Procedure The procedure of an Initial Public Offering starts with the selection of a syndicate of investment banks. These organise road shows or a series of meetings with potential investors to evaluate demand and potential price. This information is booked in a process labelled book building. IPO price tend to be under-priced. This is because investors might not show their real interest during the book building process in order to drive the price down and the fact that investment banks offer discounts to enhance demand taking advantage of the discount they have to market the securities. China has been a key player in the IPO market in both companies going public and issuing stocks and IPO being Hong Kong the market leader for 3 consecutive years (2011-2013). 2.2. Types of market • Direct Search Markets: They are the least organised markets where sellers meet directly the buyers • Brokered markets: The trade of a particular good is done through a broker who stocks a number of assets reducing the search costs for both the buyer and the seller. An example of that is the primary market where the brokers (investment banks) store the newly issued share before selling them to the public • Dealer markets (Over-the counter): When the trade level increases, dealers appear. They specialize in various types of assets, purchasing them and then selling from the inventory at a profit. The bid is the price which the dealer is willing to but while the ask is the price they are willing to sell. They reduce the searching cost also because investors go directly to the to trade and also they guaranty liquidity. An example is the NASQAD • Action markets: This markets are places where all the traders meet at the same place to buy and sell assets. An example is NYSE. They require a large number of trades to cover for the costs. 2.3. Characteristics of a good market • Liquidity: It is ability to buy and sell large quantities quickly and without moving the prices substantially. • Low transaction costs • Transparency: To be able to see the actions and intents of other traders • Regulation: Lower incidence of corners, squeezes, and fraud.
2.4. Types of orders • Market orders: They are orders to be executed immediately at market prices. • Price-Contingency order: These are orders placed subject to the asset reaching certain price. o Limit order: It specify prices at which buy or sell orders are executed o Stop-loss order: Sell stocks when price falls below a stipulated level. This is to stop further losses from a long position o Stop-buy order: Buy stocks when price rises above a stipulated level: This is to limit potential losses from a short position 2.5. Margin orders Investors may be willing to get a loan from the broker to my securities. This system is called is named Margin trading. It is a rather risky investment because it can multiply gains but also loses. The margin is the proportion of money of the purchasing price that is contributed by the investor. 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑀𝑎𝑟𝑔𝑖𝑛 =
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐸𝑞𝑢𝑖𝑡𝑦 𝑖𝑛 𝑡ℎ𝑒 𝐴𝑐𝑐𝑜𝑢𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑆𝑡𝑜𝑐𝑘
As prices change the margin also changes 𝑀𝑎𝑟𝑔𝑖𝑛 =
!"#$#%& !"#$%& !" !!! !""#$%&!∆ !" !!! !"#$% !" !!! !"#$%! !"#$% !" !!! !"#
%$The maintenance margin is the minimum amount required by the broker to hold the position. If the margin of the account goes dangerously low we will receive a margin call to add equity or liquidate our position. The return will be the following: 𝑅𝑒𝑡𝑢𝑟𝑛 =
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑡𝑜𝑐𝑘 − 𝐿𝑜𝑎𝑛 − 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 + 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑡𝑠 − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐼𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐼𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
Example on Margin Trading. Suppose an investor buys 1000 shares of Company A at a price of £10. The initial margin required is 60%. Assets Value of the Stock=1000*£10
£10,000
£10,000
Liabilities Equity=0.6*£10,000 Loan
£6000 £4000
£10,000 6000 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑀𝑎𝑟𝑔𝑖𝑛 = = 60% 10000 Suppose the maintenance margin is 30%, how much would the price have to go down to trigger a margin call? 1000𝑃 − 4000 𝑀𝑎𝑟𝑔𝑖𝑛 = = 0.3; 𝑃 = 5.71 1000𝑃 How much it has been lost (assuming 10% interest)? 5710 − 4400 − 10000 𝑅𝑒𝑡𝑢𝑟𝑛 = = −86.9% 10000
2.6. Short Sales Short sales are an speculative investment that relies on the price of a share to go down. An investor borrows a share from a broker and sells it. After a period of time agreed, the short seller buys the share again and returns it to the broker. The difference is the profit or loss. Here is an explanation of the cash flows: Time Action Cash Flow 0 Borrow and sale share + Initial Price 1 Repay dividend, buy the share and return it -(Ending Price + Dividend) In order to short sale, brokers require a minimum amount of cash or equivalents, similarly to the margin trading. Example on Short selling. You instruct your broker to short 500 shares currently worth £15. The initial margin is 50% covered using TBills you own. Assets Liabilities Cash=500*£15 £7,500 Shares you borrow=500*£15 £7,500 T-bills £3,750 Equity £3,750 £11,250 £11,250 3750 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑀𝑎𝑟𝑔𝑖𝑛 = = 50% 7500
2.7. Regulations in the financial markets • Self-regulation: The financial industry has enforced self-regulation to reduce the chance of shocks. This includes CFA certification and non-governmental bodies such as Financial Industry Regulatory Authority. • The Sarbanes-Oxley Act was created in response to large financial scandals such as Enron. o Creation of a regulatory watchdog for Auditing companies o Require independent financial experts in the audit board comities o CEO and CFO must certify the financial statements at a personal expense o Auditing companies can only audit o The independent members of the board must meet regularly and privately. • Insider trading is forbidden.
3. Return and Risk relationship After a look of the financial data, seams obvious that there is a relationship between risk and return. We going to start discussing what determine the return of the “risk-free assets” and then we are going to move on to a more formal study between risk and return. Risk can be defined as the uncertainty of a predicted return to be realised. That is, when we invest we expect to win money but this might not be the case. 3.1. Real vs Nominal Interest Rates The first think we want to analyse is the difference between nominal and real returns (or interest rates). The nominal interest rate is the growth on money while the real interest rate is the increase on purchasing power. In other words, real interest rate measure what you can buy with the returns of the investment given the inflation of the period. Mathematically, 𝑅−𝑖 𝑟= 1+𝑖 where R is the Nominal interest rate, r is the real interest rate and I is inflation 3.2. What is return? Returns can be defined as all the gains perceived in an investment. In other words, it includes not only capital gains (the change in the price) but also dividends. Mathematically, 𝐸𝑛𝑑𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒 − 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑝𝑟𝑖𝑐𝑒 + 𝐶𝑎𝑠ℎ 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝐻𝑜𝑙𝑑𝑖𝑛𝑔 − 𝑃𝑒𝑟𝑖𝑜𝑑 𝑅𝑒𝑡𝑢𝑟𝑛𝑠 𝐻𝑃𝑅 = 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒
3.3. Rates of Return for Different Holding Periods Now that we know formally what a return is, we may be interested in comparing different investments. • Zero Coupon Bonds (Defined later on): 𝑃𝐴𝑅 𝑟! 𝑇 = −1 𝑃 𝑇 where rf(T) is the total return, PAR is the face value, P(T) is the current price A problem we may encounter is that the time horizon is not the same. If we use empirical data, we see that the longer we hold a zero coupon bond, the higher are the returns and thus, we need a measure to compare. • Annual Percentage Rate (APR): It is just adjusting the interest to fit a year period. For example, if they offer 6% seminally, we just multiply by 2. • Effective Annual Rate (EAR): It is a measure that shows the return adjusted in annual periods assuming we reinvest the proceeds at the same interest. (Compound Interest) 𝑠𝑡𝑎𝑡𝑒𝑑 𝑎𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 (𝐴𝑃𝑅) !"#$%& !" !"#$%&'(&) !"#$%&' 𝐸𝐴𝑅 = (1 + ) −1 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑢𝑛𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 3.4. Returns more closely Returns largly depened on the revelation of the state of the nature. That is, events that are beyond the control of the investors. Knowing that, we can express the return as: 𝐸 𝑟 =
𝑝! 𝑟!
where E® is the expected return, pi is the probability of the state i to occur and ri is the return if the state i occurs. Yet, in the real world is impossible to predict all the state that can affect the price of a financial asset and associate them a probability. Because of that, we use historical data to express the return: • Arithmetic mean: It is used when we assume that there is no relation between observations.
r= •
1 n ∑ r ( s) n s =1
Geometric mean: Most of the cases, it is fair to assume that returns may be related. Then, we use geometric men.
r = {[(1 + r1 ) (1 + r2 ) .... (1 + rn )]} 1/n - 1 3.5. Historic Perspective & Risk Historically, stocks have doubled in performance bonds as we can see in the table below:
Then, a question may arise. Why do people invest in bonds? An answer can be risk. A common agreement says that the risk can be defined as the movement in price of a given asset. Mathematically, we use variance to calculate the risk: 𝜎! =
𝑝! ∗ (𝑟! − 𝜇)!
𝜎 = 𝜎! If we do not have the whole set of data, we can use sample variance to approximate the real ones.
σˆ 2 =
1 n ⎡ ⎤ ∑ ⎢r (s) − r ⎥⎦⎥ n − 1 s =1 ⎣⎢
σˆ = σˆ
2
2
If we take a historical perspective of risk, we can state that bonds are less risky than stocks: Period: 1926 - 2001 Average Real Returns Standard Deviation
T-bills 0.70% 4.10%
T-bonds 2.10% 10.70%
Stocks 6.90% 20.30%
Hence, we may assume those investors are risk averse. That is, in order to assume more risk, they require more return. This is called risk premium (difference between expected return and the risk free return). You may argue that volatility is not enough to measure risk. For example, mean and variance assume a normal distribution. This assumption might not be fulfilled. In order to check for normality, we use skewness. If a distribution is left-skewed, large negative returns are more likely and bad surprises are more likely to be extreme. On the other hand, right-skewed distribution large positive returns possible and bad surprises are less likely. Kurtosis is another measure investors may be interested in. It measures the umber of observations centred around the mean.
4. Risk Aversion and Capital Allocation to Risky Assets So far, it seams obvious looking into the empirical data that the economy in general is risk adverse, that is, they require a premium to assume risk. Yet, this may not be the case. In general there are three different attitudes towards risk: • Risk averse: The investor dislikes risk and requires an extra premium to assume risk • Risk neutral: The amount of risk is indifferent to the investor • Risk lover (or seeker): The investor is willing to give up some return to assume risk. In order to research on risk attitudes, economists have created the concept of utility function. Broadly, a utility function describes how attractive is an investment to particular investor given a level of risk. The most common form of utility function is: 𝑈 = 𝐸 𝑟 − 0.5𝐴𝜎 ! where U is the utility, E(r) is the expected return, A is the risk aversion corficient and 𝜎 ! is the variance. A positive coefficient of risk aversion A implies a risk adverse investor. This is the base case in Modern portfolio theory. The dominance principle states that an investor will always minimize risk given a return or maximize return given a risk. An indifference curve can be defined as the set of all possible combinations of risk and return given a certain level of utility (i.e fixing A and U in the utility function and solving for E(r)). 4.1. Capital Allocation Now we know more about how our personal appreciation of risk affects our investment decisions. Yet, there are a lot of assets in the financial markets and we might want to invest in more than one. A portfolio is defined as a set of different financial assets. The characteristics of a portfolio will be given by the assets included and the weight or share of budget invested in each one of them. If the portfolio includes the risk free asset, we can name it complete portfolio. The expected return of a portfolio can be calculated using the following formula: 𝑟! =
𝑤! ∗ 𝑟!
where rp is the portfolio return, wi is the weight invested in the stock i and ri is the return of the stock i Similarly we can compute the risk of a portfolio: 2 2 2 2 σ p 2 = w1 σ 1 + w2 σ 2 + 2 w1w2 cov(r1 , r2 )
4.1.1. Capital Allocation Line One of the simplest cases of portfolio is the combination of a risky asset with the risk-free asset. We have to remain that we are assuming we are risk averse. Hence, the goal will always we maximise utility given risk. This will be given by the combination of two equations: 𝐸 𝑟! = 𝑤 𝑟! + (1 − 𝑤)𝑟! 𝜎 ! = 𝑤𝜎!!
whre e(rp) is the reruen of the portfolio, w is the weight in the risky asset, ri is the return of the risky asset, rf is the return of the risk free asset and𝜎!! is the variance of the risky asset
The first equation is the expected return given the weights in each type of assets. The second one is the risk of the portfolio. Note that it only includes the risky assets because by definition the variance of the risk-free asset is 0. Re arranging terms we can state the Capital Allocation Line (CAL): 𝐸 𝑟 = 𝑟! +
𝑟! − 𝑟! 𝜎 𝜎!.
Note that the slope is the extra reward for each extra unit of risk. This ratio is called the Sharp ratio. In fact, the formula above assumes that the risk-free return is constant.
E (r − rf ) var(r − rf ) Can we get more return than the one given by the risky asset? The answer is yes. We can borrow money at the risk free rate (or short the risk-free asset) in order to invest more money in the risky asset.
Now, as economists try always to maximise utility, we can be interested in asses which is the best portfolio in the Capital Allocation Line. It is a mathematical optimisation given the utility function as a function of the risk. For the most common utility function decribed above: 𝑀𝐴𝑋 𝑈 = 𝐸 𝑟! − 0.5𝐴𝜎!! 𝑟 𝑀𝐴𝑥 𝑈 = 𝑟𝑓 + 𝑦 𝐸 𝑟𝑃 – 𝑟𝑓 – 0.5 𝐴 𝑦2 𝜎!! 𝑦 ∗ = ( 𝐸(𝑟𝑝) – 𝑟𝑓 ) / (𝐴 𝜎!! ) An imlicit assumption above is that we can borrow at the risk free rate which is not conisntant with the relaity. If that is not the case, we need to adjust the CAL aabove the risky asset for the extra interest we have to pay. Historically, a broad category of risky assets accounted for approx. 74% of the total wealth in the U.S. economy. Implying a risk aversion coefficient of approx. 2.6 Risky assets do not just mean stocks. It includes real estate, private businesses, commodities, etc. The Capital Allocation Line is top-down strategy, implying a more diversifies portfolio with lower transaction costs yet not taking into account the quality of the securities included.
5. Optimal Risky Portfolios Now, we are going to analyse the risky assets. We are going to start going deeper into the conept of risk. We can have two types of risk: • General economic uncertainty: It includes factors such as business cycle, inflation rate, interest rate, exchange rate, etc. In other words, it affect all firms in the economy and thus, cannot be diversified away. (Alternative names: Non-diversifiable risk, systematic risk, or market risk) • Firm-specific uncertainty. It includes risk inherent to the particular firm like Sales growth, R&D, relative performance, etc. It can be eliminated by diversification in a portfolio (Alternative names: Diversifiable risk, unsystematic risk, or idiosyncratic risk). When we invest in a portfolio, we are diversifying risk. In other words, we are choosing to be exposed to different type of risks. If we look to empirical data, we can observe that the larger number of securities a portfolio has, the lower the overall risk. This reduction is not infinite. It approaches the market risk. 5.1. The simplest case: Two-Risky-Asset When we construct a portfolio with two different risky assets, it has the following characteristics:
E ( rp ) = wD E ( rD ) + wE E ( rE ) 2 2 2 2 σ p 2 = wD σ D + wE σ E + 2 wD wE cov(rD , rE )
Let’s look closer to the concept of Covariance: D and E are the two risky assets (denoting two portfolios of bonds and stocks). Recall also that:
cov(rD , rE ) = ρDEσ Dσ E The correlation coefficient measure how related are the two risky assets. Note that, if the variables are not correlated (I.e. the correlation coefficient is 0) a reduction in risk. Similarly, if it is negative the risk is reduced. Even if it is positive, as long as it is less 1 there is also a reduction in risk. Hence, the first type of portfolio we are interest in is the minimum variance portfolio. When there is only two risky assets, we can use the following formula:
σ 2 2 − σ 1σ 2 ρ1, 2 w1 = 2 σ 1 + σ 2 2 − 2σ 1σ 2 ρ1, 2 w2 = 1 − w1 This portfolio is the one that would minimize risk the most and thus, it will also be the efficient portfolio with less return. When there are only two risky assets, the optimal risky portfolio is determined by the minimum variance frontier portfolio opportunity set and the indifference curve. That is, there exists a portfolio that maximizes the risk reward for risk (sharp ratio) and we call it optimal risky portfolio. Graphically, this is the point on the combination line where the slope of the indifference curve is equal to the slope of the portfolio opportunity set.
5.2. Can we be even more efficient? The introduction of the risk-free asset When we introduce the risk free asset, we are able to draw a Capital Allocation Line in order to decide how much we want to invest in a risky portfolio and in the risk-free. The question is: which risky portfolio should we chose? Well, know that we are able to allocate our capital, it seams obvious that we should chose the portfolio that rewards the risk the most. That is, the Optimal Risky Portfolio. When there is only 2 assets we can use the following formula: 2
w1 =
[ E ( r1 ) − rf ]σ 2 − [ E ( r2 ) − rf ]σ 1σ 2 ρ1, 2 2
2
[ E ( r1 ) − rf ]σ 2 + [ E ( r2 ) − rf ]σ 1 − [ E ( r1 ) − rf + E ( r2 ) − rf ]σ 1σ 2 ρ1, 2 2
=
E ( R1 )σ 2 − E ( R2 )σ 1σ 2 ρ1, 2 2
2
E ( R1 )σ 2 + E ( R2 )σ 1 − [ E ( R1 ) + E ( R2 )]σ 1σ 2 ρ1, 2
R stands for excess return. w2 = 1 − w1
Know, we are able to assume the risk we want along the Capital Allocation Line combining the risk free asset and the optimal portfolio to achieve the wished risk. Now, we are able to use the utility function to choose the Portfolio that suits us the most. That is, the optimal complete portfolio. How to solve a typical problem: 1. First find the weights in the tangency portfolio 2. Then, find the expected return, variance and sharp ratio 3. After, solve the utility function to see how much an investor will invest in the risky portfolio. 4. Know, we can build the characteristics of the optimal complete poerfolio.
5.3. Multiple securities case The reasoning is the same but the statistics change a little bit. •
3 Case scenario 3
E (rp ) = ∑ wi E (ri ) =w1 E (r1 ) + w2 E (r2 ) + w3 E (r3 ) i =1
3
3
σ p2 = ∑∑ wi w j cov(ri , rj ) i =1 j =1
3
= ∑ [wi w1 cov(ri , r1 ) + wi w2 cov(ri , r2 ) + wi w3 cov(ri , r3 )] i =1
= w1w1 cov(r1 , r1 ) + w1w2 cov(r1 , r2 ) + w1w3 cov(r1 , r3 ) + w2 w1 cov(r2 , r1 ) + w2 w2 cov(r2 , r2 ) + w2 w3 cov(r2 , r3 ) + w3 w1 cov(r3 , r1 ) + w3 w2 cov(r3 , r2 ) + w3 w3 cov(r3 , r3 )
The names also change a little bit when all securities are included: • The optimal combinations result in lowest level of risk for a given return à minimum-variance frontier • The optimal trade-off is described as the efficient frontier. • These portfolios are dominant.
5.4. Separation Theorm A most interesting result is known as the Separation Property. Suppose an investment advisor has multiple clients with different levels of risk aversions. In absence of the risk-free security, the advisor will recommend different combinations of risky securities to different clients. But in presence of the risk-free security, he will recommend the same combination of risky securities to all clients. The CAL(P) thus becomes the separating line between the return dynamics and the risk aversion.
6. The Capital Asset Model (CAPM) So far, we have reached the following conclusions: • Investors choose a mean variance efficient portfolio, a portfolio that for a given level of standard deviation has the highest expected return relative to all possible portfolios. • With a risk-free asset, investors allocate money between the optimal risky portfolio and the risk-free asset. Yet, we are interested in understand how the Capital market works. In other words, so far we have been examining individual investors. Now, we are going to move to Capital Markets in general. The Capital Asset Pricing Model was build as an extension of the Markovicht Portfolio Theory. It is an equilibrium model, that is, it assumes certain behaviour of the participants and then draws conclusions about the market. 6.1. The Capital Asset Pricing Model Assumptions 1) Investors are price takers.: Personal wealth is small compared to the total wealth of all investors. Investors act as though security prices are unaffected by their own trades 2) Investors have an identical one period investment horizon. That is, Investors ignore everything that might happen after the end of the single period 3) Investors are limited to publicly traded financial assets. They have unlimited borrowing and lending opportunities at the risk free rate. 4) No taxes and transaction costs. 5) Investors are rational mean-variance optimizers. That is, investors all use the Markowitz portfolio selection model. 6) Information is costless and available to all investors. All investors analyse securities in the same way. Investors have homogeneous expectations. That is, given a set of security prices and risk free , all investors use the same expected returns and covariance matrix of security returns to generate the efficient frontier and the unique optimal risky portfolio. 6.2. The CAPM results The first result is that all investors will hold the same risky portfolio. As shown in the modern portfolio theory and based on the CAPM assumptions, there is only one optimal risky portfolio. Similarly, there will be a single Capital Allocation Line shared by all investors. This is called Capital Market Line. Another interesting result of the CAPM is that the market risk premium will be a result of the average investors’ risk aversion. Hence, the risk premium of a particular security will be given by its correlation with the market.
6.3. The Market Portfolio If all investors use identical Markowitz analysis, applied to the same universe of securities) for the same time horizon and use the same input list, they all must arrive at the same composition of the optimal risky portfolio. When we aggregate, the value of the optimal risky portfolio will equal the entire wealth of the economy. The Market Portfolio is the portfolio of all risky assets, where the assets are held in proportion to their market value. If an asset is not included, demand drops so does the price and there is an incentive to be included. The weight on a stock S is the fraction of that stock’s market value relative to the total market value of all stocks. • The total market value of all stocks: V = V1+V2+...+Vn • The weight of stock S in the market portfolio: wS,M = VS /V Example There are only three risky assets, A, B and C. Suppose that the tangent portfolio is
There are only three investors in the economy, 1, 2 and 3, with total wealth of 500, 1000, 1500 billion dollars, respectively. Their asset holdings (in billion dollars) are:
In equilibrium, the total dollar holding of each asset must equal its market value
The market portfolio is the tangent portfolio:
6.4. Market Risk Premium Let’s start with some mathematical notation: • M= Market portfolio • r = Risk-free rate • E(r ) – r =Market risk premium • [E(r ) – r ]/σ = Market price of risk = Slope of the CML We must remember that we assume that the risk aversion of the investors is given by the following equation: f
M
f
M
f
M
y=
E (rM ) − rf Aσ M2
That is, Y is the proportion that the investor will invest in the market portfolio. Since the risk-free lending and borrowing is among the investors (i.e. already issued government debt), the net borrowing (i.e. borrowing – lending) , must be zero. Then,
y=
E (rM ) − rf A σ M2
= 1 ⇒ E (rM ) − rf = A σ M2
6.5. Individual contribution of the securities Now that we know that the market portfolio includes all the assets in the economy, we are able to answer the following question: what is the fair price of an asset? Let’s start with the risk of the market portfolio: n
n
σ M2 = ∑∑ w j wk cov(rj , rk ) j =1 k =1
= w1w1 cov(r1 , r1 ) + ... + w1wn cov(r1 , rn ) ! + wS w1 cov(rS , r1 ) + ... + wS wn cov(rS , rn ) ! + wn w1 cov(rn , r1 ) + ... + wn wn cov(rn , rn )
As said before, the each asset in contributing to the market will depend on the correlation amount them and the weight in the portfolio. That is: n
n
k =1
k =1
= wS ∑ wk cov(rS , rk ) = wS cov(rS , ∑ wk rk ) = wS cov(rS , rM )
Thus, we can know analyse the reward-to-risk ratio of the individual asset as:
=
wS [ E (rS ) − rf ] wS cov(rS , rM )
=
E (rS ) − rf
Contribution to market premium
cov(rS , rM )
If we rearrange terms: E ( rS ) − rf =
cov(rS , rM )
σ M2
Contribution to variance
[E (r
M
) − rf ] = β S [E ( rM ) − rf ]
Due to the diversification effect, investors will be compensated only for bearing systematic risk.The CAPM is often stated as a expected return-beta relationship:
[
E (rS ) = rf + β S E (rM ) − rf
]
6.6. The concept of alpha If we compare the results of the CAPM with the real world data we may have discrepancies. Because of that, the concept of alpha was developed. The vertical distance between the security and the SML is known as the alpha of that security. It is a measure of the abnormal return. Security analysis is the process of identifying and buying securities with positive alpha. It can be extended to include shorting securities with negative alpha.
7. The Asset Pricing Theory The Asset Pricing Theory was constructed in order to explain why alphas exist. In other words, empirical data show discrepancies between the CAPM and real pricing. The APT approaches the asset pricing by exploring more than one systematic factor that may affect differently the assets in the market. One of the advantages of the Asset Pricing Theory is that is a statistical model rather than a equilibrium model and thus, it requires fewer assumptions. The ATP is based on the no arbitrage rule or the law of one price. That is, if a factor affects equally two assets, the affect on price must be the same. 7.1. Asset Pricing Theory Assumptions • Security returns can be described by a linear (multi-) factor model. There are a few macroeconomic factors influencing returns. • There are sufficient securities so that firm specific (idiosyncratic) risk can be diversified away. • Well-functioning security markets do not allow for persistent arbitrage opportunities. That is, there is no opportunity of risk-free profits made by investors by exploiting security mispricing, without a net investment o Under the CAPM, when a security is mispriced, almost all investors will make limited portfolio changes à restore equilibrium o Under the APT, any investor who identifies an arbitrage opportunity will want an infinite position in the risk-free arbitrage portfolio à restore equilibrium. 7.2. Single Factor Model The mathematical expression for a single factor APT model would be: 𝐸 𝑟! = 𝑟! + 𝛽! (𝑟! − 𝑟! ) where E(ri) is the expected return on the security I, rf is the risk-free rate, bi is the sensivity of the stock I over the factor F and rF is the return of the factor F Whereas the real return is described by: E (r ) = r i
f
+ β i RP( Factor )
ri = E (ri ) + β i F + ei Where F is the revelation of the state of the nature of factor F. Since we assume that is something abnormal, F has always-mean 0. F is affecting everyone in the economy with a different degree. Ei is the firm specific state of the natures
7.3. Well diversified portfolio The return of a well-diversified portfolio will be given by: The firm specific component A can be ignored since we are assuming a well-diversified portfolio. The error in prediction becomes lower as the number of securities increases. The well-diversified portfolio’s return is determined completely by the systematic factor. The undiversified stock is subject to non-systematic risk. 7.4. CAPM vs APT • •
•
APT APT equilibrium means no arbitrage opportunities. APT equilibrium is quickly restored even if only a few investors recognize an arbitrage opportunity. The expected return–beta relationship can be derived without using the true market portfolio.
• •
•
CAPM Model is based on an inherently unobservable “market” portfolio. Rests on mean-variance efficiency. The actions of many small investors restore CAPM equilibrium. CAPM describes equilibrium for all assets.
7.5. Fama-French Three Factor Model
ri − rf = α i + bi (rM − rf ) + si SMB + hi HML + ei •
SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks.
•
HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book-to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio.
8. Bond prices and Yield 8.1. What is a Bond? A bond is a periodic stream of interest payments for a certain number of periods combined with a principal payment at the end of last period. • Bond indenture: The indenture specifies all the important features of a bond, such as its maturity date, timing of interest payments, method of interest calculation, callable/convertible features if applicable and so on. The indenture also contains all the terms and conditions applicable to the bond issue. • Face value or par value: The nominal value or dollar value of a security stated by the issuer. For bonds, it is the amount paid to the holder at maturity (generally $1,000). • Coupon rates or coupon payments: The yield paid by a fixed income security. A fixed income security's coupon rate is simply just the annual coupon payments paid by the issuer relative to the bond's face or par value. Normally, payments are made semi-annually. • Zero-coupon bond: A debt security that doesn't pay interest (a coupon) but is traded at a deep discount, rendering profit at maturity when the bond is redeemed for its full face value. 8.2. Bond Market Bond market includes not only Government but also, Financial organizations, Corporations and International organizations and has a size of approximately 100 Trillion US dollars.
8.3. Interpreting treasury issues
Current date
Prices as of last closing
Oct 30, 2009.
Matures
When the Bond has to be repaid Interest Rate (Annually). Payments are made semi annually unless stated Price at which you can sell the bond from a dealer. Decimals are over 32 as a % of the PAR Price at which you can buy the bond from a dealer. Decimals are over 32 as a % of the PAR Dealer’s commission. Decimals are over 32 as a % of the PAR Change since previous closing. Decimals are over 32 as a % of the PAR It is the return of the bond if it bought at the asked price. Semi-annually compounded
Oct 2014
Coupon
Bid price
Ask price
Bid-ask spread Change since previous closing
Ask yield
2.375%
100:08 = 1008/32= par 100:09 = 1009/32 of par
100.25% of
= 100.281%
1
/32 = 0.03125%
18
/32 = 0.5625
2.32%,
We must remember that there is a difference between the ask price and the actual price paid. That is because the selling investor will require us to pay for the interest accrued sine the last interest payment. After that, we will recive the full payment of interest. Suppose coupon = 8%; paid semiannually 40 days have passed since last interest payment Quoted price = $990 Accrued interest = $40´(40/182) = $8.79 Actual price paid = $990+8.79 = $998.79 The bond purchase entitles the investor to receive the full 6-month coupon after holding it for just 142 days
8.4. Type of Bonds • Callable bonds: Issuer can repurchase the bond at a specified call price before the maturity date • Puttable bonds: Bondholders have the option to retire or extend the bond • Convertible bonds: Bondholders can exchange each bond for a specified number of shares of common stock. Convertible bonds offer lower coupon rates or promised yields to maturity than do nonconvertible bonds. • Floating/adjustable rate bonds: They have an adjustable coupon rate tight to some index such as Libor or Euribor. 8.5. Bond Pricing. A bond has two components that affect its price. The first one is the right over the payments. That is, as bonds pay interest over time, it is a component to take into account. The second one is the right of the repayment at the maturity. T
PB = ∑ t =1
Ct Par Value + t (1 + r ) (1 + r )T
1 ⎡ 1 ⎤ Par Value = C × ⎢1 − + when C is constant over time. r ⎣ (1 + r )T ⎥⎦ (1 + r )T Where PB is Price of the bond, Ct is interest or coupon payments, T is the number of periods to maturity and r is interest rate over one payment period Example: 8% Coupon bond, paid semiannually, 30 years to maturity Interest rate (or yield) is 8% (or 4% per 6 month) 60 40 1000 P =∑ + t 1 . 04 1 .04 60 t =1 1 ⎡ 1 ⎤ 1000 = 40 × 1− + 0.04 ⎢⎣ 1.04 60 ⎥⎦ 1.04 60 = $1000 When interest rate is 10%, what’s the price? 60
40 1000 + = $810.71 t 1 . 05 1 .0560 t =1
P =∑
8.6. Relationship between interest rates and Bond pricing Bonds are assets tightly related with interest rate. When interest rate increases the value of the Bond decreases because the investor requires more return to hold the bond. Thus: • If r = coupon rate, then par value = bond price • If r > coupon rate, then bond price < par valueà discount bond • If r < coupon rate, then par value < bond price à premium bond Hence, is obvious that price of bonds follow a convexity function. The longer the bond maturity, the more sensitive to interest changes.
Calculator: 40 PMT 60 N 4 I/Y 1000 FV CPT PV
8.7. The Yield-to-Maturity Concept Similarly to the concept of APR, we may be interested in a measure that allows us to compare between bonds. The Yield-to-Maturity will indicates us the interest rate that that makes the present value of the bond’s payments equal to its price. Mathematically, we should solve the bond formula for r: T
PB = ∑ t =1
Ct Par Value + (1 + r )t (1 + r )T
Example: Given: 8% coupon bond, payable semiannually 30 year maturity Current price P0 = $1,276.76 YTM = r = ? 60
1,276.76 = ∑ t =1
40 1,000 + (1 + r ) t (1 + r ) 60
Calculator: 40 PMT 60 N -1,276.76 PV 1000 FV CPT I/Y
A hit and trial approach shows YTM = r = 3% per half year. Annual YTM = 3% x 2 = 6%
8.8. Callable bonds As explained before, callable bonds can be repaid before the maturity. Thus, the price will behave differently. Hence, additionally to the Year-To-Maturity, we would calculate the yield-to-call (sing only the period until the call period.
8.9. Zero-Coupon bond Zero-Coupon bonds provide only one cash flow to their owners, on the maturity date of the bonds. Long-term zero-coupon bonds are usually created from the existing coupon bonds. Fore most example are the Treasury strips (Separate Trading of Registered Interest and Principal Securities ).T-bills are examples of short-term zero-coupon instruments. Consider a zero coupon bond with a par value of $1,000. Suppose the YTM remains a constant 10% annually compounded.
8.10. Promised vs. Actual Yield to Maturity and junk bonds It is important to keep in mind that Yield –to maturity assumes full repayment of the principal. Thus, it should be adjusted (Changing the PAR) if we expect not to be fully repaid. In order to avoid not being repaid, there are number of mechanisms. • Sinking funds o Spread the payment burden over several years o Repurchase a fraction of the outstanding bonds in the open market each year • Subordination of further debt o Restrict the amount of additional borrowing o Junior bondholders will not be paid unless the prior senior debt is fully paid off. • Dividend restrictions o Limit the dividends firms may pay • Collateral o Particular asset bondholders receive if the firm defaults.
9. Managing Bond Portfolio 9.1. Risk involved in Bond portfolios As interest rates rise and fall, bondholders experience capital looses and gains. Therefore, the sensitivity of bond prices to changes in market interest rates is of great concern to investors o A:B - different maturity o B:C - different coupon rate o C:D - different YTM at which the bond currently is selling
9.2. Bond pricing relationship o Bond prices and yields are inversely related. An increase in yield results in a smaller price decline than the gain associated with an equal magnitude decrease in yield (convexity). o Long-term bonds are more price sensitive to interest rate changes than short-term bonds. o As maturity increases, price sensitivity increases, but at a decreasing rate. o Prices of low coupon bonds are more sensitive than prices of high coupon bonds. o The price sensitivity is higher when the current YTM is lower. 9.3. The coupon effect As mentioned before, there are two factors affecting the price: PAR and Coupon payment. A higher coupon payment implies that we will recover the capital quicker the capital thus, shortening the effective maturity. 9.4. The Yield-To-Maturity effect When the Yield-to-Maturity increases, it implies that the present value of the cash flow is reduced. The effect is grater on distant payments as there is more periods to discount. Thus, the price relies more on earlier payments, which reduces the interest change risk. 9.5. Duration of a bond The term duration has a special meaning in the context of bonds. It is a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows (Not considering principal repayment). It is an important measure for investors to consider, as bonds with higher durations carry more risk and have higher price volatility than bonds with lower durations.
wt =
CFt /(1 + y )t T
∑1 CFt /(1 + y)t
=
CFt /(1 + y )t ; bond price
T
∑w 1
t
=1
y is the bonds YTM T
D = ∑1 t × wt 9.5.1. Duration and price relationship Let’s analyse the interest sensitivity of the price of a bond: T
P=∑ t =1
CFt (1 + y )t
CF dP − 1 T ⎛ = ∑ ⎜ t ⋅ t dy 1 + y t =1 ⎜⎝ (1 + y )t
⎞ − 1 T ⎛ ⎞ CFt ⎟⎟ = ⎜⎜ t ⋅ P ⎟⎟ ⋅ P ∑ t ( 1 + y ) 1 + y t =1 ⎝ ⎠ ⎠
dP −1 = ⋅ D⋅P dy 1 + y
dP dy = −D ⋅ P 1+ y ΔP Δy = −D ⋅ P 1+ y If we want a more direct measure of the relationship between changes in bond prices and interest rates, we can use Modified Duration, defined:
D* =
D (1 + y )
ΔP = − D *Δy P
o o o o o
9.5.2. Duration Rules Duration is shorter than maturity for all bonds except zero coupon bonds. Duration of a zerocoupon bond equals maturity. Holding time to maturity and YTM constant, duration is higher when coupons are lower. Holding coupon and YTM constant, duration generally (but not always) increases with time to maturity. Holding coupon and time to maturity constant, duration is higher when YTM is lower. Duration of a perpetuity is (1+y)/y.
Duration is an important concept in bond management. First, it Measures sensitivity of bond price to interest rate changes (price volatility). It’s important to remember that, since the price of a bond follows a convexity formula and duration is a straight relationship, duration is a local concept. 9.6. Convexity The convexity of the price formula of a bond can be approximated by the second derivety with respect to yield divided by the bond price. That is: ∂2P 1 = ∂ 2 y (1 + y ) 2
T
⎡ CFt
⎤ ⋅ (t 2 + t )⎥ t =1 ⎣ ⎦ T ⎡ CFt ⎤ 1 ∂2P 1 Convexity = = ⋅ (t 2 + t )⎥ 2 2 ∑ ⎢ t P ∂ y P(1 + y ) t =1 ⎣ (1 + y ) ⎦
∑ ⎢ (1 + y)
t
Thus, the real change in price will be:
ΔP ⎛ 1 ⎞ = − D* × Δy + ⎜ × Convexity × (Δy ) 2 ⎟ P ⎝ 2 ⎠
(
)
Bonds with greater curvature gain more in price when yields fall than they lose when yields rise. The more volatile interest rates, the more attractive this asymmetry. Bonds with greater convexity tend to have higher prices and/or lower yields, all else equal. 9.7. Passive vs Active management There are two passive bond portfolio strategies. Both strategies see market prices as being correct, but the strategies have very different risks. o Indexing: have the same risk-reward profile as the bond market index to which it is tied o Immunization: seek to establish a virtually zero-risk profile Active management o Interest rating forecasting o Identification of relative mispricing within the fixed-income market
9.8. Immunization Immunization is a strategy that matches the durations of assets and liabilities thereby minimizing the impact of interest rates on the net worth. Example An insurance company must make a payment of $19,487 in 7 years. The market interest rate is 10%, so the present value of the obligation is $10,000. The company’s portfolio manager wishes to fund the obligation using 3-year zero-coupon bonds and perpetuities paying annual coupons. How can the manager immunize the obligation? 1) Calculate the duration of the liability o one single payment: duration = 7 years 2)
Calculate the duration of the asset portfolio o duration of the zero-coupon bond = 3 years o duration of the perpetuity is 1.1/0.1 = 11 years o assume the fraction in the zero is w, then the portfolio duration = w * 3 + (1 - w) * 11 3) Find the asset mix that sets the duration of assets equal to the 7-year duration of liabilities o w * 3 + (1 - w) * 11 = 7, implying w = 0.5 4) Fully fund the obligation. o The manager must purchase $5,000 of the zero and $5,000 of the perpetuity. o Face value of zero = $5,000 * (1.1)3 = $6,655 Suppose that 1 year has passed, and the interest rate remains at 10%. The portfolio manager needs to reexamine her position. Is the position still fully funded (i.e., value of the asset = value of the obligation) ? Is it still immunized? • We first need to calculate the PV of the asset and the obligation. PV of the obligation: FV = $19,487; I/Y = 10; PMT = 0; N = 6; CPT PV = $11,000 PV of the asset: Zero: PV = 6,655/ (1.1) = 5,500 Perpetuity : Get paid $500, and remain worth 5,000 Value of the asset = Value of the obligation à fully funded 2
We next need to find the asset mix that sets the duration of assets equal to the duration of liabilities Immunization: w * 2 + (1 - w) * 11 = 6, w = 5/9 The manager now must invest a total of $11,000*(5/9)= $6,111.11 in the zero.
10. The Efficient Market Hypothesis Economists for a long time have tried to use computers to discover patterns in data. A first attempt in 1953 by a British Economist, Maurice Kendall, found that there were no patterns in data. The stock price series behave as if successive prices differ by a random number, having no correlation with past prices. Enter the random walk hypothesis. Economists realized that randomness of returns was a result of rationality and not irrationality. It is best summarized as follows: • Current stock prices reflect the currently known information. • Stock prices change when new information arrives. • New information, or news, by definition, cannot be predicted. Therefore, changes in stock price cannot be predicted. 10.1. What is market efficiency? Market efficiency is one of the most commonly used terms. Yet attempts to define it precisely have failed. Fortunately, the implications of market efficiency in a given context are usually easy to understand. • Information: The most precious commodity o Strong competition (i.e. analysts) assures prices reflect information. o Information is costly to uncover and analyze. Information-gathering is motivated by desire for higher investment returns. 10.2. Types of analysis A. Technical Analysis: The adjustment period is long enough that the analyst will be able to identify a trend that can be exploited. Sluggish response of stock prices to fundamental supply-and-demand factors. a. Support level and resistance level: Price below which it is unlikely for them to fall, or levels above which it is difficult for stock prices to rise. b. Use prices and volume information to predict future prices c. Mainly for market timing purpose B. Fundamental Analysis: Fundamentalists identify undervalued and overvalued securities a. Use economic and accounting information to predict stock prices b. Mainly for stock selection purpose 10.3. Levels of market efficiency 1. Weak-form market efficiency: Prices today reflect all information that went into the past prices: All the information contained in past prices of the stock is impounded into the current price. Therefore you cannot predict future prices by observing past prices/returns. Thus, Technical analysis (i.e., trading rules based on historical prices/returns) does not work and prices follow random walk 2. Semi-strong-form market efficiency: The current price reflects all public information. All the public information is impounded into prices immediately once released. Only new information moves the prices. The speed prices adjust to new information is a measure for market efficiency: the faster the adjustment, the more efficient the market is. 3. Strong-form market efficiency: The stock prices fully reflect all information. All information (public and private) is impounded into prices. No room for arbitrage or insider trading.
10.4. Behavioural Finance Behavioural finance deals with the influence of psychology on various aspects of financial markets, including the behaviour of individual and institutional investors. There are two approaches: • Investors do not always process information correctly. o Result: Incorrect probability distributions of future returns, i.e., E(r) or σ § Forecasting Errors: Too much weight is placed on recent experiences. § Overconfidence: Investors overestimate their abilities and the precision of their forecasts. § Conservatism: Investors are slow to update their beliefs and underreact to new information. § Sample Size Neglect and Representativeness: Investors are too quick to infer a pattern or trend from a small sample o • Even when given a probability distribution of returns, investors may make inconsistent or suboptimal decisions. o Result: They have behavioural biases. § Framing: How the risk is described, “risky losses” vs. “risky gains”, can affect investor decisions. § Mental Accounting: Investors may segregate accounts and take risks with their gains that they would not take with their principal. § Regret Avoidance: Investors blame themselves more when an unconventional or risky bet turns out badly. § Prospect Theory: • Conventional view: Utility depends on level of wealth. • Behavioural view: Utility depends on changes in current wealth.
