Uncertainty, Risk, and Risk Management

OutlineGoalsRisk

Risk measurement: Single BondRisk Measurement: A Portfolio

Sources of RiskCapital Asset Pricing Model (CAPM)

CAPM Econometrics

Uncertainty, Risk, and Risk Management

SFC - [email protected]

Fall 2012

SFC - [email protected] Uncertainty, Risk, and Risk Management

OutlineGoalsRisk



CAPM Econometrics

Goals

Risk

Risk measurement: Single Bond

Risk Measurement: A Portfolio

Sources of Risk

Capital Asset Pricing Model (CAPM)

CAPM Econometrics


OutlineGoalsRisk



CAPM Econometrics

Goals

My goals are to help you address the following questions:

I What is risk? What’s risk aversion?

I How do we measure the risk of a bond?

I How can we use statistics to quantify the risk of a bond?

I What is the effect on risk of diversifying a bond portfolio?

I What are the different sources of risk?

I What’s the beta of a bond?

I What is the capital asset pricing model (CAPM) and what is it for?


OutlineGoalsRisk



CAPM Econometrics

Uncertainty and Risk

The future is fundamentally unknown. Therefore, there is uncertaintyabout the future consequences of choices we make today.The past is a guide to the future only if the future looks like the past insome way. But if the future doesn’t look like the past, then the past isnot necessarily a good guide.One way to measure risk in bonds is to look at the variability of returns.Think of bonds as lotteries. For example, a lottery gives you return X ifthe state of the world is A and a return Y if the state of the world isnon − A.


OutlineGoalsRisk



CAPM Econometrics

Uncertainty and Risk

We know that return is the total gain/loss experienced on investing in abond over a period of time expressed as a percentage of the value of theinvestment at the beginning of the period.

kt =Ct + Pt − Pt−1

Pt−1

where kt is the actual, expected, or required return rate (or just return)over period t, Ct is the cash flow (coupon, income, dividend, rent, orinterest) received from the investment in the time period from t − 1 to t,Pt is the price (value) of the bond at time t, and Pt−1 is the price(value) of the bond at time t − 1.


OutlineGoalsRisk



CAPM Econometrics

Risk Preferences

People have “preferences” for risk. Some like it or tolerate it better thanothers (“risk lovers”). Some are “risk neutral” (indifferent to bonds ofsame return but different return variation). The observed behavior ofcrowds (e.g. markets) indicates that most people are risk averse as theyare willing to pay a return premium on risky bonds.


OutlineGoalsRisk



CAPM Econometrics

Measuring risk of a single bond

To measure the risk of a single bond, we need to learn aboutstatistics, a branch of applied math that uses a mathematicaltheory of uncertainty (probability theory) plus some additionalassumptions to extract information from a data set.Let’s introduce the concept of a probability distribution.


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CAPM Econometrics

Probability distributions

The return on an investment is a random variable because, in advance,we don’t know for sure what its value will be. Its value is contingentupon the particular state of the world realized.Say we can list all the possible states of the world. Say there are onlythree equally-possible states of the world: (a) Good, (b) Regular, and (c)Bad. And, under each of th.ese states of the world, we know (or canguess) the return rate. Then we can form expectations on the return.Also, we can compute different measures of dispersion or variability thatcould give us a measure of risk.Discrete and continuous distributions. Show examples.


OutlineGoalsRisk



CAPM Econometrics

Probability distributions

Expected return: Mean of returns.

k̄ =n∑

i=1

kipi

where k̄ is the expected return, kj for j = 1, . . . , n is the list of returnsobserved under different states of the world, and Prj is the probabilitythat kj occurs – being the probability a number between 0 and 1, where0 means absolute impossibility of occurrence and 1 means absolutecertainty that it will occur.The measures of dispersion or variability used as proxies for risk areformulas (3) the variance, (4) the standard deviation, and (5) thecoefficient of variation – of the returns. The higher each of these is, thegreater the variability of returns and the higher the risk.

σ2k =

n∑i=1

(ki − k̄)2pi ,

σk =

√√√√ n∑i=1

(ki − k̄)2pi ,

CVk =σk

k̄.

Examples.


OutlineGoalsRisk



CAPM Econometrics

Portfolio risk

A portfolio is a collection of bonds.Correlation is the statistical measure of the association between any twoseries of numbers, e.g. returns of two bonds under different states of theworld.The degree of correlation is measured by the Pearson coefficient:−1 ≤ ρ ≤ 1, where ρ = −1 means perfect negative correlation, ρ = 0means no correlation at all, and rho = 1 means perfect positivecorrelation.Show in Excel.


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CAPM Econometrics

Sources of risk

Say, we measure the risk of a portfolio by its standard deviation, σk .Start with a portfolio with one single bond and add bonds randomly tothe portfolio, one at a time. What happens to risk as you keep addingbonds? It tends to decline towards a lower limit or baseline risk.On average, portfolio risk approaches the lower limit when you’ve added15-20 randomly selected securities to your portfolio. So, total risk can beviewed as the sum of two types of risk:Total security risk = Nondiversifiable risk + Diversifiable risk.Diversifiable risk is also called ‘unsystematic,’ because it comes fromrandom factors that can be eliminated by diversifying the portfolio.Nondiversifiable risk is also known as ‘systematic’ and it comes frommarket-wide factors that affect all securities. It may also be called‘market’ or ‘macroeconomic’ risk.


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CAPM Econometrics

Diversification

However correlated the returns of two bonds may be, the expected returnof a portfolio with the two bonds will fall in some midpoint between thereturns of the two bonds held in isolation.If the bonds are highly positively correlated, the risk is some midpointbetween the risk of each bond in isolation.If the bonds are largely uncorrelated, the risk is some midpoint betweenthe risk of the most risky bond and less than the risk of the least riskybond, but greater than zero.If the bonds are highly negatively correlated, the risk is some midpointbetween the risk of most risky bond and zero.In no case will a portfolio be riskier than the riskiest bond included in it.


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CAPM Econometrics

CAPM model

The model links nondiversifiable risk to returns for all bonds.First, we will discuss the beta, an element of the model. Second, we willintroduce the CAPM equation. Third, we will show how to use it inconcrete applications.The beta is a measure of the nondiversifiable risk. It indicates to whatextent the return on a bond responds to a change in the market oraverage return of all bonds.Who knows about ‘all bonds,’ but there are broad indices of securities,e.g. S&P 500. How do we estimate the beta of a given bond? We needdata on the returns on that bond and data on the returns of awell-diversified portfolio representative of the market as a whole. We useregression analysis.


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CAPM Econometrics

Regression analysis

Consider the plot of the returns on two bonds. The horizontal axis showsthe return on an bond representative of the whole market, e.g. the S&P500. The vertical axis shows the return on a particular bond, e.g. thereturn on Google.Show plot in Excel.


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CAPM Econometrics

Regression analysis

Consider the simple bi-variate linear equation:

y = a + bx

The equation says that the value of the variable y (the dependentvariable) depends on the value of the variable x (the independentvariable).The literals a and b are called, respectively, the intercept and the slope.The intercept indicates the value of y when x = 0. The slope indicatesthe change in y associated to a unit change in x or b = ∆Y

∆X .


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CAPM Econometrics

Regression analysis

Regression is a statistical procedure to fit a curve in a scatterplot. Forexample, a straight line.

yi = α + βxi + εi .

Show regression in Excel.The beta coefficient (slope) indicates the change in the return on aparticular bond (e.g. Google or GE) when the return on a marketportfolio (e.g. S&P 500) changes in one unit. The beta shows howsensitive the return on the bond is to performance of the market as awhole.What do the betas of Google and GE say?The beta of a portfolio is the weighted average of its individual bonds’betas:

βp =n∑

i=1

wiβi ,

where∑n

i=1 = 1. The beta of a portfolio indicates how responsive theportfolio’s return is to changes in the market return.Example.


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CAPM Econometrics

CAPM Model

The capital asset pricing model (CAPM) gives us the return that wewould require in order to compensate for the risk involved in holding agiven bond i . The model helps us determine the return over and above arisk-free return that would offset the risk inherent to that bond. And byrisk inherent to that bond, we mean the risk of holding that bond that isnot diversifiable, i.e. the risk that it shares with a well-diversified marketportfolio.So, to determine that extra return or risk premium, we need to measurethe extent to which an bond is exposed to nondiversifiable risk. Butinstead of pulling that measure out of our own heads, we consider theway crowds (e.g. markets) determine that extra return.


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CAPM Econometrics

CAPM Model

We now remember that the beta of a bond tells us the change in thereturn on the bond that is due to a one-percent change in the marketreturn. That beta can be used as an index or measure of nondiversifiablerisk, since it reflects to covariation of the return on that bond and themarket return. So, we plug the beta in the following equation and get therequired return, ki :

ki = RF + [βi (km − RF )],

where, again, ki is the required return on bond i , RF is the risk-free rateof return (usually the return on a U.S. Treasury bill), βi is the betacoefficient or index of nondiversifiable risk for bond i , and km is thereturn on a market portfolio.


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CAPM Econometrics

CAPM Model

ki = RF + [βi (km − RF )]. (1)

Under the CAPM model, the required return on bond i has twoparts: (1) the risk-free rate of return – something like a baselinerate of return (say, the return or interest on a 3-month T bill) –and (2) the risk premium. In turn, (2) has two parts: (a) the betaor index of nondiversifiable risk and (b) (km − RF ) or market riskpremium. The market risk premium represents the premiuminvestors require for taking the average amount of risk associatedwith holding a well-diversified market portfolio of bonds.


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CAPM Econometrics

CAPM Model

The graphical representation of the CAPM model is called the securitymarket line (SML). We plot our measure of nondiversifiable risk on thehorizontal axis and the required return on the vertical axis. Note thatbeta is our independent variable.Suppose the risk-free return (RF ) is 7% and the expected market return(km) is 11%. Then, (km − RF ) = 4%. The intercept of our graph will beRF . The slope will indicate the change in our required return when wevary beta in one unit. The slope is given by (km − RF ) = 4%.Example. If beta goes from 1 to 1.5, i.e. ∆β = 0.5, the required returnincreases from 11% to 13% or 2% = 0.5 × 4%.


OutlineGoalsRisk



CAPM Econometrics

CAPM Model

In brief, the CAPM model provides us with a way to determine a returnadequate to the (nondiversifiable) risk involved in holding a given bond.And determining an adequate return is essential to value bonds. Ittranslates (nondiversifiable) risk into a risk premium or additional returnthat would compensate (according to the market) for our taking the riskof holding that given bond.The CAPM is not foolproof. We can only use historical data to estimatethe betas. But past variability may not reflect future variability. So, wehave to be careful and make adjustments if we have additionalinformation.The CAPM, if it is to function well, requires that the market that pricesbonds and, therefore, determines returns, be competitive and efficient –in the sense of being made up by many buyers and sellers, and capable ofabsorbing available information quickly. It is also assumed thatgovernment or other types of restrictions don’t exist, that there are notaxes or transaction costs, and that investors are rational. Real life ismore complicated. Still, the CAPM is a nice, useful benchmark.The CAPM performs well with bonds such as stocks of highly tradedcompanies. It doesn’t do as well with real corporate assets, such asbuildings, equipment, etc. But that may be due to the lack of efficientmarkets.All in all, the CAPM is a good starting point. And it is widely used byfinance managers in large companies.


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CAPM Econometrics

What have we learned?

I What is risk? What is return? What’s risk aversion?

I How do we measure the risk of a given bond?

I How can we use statistical measures to quantify the risk of a givenbond?

I What is the effect on risk of diversifying a portfolio of bonds?

I Are there different types of risk and what are they?

I What’s the beta of a bond?

I What is the capital asset pricing model (CAPM)?

I What is the security market line?


Uncertainty, Risk, and Risk Management

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