Uncertainty, Risk, and Risk Management

Outline
Goals
Risk
Risk measurement: Single Bond
Risk Measurement: A Portfolio
Sources of Risk
Capital Asset Pricing Model (CAPM)
CAPM Econometrics

Uncertainty, Risk, and Risk Management

SFC - jhuato.sfc@gmail.com

Fall 2012

SFC - jhuato.sfc@gmail.com Uncertainty, Risk, and Risk Management

Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

Goals

Risk



Sources of Risk


CAPM Econometrics


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

Goals

My goals are to help you address the following questions:
What is risk? What’s risk aversion?
How do we measure the risk of a bond?
How can we use statistics to quantify the risk of a bond?
What is the eﬀect on risk of diversifying a bond portfolio?
What are the diﬀerent sources of risk?
What’s the beta of a bond?
What is the capital asset pricing model (CAPM) and what is it for?


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

Uncertainty and Risk

The future is fundamentally unknown. Therefore, there is uncertainty
about the future consequences of choices we make today.
The past is a guide to the future only if the future looks like the past in
some way. But if the future doesn’t look like the past, then the past is
not 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 if
the state of the world is A and a return Y if the state of the world is
non − A.


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

Uncertainty and Risk

We know that return is the total gain/loss experienced on investing in a
bond over a period of time expressed as a percentage of the value of the
investment at the beginning of the period.
Ct + Pt − Pt−1
kt =
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, or
interest) 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.


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

Risk Preferences

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


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

Measuring risk of a single bond

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


Outline
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Risk
Sources of Risk
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 contingent
upon the particular state of the world realized.
Say we can list all the possible states of the world. Say there are only
three 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 can
guess) the return rate. Then we can form expectations on the return.
Also, we can compute diﬀerent measures of dispersion or variability that
could give us a measure of risk.
Discrete and continuous distributions. Show examples.


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Risk
Sources of Risk
CAPM Econometrics

Probability distributions
Expected return: Mean of returns.
n
¯
k= ki pi
i=1
¯
where k is the expected return, kj for j = 1, . . . , n is the list of returns
observed under diﬀerent states of the world, and Prj is the probability
that kj occurs – being the probability a number between 0 and 1, where
0 means absolute impossibility of occurrence and 1 means absolute
certainty that it will occur.
The measures of dispersion or variability used as proxies for risk are
formulas (3) the variance, (4) the standard deviation, and (5) the
coeﬃcient of variation – of the returns. The higher each of these is, the
greater the variability of returns and the higher the risk.
n
2
σ =
SFC - jhuato.sfc@gmail.com ¯
− k)2 p
(k Uncertainty,, Risk, and Risk Management

Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

Portfolio risk

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


Outline
Goals
Risk
Sources of Risk
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 to
the portfolio, one at a time. What happens to risk as you keep adding
bonds? It tends to decline towards a lower limit or baseline risk.
On average, portfolio risk approaches the lower limit when you’ve added
15-20 randomly selected securities to your portfolio. So, total risk can be
viewed as the sum of two types of risk:
Total security risk = Nondiversifiable risk + Diversifiable risk.
Diversifiable risk is also called ‘unsystematic,’ because it comes from
random factors that can be eliminated by diversifying the portfolio.
Nondiversifiable risk is also known as ‘systematic’ and it comes from
market-wide factors that affect all securities. It may also be called
‘market’ or ‘macroeconomic’ risk.


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

Diversiﬁcation

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


Outline
Goals
Risk
Sources of Risk
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 will
introduce the CAPM equation. Third, we will show how to use it in
concrete applications.
The beta is a measure of the nondiversifiable risk. It indicates to what
extent the return on a bond responds to a change in the market or
average 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 need
data on the returns on that bond and data on the returns of a
well-diversified portfolio representative of the market as a whole. We use
regression analysis.


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Risk
Sources of Risk
CAPM Econometrics

Regression analysis

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


Outline
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Risk
Sources of Risk
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 dependent
variable) depends on the value of the variable x (the independent
variable).
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 indicates
the change in y associated to a unit change in x or b = ∆Y .∆X


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

Regression analysis
Regression is a statistical procedure to ﬁt a curve in a scatterplot. For
example, a straight line.

yi = α + βxi + i .
Show regression in Excel.
The beta coeﬃcient (slope) indicates the change in the return on a
particular bond (e.g. Google or GE) when the return on a market
portfolio (e.g. S&P 500) changes in one unit. The beta shows how
sensitive the return on the bond is to performance of the market as a
whole.
What do the betas of Google and GE say?
The beta of a portfolio is the weighted average of its individual bonds’
betas:
n
βp = w i βi ,

Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

CAPM Model

The capital asset pricing model (CAPM) gives us the return that we
would require in order to compensate for the risk involved in holding a
given bond i. The model helps us determine the return over and above a
risk-free return that would offset the risk inherent to that bond. And by
risk inherent to that bond, we mean the risk of holding that bond that is
not diversifiable, i.e. the risk that it shares with a well-diversified market
portfolio.
So, to determine that extra return or risk premium, we need to measure
the extent to which an bond is exposed to nondiversifiable risk. But
instead of pulling that measure out of our own heads, we consider the
way crowds (e.g. markets) determine that extra return.


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

CAPM Model

We now remember that the beta of a bond tells us the change in the
return on the bond that is due to a one-percent change in the market
return. That beta can be used as an index or measure of nondiversifiable
risk, since it reflects to covariation of the return on that bond and the
market return. So, we plug the beta in the following equation and get the
required return, ki :
ki = RF + [βi (km − RF )],
where, again, ki is the required return on bond i, RF is the risk-free rate
of return (usually the return on a U.S. Treasury bill), βi is the beta
coefficient or index of nondiversifiable risk for bond i, and km is the
return on a market portfolio.


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

CAPM Model

ki = RF + [βi (km − RF )]. (1)

Under the CAPM model, the required return on bond i has two
parts: (1) the risk-free rate of return – something like a baseline
rate 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 beta
or index of nondiversiﬁable risk and (b) (km − RF ) or market risk
premium. The market risk premium represents the premium
investors require for taking the average amount of risk associated
with holding a well-diversiﬁed market portfolio of bonds.

Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

CAPM Model

The graphical representation of the CAPM model is called the security
market line (SML). We plot our measure of nondiversiﬁable risk on the
horizontal axis and the required return on the vertical axis. Note that
beta 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 be
RF . The slope will indicate the change in our required return when we
vary 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 return
increases from 11% to 13% or 2% = 0.5 × 4%.


Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

CAPM Model
In brief, the CAPM model provides us with a way to determine a return
adequate to the (nondiversifiable) risk involved in holding a given bond.
And determining an adequate return is essential to value bonds. It
translates (nondiversifiable) risk into a risk premium or additional return
that would compensate (according to the market) for our taking the risk
of holding that given bond.
The CAPM is not foolproof. We can only use historical data to estimate
the betas. But past variability may not reflect future variability. So, we
have to be careful and make adjustments if we have additional
information.
The CAPM, if it is to function well, requires that the market that prices
bonds and, therefore, determines returns, be competitive and efficient –
in the sense of being made up by many buyers and sellers, and capable of
absorbing available information quickly. It is also assumed that
government or other jhuato.sfc@gmail.com
SFC - types of restrictions don’t Risk, and that there are no
Uncertainty, exist, Risk Management

Outline
Goals
Risk
Sources of Risk
CAPM Econometrics

What have we learned?

What is risk? What is return? What’s risk aversion?
How do we measure the risk of a given bond?
How can we use statistical measures to quantify the risk of a given
bond?
What is the eﬀect on risk of diversifying a portfolio of bonds?
Are there diﬀerent types of risk and what are they?
What’s the beta of a bond?
What is the capital asset pricing model (CAPM)?
What is the security market line?


Uncertainty, Risk, and Risk Management

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