Title: An Introduction to Risk and Risk Premiums, and The Historical Record
1An Introduction to Risk and Risk Premiums, and
The Historical Record
B,K M Chapter 5 End-of-chapter problems 1-15
2Determinants of the Level of Interest Rates
- Interest rates, of course, are important inputs
to many economic decisions - Forecasting interest rates is difficult, but
they are determined by the forces of supply and
demand (as would be expected in any competitive
market) - Inflationary expectations are critical since
lenders will demand compensation for anticipated
losses in purchasing power
3Real and Nominal Interest Rates
- The real rate of interest is the nominal
(reported) interest rate reduced by the loss of
purchasing power due to inflation - - r is (approximately) R- inf
- Where r is the real interest rate
- R is the nominal interest rate
- inf is the inflation rate
- The exact relationship (when reported rates are
compounded annually) is given below - although the approximation is good if the
inflation rate is not too high
4The Equilibrium Real Rate of Interest
- Supply, demand, and government actions determine
the real rate while the nominal rate is the real
rate plus the expected rate of inflation - The fundamental determinants of the real rate
are the propensity of households to borrow and to
save, the expected productivity (profitability)
of physical capital, and the propensity of the
government to borrow or save - In Class Exercise
- - Use demand and supply analysis to predict the
change in the real rate given an increase in each
of its above mentioned determinants
5The Equilibrium Nominal Rate of Interest
- As noted, the nominal rate differs from the real
rate because of inflation - The Treasury and the Fed have the ability to
influence short-term interest rates by
controlling the flow of new funds into the
banking system, however the influence on
long-term rates is not always favorable because
of the potential impact of expansionary monetary
policy on expected inflation - It is, of course, an over-simplification to
speak of a single interest rate because in
reality there are many rates which depend on term
to maturity and default risk
6Fisher Effect
- The fisher effect is the relationship between
real and nominal rates - The basic intuition is that investors will
require compensation for inflation in order to
hold securities whose returns are in nominal
terms. The expected real rate is thus the nominal
rate minus expected inflation - If real interest rates are relatively constant,
then fluctuations in nominal rates will be due to
changes in expected inflation
7Fisher Effect
- In fact, short-term realized real rates are quite
variable (Graph below)
8Taxes, the Real Rate of Interest, and Realized
Returns
- In Class Exercise
- Are you indifferent between earning 10 when
inflation is 8 and 2 when inflation is 0? - It is critical to remember that the real
after-tax rate is (approximately) the after-tax
nominal rate minus the inflation rate
9Risk and Risk Premiums
10Holding Period Returns (HPR)
- Risk means uncertainty about what your realized
holding period return will be - We can quantify the uncertainty using probability
distributions
11Holding Period Returns (HPR)
- Example Assume there is considerable
uncertainty with respect to the end of year price
of an index stock fund which currently sells for
100, although we expect a dividend of 4 - If the normal growth prevails, then the HPR is
- (110 4 - 100) / (100) .14 or 14
- The expected return is the probability weighted
average of all possible outcomes
12Holding Period Returns (HPR)
- In the above example, the expected return is
calculated as follows - The standard deviation (?) is a measure of risk.
It is defined as the square root of the variance - Which in this example is calculated as follows
- Therefore the standard deviation (?) is 21.21
13Holding Period Returns (HPR)
Would this investment be attractive to a risk
averse investor? This will generally depend on
the risk premium it affords, where the risk
premium is the excess of the expected return over
the risk-free rate. The risk-free rate is the
return on competitive risk-free assets such as
T-Bills.
14The Historical Record
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18Risk in a Portfolio Context
- A fundamental principle of financial economics
is that you cannot assess the riskiness of an
investment by examining only its own standard
deviation! - Risk must always be considered in a portfolio
context, that is, taking into account the
standard deviation of your entire portfolio after
adding the asset in question - Example
- Your 100,000 home will burn down with a prob.
.002. Your expected loss (due to your home
burning down) is .002 x 100,000, or 200. - An insurance policy (no deductible) costs 220
19Risk in a Portfolio Context
- A. What is the expected profit of the
investment in the policy? - The expected profit is -20, with an expected
return of 20/220 or 9.09 - B. What is the standard deviation of profit of
an investment in the policy? -
- The standard deviation (?) is therefore 4467.8
20Risk in a Portfolio Context
- Who wants to buy an asset with a negative
expected return and a high standard deviation? - In fact, this may be a valuable addition to a
portfolio because of its impact on portfolio risk - In Class Exercise
- - What is the standard deviation of the value of
the complete portfolio which includes the
insurance policy?
21Risk, Risk Aversion, and Portfolio Risk and Return
22Risk and Risk Aversion
- Investors avoid risk and demand a reward for
investing in risky investments - The proper measure of the risk of an asset is
the marginal impact of the asset on the riskiness
of the entire portfolio in which it is held - A Simple Example
- Assume you have initial wealth of 100,000
- You can invest it in a risky portfolio or in
risk-free T-Bills
23Risk and Risk Aversion
- The risky portfolio has an expected return of
- .6 x 50 .4 x 25 20
- The risk-free portfolio has an expected return
of 8 - The risk premium is therefore 20 - 8 12
- The investors choice will depend upon his/her
attitude toward risk
24Overview
- In a world of certainty, rational choice entails
choosing the bundle of consumption that maximizes
utility subject to budget constraint - In finance we focus on the utility of
end-of-period wealth (or rate of return given the
current level of wealth) - The Utility Function characterizes the
preferences of an individual investor over the
distribution of the rate of return on the
portfolio. - The individual chooses the portfolio to maximize
utility. - An example of a simple utility function follows
- where U is the utility value, A is the investors
degree of risk aversion, ER is the expected
rate of return on the portfolio, and ?r2 is the
variance of the rate of return on the portfolio
25Overview
- If an investor is risk averse
- - He/she prefers a certain outcome to an
uncertain outcome with the same rate of return - The utility function is concave (U(W) as a
function of W) - This representation of utility ignores higher
moments of the return distribution such as
skewness and kurtosis to simplify the math - In fact, investors probably prefer skewed
distributions with long positive tails
26Certainty Equivalent Rate
- A risky portfolio utility value is the rate that
a risk-free portfolio would have to earn to be
equally attractive to the risky portfolio. - The risky portfolio is only desirable if its
certainty-equivalent is equal to or higher than
the risk-free rate - A less risk-averse investor would assign a
higher certainty-equivalent to the same risky
portfolio - A risk-neutral investor (A 0) cares only about
the expected rate of return
27Certainty Equivalent Rate
- Without knowing more about an investor than
he/she is risk averse, any portfolio to the
northwest of another portfolio will be
preferred because it has both higher expected
return and lower risk. Any portfolio to the
southeast is dominated and is preferred by the
mean variance criterion
28Empirical Evidence on Us
- Very strong evidence that investors prefer more
to less - Very strong evidence that investors are risk
averse (Agt0) - Some view casino gambling that has negative
expected return as consumption rather than
investment
29Portfolio Returns
30Portfolio Returns
- To compute the return on a portfolio, use the
same formula you use for the return on a single
asset - is the period t return on asset A
- The return on the portfolio is the weighted
average of the individual security returns - If we are concerned with the ex ante expected
return of a portfolio, the above formula applies
as well - The historical average return is often used as a
proxy for expected return
31The Variance and Standard Deviation of an
individual Security
- The variance and standard deviations are
measures of the dispersion of returns from its
expected value - If we do not know the probability of each state
of the world, we can estimate the variance of an
asset using historical date (sample variance) - where T is the number of time periods of data
- In class exercise
- - Why is the denominator in the above formula T-1
rather than T?
32Portfolio Variance
33Portfolio Variance
- Variance of the portfolio is more complicated
because the extent to which the randomness in the
different securities tends to reduce overall risk
must be accounted for - Example Consider 2 stocks, A B. Both have
identical variances and expected returns. If we
hold a portfolio that consists of equal weights
of both stocks, the variance of the portfolio
will depend upon the extent to which the two
stock returns move together. If A has below
normal returns when B has above normal return,
and vice verse, then it will be possible to form
a portfolio with very low variance relative to
the variance of either A or B - If both tend to be above average or below
average together, portfolio variance may not be
appreciably less than the variance of either A or
B
34Portfolio Variance
- The formula for the variance of a portfolio is
derived by evaluating the following expectation,
using the definitions we developed above for the
return and expected return of a portfolio - The measure of co-movement which emerges is the
covariance, where the covariance is the expected
value of the products of deviations from the
sample means - In practice we must estimate the covariance
- We will first see how to estimate the covariance
between two assets in a portfolio, then we show
how we use these covariances to estimate the
portfolio variance
35Portfolio Variance
- The sample covariance of stock returns is the
average of the products of the deviations from
the sample means - What is the relation between the sample
covariance and the sample correlation coefficient
(a statistic you may be more familiar with)? - The sample correlation coefficient of two assets
is simply the scaled covariance
36Portfolio Variance
- Note that
- The covariance and the correlation coefficient
have the same sign - ?A,B gt 0 when A and B tend to move together
- ?A,B lt 0 when A and B tend to move in opposite
directions - ?A,B 0 when A and B are independent
- The larger ?A,B , the more closely related are
the returns of A and B - The larger is ?A,B , the narrower is the
cloud formed by plotting the returns of A on
the horizontal axis, and B on the vertical axis.
The cloud is a straight line when ?A,B 1 - Finally, note that covA,B ?A,B?A?B. We will
use this later on
37Portfolio Variance
- Following is the formula for the variance of
portfolio returns in a form you can use - Where
- wi is the proportion of the portfolio invested
in asset i at the beginning of the period - ?i,j is the covariance of returns between i and
j - ?i,j ?i2 if and only if i j
- ?i is the standard deviation of asset i
- ?i,j is the correlation of returns between
assets i and j
38Portfolio Variance
- Some find matrix notation to be easier. First
form a matrix with the covariances between the
assets returns and the portfolio weights - Below is an example for a 3 asset portfolio
39Portfolio Variance
- Each element gives the covariance of returns for
the intersection of the columns stock and the
rows stock. (The diagonal from the NW to the SE
is the covariance between each stock and itself.
This is, by definition, the stocks variance.) - Note as well that the matrix is symmetric for
each weight and covariance above the diagonal
there is an equal covariance and weight below the
diagonal - Calculating the variance of the portfolio is
done by multiplying each of the covariances in
the matrix by the weight at the top of its column
and the weight at the left side of its row - You then obtain the variance of the portfolio by
summing these products
40Portfolio Variance
- Note that the formula for a two asset risky
portfolio is therefore as follows - In class exercise What is the variance of the
three asset portfolio? - Note that the number of elements is the square
of the number of assets in the portfolio. How
many covariance estimates are necessary to
estimate the variance of a 100 asset portfolio? - Note as well that as the number of assets in the
portfolio increases, the relative importance of
the variance terms diminishes (because the ratio
of stocks on the diagonal to stocks off the
diagonal diminishes).
41The Case of Independent Stock Returns An Example
42An Example
- Assume returns are independent (?i,j 0 if i is
not equal to j), and all standard deviations are
30 - What is the standard deviation of a portfolio of
three stocks with equal weights? - (Note that all covariance terms are zero by
assumption)
.17
43Example (cont.)
- As we increase the number of assets in a
portfolio, the importance of the variance terms
diminishes, but that of the covariance term does
not. If all covariance terms are 0, then the
standard deviation of the portfolio approaches
zero as the number of assets becomes large, or in
the above example - N ?p
- 10 9.4868
- 100 3.0
- 1000 0.94868
- In fact, most covariances between pairs of
stocks are positive, so we are not able to ignore
the covariance terms - This means that it is virtually impossible to
reduce portfolio variance to zero (for portfolios
of risky assets)