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summary of managerial finance
- Instelling
- Maastricht University (UM)
detailed summary of the chapters mentioned.
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CHAPTER 10: Capital Markets and the Pricing of Risk10.1 Risk and Return: Insights from 89 Years of Investor History
During the 2008 financial crisis, all of the stock portfolios declined by more than 50%, with the small stock
portfolio declining by almost 70% (over $1.5 million) from its peak in 2007 to its lowest point in 2009. Many
investors faced a double whammy: an increased risk of being unemployed (as firms started laying off
employees) precisely when the value of their savings eroded. Thus, while the stock portfolios had the best
performance over this 89-year period, that performance came at a cost––the risk of large losses in a
downturn. On the other hand, Treasury bills enjoyed steady––albeit modest–– gains each year.
Panel (a) shows the value of each investment after one
year and illustrates that if we rank the investments by
the volatility of their annual increases and decreases in
value, we obtain the same ranking we observed with
regard to performance: small stocks had the most
variable returns, followed by the S&P 500, the world
portfolio, corporate bonds, and finally Treasury bills.
Panels (b), (c), and (d) of Figure 10.2 show the results for
5-, 10-, and 20-year investment horizons, respectively.
Note that as the horizon lengthens, the relative
performance of the stock portfolios improves.
Greatest variation:
1. Small stocks
2. Large stocks
3. Corporate bonds
4. Treasury bills
10.2 Common Measures of Risk and Return
When a manager makes an investment decision or an investor purchases a security, they have some view as to
the risk involved and the likely return the investment will earn.
Probability Distributions
Different securities have different initial prices, pay different cash flows, and sell for different future amounts. To
make them comparable, we express their performance in terms of their returns. The return indicates the
percentage increase in the value of an investment per dollar initially invested in the security. When an investment
is risky, there are different returns it may earn. Each possible return has some likelihood of occurring. We
summarize this information with a probability distribution, which assigns a probability, pR, that each possible
return, R, will occur.
We can also represent the probability distribution with a histogram, in which the height of a bar indicates the
likelihood of the associated outcome.
Expected Return
Given the probability distribution of returns, we can compute the expected return. We calculate the expected (or
mean) return as a weighted average of the possible returns, where the weights correspond to the probabilities:
Expected (Mean) Return: E[𝑅] = ∑! 𝑃! × 𝑅
The notation ∑! means that we calculate the sum of the expression (in this case, PR * R) over all possible
returns R.
The expected return is the return we would earn on average if we could repeat the investment many times,
drawing the return from the same distribution each time. In terms of the histogram, the expected return is the
“balancing point” of the distribution, if we think of the probabilities as weights.
Variance and Standard Deviation
Two common measures of the risk of a probability distribution are
its variance and standard deviation. The variance is the expected
squared deviation from the mean, and the standard deviation is
the square root of the variance:
,If the return is risk-free (so never deviates from its mean), the variance is zero. Otherwise, the variance increases
with the magnitude of the deviations from the mean. Therefore, the variance is a measure of how “spread out”
the distribution of the return is. In finance, we refer to the standard deviation of a return as its volatility. While
the variance and the standard deviation both measure the variability of the returns, the standard deviation is
easier to interpret because it is in the same units as the returns themselves.
If we could observe the probability distributions that investors anticipate for different securities, we could
compute their expected returns and volatilities and explore the relationship between them. In most situations,
however, we do not know the explicit probability distribution. Without that information, how can we estimate
and compare risk and return? A popular approach is to extrapolate from historical data, which is a sensible
strategy if we are in a stable environment and believe that the distribution of future returns should mirror that of
past returns.
10.3 Historical Returns of Stocks and Bonds
Computing Historical Returns
Of all the possible returns, the realized return is the return that actually occurs over a particular time period. How
do we measure the realized return for a stock? Suppose you invest in a stock on date t for price Pt. If the stock
pays a dividend, Divt +1, on date t + 1, and you sell the stock at that time for price Pt +1, then the realized return
from your investment in the stock from t to t + 1 is
"#$ %& "#$ & '&
Realized Return: Rt+1= !"# !"# − 1 = $"# + !"# ! = 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 + 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 𝑅𝑎𝑡𝑒
&! &! &!
Ø The realized return, Rt+1, is the total return we earn from dividends and capital gains, expressed as a
percentage of the initial stock price.
Calculating Realized Annual Returns: if you hold the stock beyond the date of the first dividend, then to compute
your return you must specify how you invest any dividends you receive in the interim. To focus on the returns of a
single security, let’s assume that you reinvest all dividends immediately and use them to purchase additional
shares of the same stock or security. In this case, we can use the equation above to compute the stock’s return
between dividend payments, and then compound the returns from each dividend interval to compute the return
over a longer horizon. For example, if a stock pays dividends at the end of each quarter, with realized returns
RQ1, …, RQ4 each quarter, then its annual realized return, Rannual, is:
1+Rannual= (1+ RQ1)(1+ RQ2)(1+ RQ3)(1+ RQ4)
Two features of the returns from holding a stock:
1. Both dividends and capital gains contribute to the total realized return.
2. The returns are risky.
We can compute realized returns in this same way for any investment. We can also compute the realized returns
for an entire portfolio, by keeping track of the interest and dividend payments paid by the portfolio during the
year, as well as the change in the market value of the portfolio.
Comparing Realized Annual Returns: once we have calculated the realized annual returns, we can compare them
to see which investments performed better in a given year. Over any particular period we observe only one draw
from the probability distribution of returns. However, if the probability distribution remains the same, we can
observe multiple draws by observing the realized return over multiple periods. By counting the number of times
the realized return falls within a particular range, we can estimate the underlying probability distribution.
In a histogram, the height of each bar represents the number of years that the annual returns were in each range
indicated on the x-axis. When we plot the probability distribution in this way using historical data, we refer to it as
the empirical distribution of the returns.
Average Annual Returns
The average annual return of an investment during some historical period is simply the average of the realized
returns for each year. That is, if Rt is the realized return of a security in year t, then the average annual return for
years 1 through T is
( (
Average Annual Return of a Security: 𝑅9 = (𝑅( + 𝑅* + ⋯ + 𝑅) ) = ∑)+,( 𝑅+
) )
The average annual return is the balancing point of the empirical distribution— in this case, the probability of a
return occurring in a particular range is measured by the number of times the realized return falls in that range.
,Therefore, if the probability distribution of the returns is the same over time, the average return provides an
estimate of the expected return.
Variance: measure of how spread out payoffs are
The Variance and Volatility of Returns
To quantify the difference in variability, we can estimate the standard deviation of the probability distribution. As
before, we will use the empirical distribution to derive this estimate. Using the same logic as we did with the
mean, we estimate the variance by computing the average squared deviation from the mean. We do not actually
know the mean, so instead we use the best estimate of the mean—the average realized return:
(
Variance Estimate Using Realized Returns: 𝑉𝑎𝑟(𝑅) = ∑) (𝑅 − 𝑅9)*
)'( +,( +
We estimate the standard deviation or volatility as the square root of the variance à The volatility or standard
deviation is therefore SD(R ) = ?Var (R )
Estimation Error: Using Past Returns to Predict the Future
To estimate the cost of capital for an investment, we need to determine the expected return that investors will
require to compensate them for that investment’s risk. If the distribution of past returns and the distribution of
future returns are the same, we could look at the return investors expected to earn in the past on the same or
similar investments, and assume they will require the same return in the future. However, there are two
difficulties with this approach.
1. We do not know what investors expected in the past; we can only observe the actual returns that were
realized.
If we believe that investors are neither overly optimistic nor pessimistic on average, then over time, the average
realized return should match investors’ expected return. Armed with this assumption, we can use a security’s
historical average return to infer its expected return. But now we encounter the second difficulty:
2. The average return is just an estimate of the true expected return, and is subject to estimation error.
Given the volatility of stock returns, this estimation error can be large even with many years of data.
Standard Error: we measure the estimation error of a statistical estimate by its standard error. The standard error
is the standard deviation of the estimated value of the mean of the actual distribution around its true value; that
is, it is the standard deviation of the average return. The standard error provides an indication of how far the
sample average might deviate from the expected return. If the distribution of a stock’s return is identical each
year, and each year’s return is independent of prior years’ returns, then we calculate the standard error of the
estimate of the expected return as follows:
Standard Error of the Estimate of the Expected Return:
Because the average return will be within two standard errors of the true expected return approximately 95% of
the time, we can use the standard error to determine a reasonable range for the true expected value. The 95%
confidence interval for the expected return is 𝐻𝑖𝑠𝑡𝑜𝑟𝑖𝑐𝑎𝑙 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑅𝑒𝑡𝑢𝑟𝑛 ± (2 × 𝑆𝐸)
Limitations of Expected Return Estimates: individual stocks tend to be even more volatile than large portfolios,
and many have been in existence for only a few years, providing little data with which to estimate returns.
Because of the relatively large estimation error in such cases, the average return investors earned in the past is
not a reliable estimate of a security’s expected return.
10.4 The Historical Trade-Off Between Risk and Return
The Returns of Large Portfolios
The excess return is the difference between the average return for the investment and the average return for
Treasury bills, a risk-free investment, and measures the average risk premium investors earned for bearing the
risk of the investment.
In general, the investments with higher volatility reward investors with higher average returns to compensate
them for the extra risk they are taking on.
The Returns of Individual Stocks
Investments with higher volatility should have a higher risk premium and therefore higher returns. However, if we
look at the volatility and return of individual stocks, we do not see any clear relationship between them.
, There is a relationship between size and risk: larger stocks have lower volatility overall. Even the largest stocks are
typically more volatile than a portfolio of large stocks. Moreover, there is no clear relationship between volatility
and return.
Thus, while volatility is perhaps a reasonable measure of risk when evaluating a large portfolio, it is not adequate
to explain the returns of individual securities.
10.5 Common Versus Independent Risk
Theft Versus Earthquake Insurance: An Example
Consider two types of home insurance: theft insurance and earthquake insurance. Let us assume that the risk of
each of these two hazards is similar for a given home in the San Francisco area. Each year there is about a 1%
chance that the home will be robbed and a 1% chance that the home will be damaged by an earthquake.
So, the chance the insurance company will pay a claim for a single home is the same for both types of insurance
policies. Suppose an insurance company writes 100,000 policies of each type for homeowners in San Francisco.
We know that the risks of the individual policies are similar, but are the risks of the portfolios of policies similar?
• Theft insurance: because the chance of a theft in any given home is 1%, we would expect about 1% of the
100,000 homes to experience a robbery. Thus, the number of theft claims will be about 1000 per year,
even if the actual number of claims may be a bit higher or lower each year. We can estimate the
likelihood that the insurance company will receive different numbers of claims, assuming that instances
of theft are independent of one another. The number of claims will almost always be between 875 and
1125 (0.875% and 1.125% of the number of policies written). In this case, if the insurance company holds
reserves sufficient to cover 1200 claims, it will almost certainly have enough to meet its obligations on its
theft insurance policies.
• Earthquake insurance: most years, an earthquake will not occur. But because the homes are in the same
city, if an earthquake does occur, all homes are likely to be affected and the insurance company can
expect as many as 100,000 claims. As a result, the insurance company will have to hold reserves sufficient
to cover claims on all 100,000 policies it wrote to meet its obligations if an earthquake occurs.
Ø Thus, although the expected numbers of claims may be the same, earthquake and theft insurance lead to
portfolios with very different risk characteristics. For earthquake insurance, the number of claims is very
risky. It will most likely be zero, but there is a 1% chance that the insurance company will have to pay
claims on all the policies it wrote. In this case, the risk of the portfolio of insurance policies is no different
from the risk of any single policy— it is still all or nothing. Conversely, for theft insurance, the number of
claims in a given year is quite predictable. Year in and year out, it will be very close to 1% of the total
number of policies, or 1000 claims. The portfolio of theft insurance policies has almost no risk.
Types of Risk: intuitively, the key difference between them is that an earthquake affects all houses
simultaneously, so the risk is perfectly correlated across homes à common risk.
In contrast, because thefts in different houses are not related to each other, the risk of theft is uncorrelated and
independent across homes à independent risks.
When risks are independent, some individual homeowners are unlucky and others are lucky, but overall the
number of claims is quite predictable. The averaging out of independent risks in a large portfolio is called
diversification.
The Role of Diversification
We can quantify this difference in terms of the standard deviation of the percentage of claims. First, consider the
standard deviation for an individual homeowner. At the beginning of the year, the homeowner expects a 1%
chance of placing a claim for either type of insurance. But at the end of the year, the homeowner will have filed a
claim (100%) or not (0%). The standard deviation is
SD (Claim) = ?Var (Claim ) =?0.99 × (0 − 0.01)* + 0.01 × (1 − 0.01)* = 9.95%
For the homeowner, this standard deviation is the same for a loss from earthquake or theft.
Now consider the standard deviation of the percentage of claims for the insurance company. In the case of
earthquake insurance, because the risk is common, the percentage of claims is either 100% or 0%, just as it was
for the homeowner à The percentage of claims received by the earthquake insurer is also 1% on average, with a
9.95% standard deviation.
While the theft insurer also receives 1% of claims on average, because the risk of theft is independent across
households, the portfolio is much less risky.
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