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ERIC WANG
Chapter5.pptx
5. Risk, return, and the historical record
Instructor: Seongcheol Paeng
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5.1 Determinants of the Level of Interest Rates
Unfortunately, forecasting interest rates is one of the most notoriously difficult parts of applied macroeconomics. Nonetheless, we do have a good understanding of the fundamental factors that determine the level of interest rates:
1. The supply of funds from savers, primarily households.
2. The demand for funds from businesses to be used to finance investments in plant, equipment, and inventories (real assets or capital formation).
3. The government’s net demand for funds as modified by actions of the Federal Reserve Bank.
4. The expected rate of inflation.
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5.1 Determinants of the Level of Interest Rates
Real and Nominal Rates of Interest
We need to distinguish between a nominal interest rate()—the growth rate of your money—and a real interest rate()—the growth rate of your purchasing power.
i: inflation rate
(5.1)
(5.2)
(5.3)
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5.1 Determinants of the Level of Interest Rates
Real and Nominal Rates of Interest
Ex 5.1) If the nominal interest rate on a 1-year CD is 8%, and you expect inflation to be 5% over the coming year, then using the approximation formula, you expect the real rate of interest to be
8% − 5% = 3%.
Using the exact formula
= (.08 − .05)/(1 + .05) = .0286, or 2.86%.
The Equilibrium Real Rate of Interest
Figure 5.1 shows a downward-sloping demand curve and an upward-sloping supply curve.
On the horizontal axis, we measure the quantity of funds, and on the vertical axis, we measure the real rate of interest.
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5.1 Determinants of the Level of Interest Rates
The Equilibrium Real Rate of Interest
The government and the central bank (the Federal Reserve) can shift these supply and demand curves either to the right or to the left through fiscal and monetary policies.
For example, consider an increase in the government’s budget deficit.
This increases the government’s borrowing demand and shifts the demand curve to the right, which causes the equilibrium real interest rate to rise to point E′.
The Fed can offset such a rise through an expansionary monetary policy, which will shift the supply curve to the right.
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5.1 Determinants of the Level of Interest Rates
The Equilibrium Nominal Rate of Interest
Irving Fisher (1930) argued that the nominal rate ought to increase one-for-one with expected inflation, E(i). The so-called Fisher hypothesis is
(5.4)
Taxes and the Real Rate of Interest
The real after-tax rate is approximately the after-tax nominal rate minus the inflation rate:
(5.5)
Equation 5.5 tells us that, because you pay taxes on even the portion of interest earnings that is merely compensation for inflation, your after-tax real return falls by the tax rate times the inflation rate.
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5.1 Determinants of the Level of Interest Rates
Taxes and the Real Rate of Interest
(5.5)
Ex) tax bracket: 30%, nominal return: 12%
Inflation rate: 8%, before-tax real rate: approximately 4%
after-tax real return: 4%(1 − .3) = 2.8%
But the tax code does not recognize that the first 8% of your return is only compensation for inflation—not real income. Your
after-tax nominal return: 12%(1 − .3) = 8.4%
after-tax real interest rate: 8.4% − 8% = .4%.
it = 8% × .3 = 2.4%.
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5.2 Comparing Rates of Return for Different Holding Periods
Given the price, P(T), of a Treasury bond with $100 par value and maturity of T years, we calculate the total risk-free return available for a horizon of T years as the percentage increase in the value of the investment.
(5.6)
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5.2 Comparing Rates of Return for Different Holding Periods
We typically express all investment returns as an effective annual rate (EAR), defined as the percentage increase in funds invested over a 1-year horizon.
For the 6-month bill in Example 5.2, we compound 2.71% half-year returns over two semiannual periods to obtain a terminal value of , implying that EAR = 5.49%.
For example, the investment in the 25-year bond in Example 5.2 grows by its maturity by a factor of 4.2918 (i.e., 1 + 3.2918), so its EAR is found by solving
In general, we can relate EAR to the total return, , over a holding period of length T by using the following equation:
(5.7)
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5.2 Comparing Rates of Return for Different Holding Periods
Annual Percentage Rates
Annualized rates on short-term investments (by convention, T < 1 year) often are reported using simple rather than compound interest. These are called annual percentage rates, or APRs.
(5.8)
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5.2 Comparing Rates of Return for Different Holding Periods
Continuous Compounding
As T approaches zero, we effectively approach continuous compounding (CC), and the relation of EAR to the APR, denoted by for the continuously compounded case, is given by the exponential function
(5.9)
where e is approximately 2.71828.
To find from the effective annual rate, we solve Equation 5.9 for as follows:
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Definition:
5.3 Bills and Inflation, 1926-2015
Table 5.2 summarizes the history of returns on 1-month U.S. Treasury bills, the inflation rate, and the resultant real rate. You can find the entire post-1926 history of the monthly rates of these series in Connect (link to the material for Chapter 5).
The first set of columns of Table 5.2 lists average annual rates for three periods. The average interest rate over the more recent portion of our history, 1952–2015 (essentially the post-war period), 4.45%, was noticeably higher than in the earlier portion, 1.04%.
The reason is inflation, the main driver of T-bill rates, which also had a noticeably higher average value, 3.53%, in the later portion of the sample than in the earlier period, 1.68%.
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5.3 Bills and Inflation, 1926-2015
Figure 5.2 shows why we divide the sample period at 1952.
After that year, inflation is far less volatile, and, probably as a result, the nominal interest rate tracks the inflation rate with far greater precision, resulting in a far more stable real interest rate.
This shows up as the dramatic reduction in the standard deviation of the real rate documented in the last column of Table 5.2.
Whereas the standard deviation is 6.27% in the early part of the sample, it is only 2.13% in the later portion.
The lower standard deviation of the real rate in the post-1952 period reflects a similar decline in the standard deviation of the inflation rate.
We conclude that the Fisher relation appears to work far better when inflation is itself more predictable and investors can more accurately gauge the nominal interest rate they require to provide an acceptable real rate of return.
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5.4 Risk and Risk Premiums
Holding-Period Returns
The realized return, called the holding-period return, or HPR (in this case, the holding period is 1 year), is defined as
. (5.10)
Expected Return and Standard Deviation
Expected Return() and Standard Deviation()
(5.11)
. (5.12)
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5.4 Risk and Risk Premiums
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5.4 Risk and Risk Premiums
Excess Returns and Risk Premiums
We measure the reward as the difference between the expected HPR on the index stock fund and the risk-free rate, that is, the rate you would earn in risk-free assets such as T-bills, money market funds, or the bank.
We call this difference the risk premium on common stocks.
HPR = risk-free rate + risk premium
The risk-free rate in our example is 4% per year, and the expected index fund return is 9.76%, so the risk premium on stocks is 5.76% per year.
The difference in any particular period between the actual rate of return on a risky asset and the actual risk-free rate is called the excess return.
Excess return = actual rate of return – actual risk-free rate
The degree to which investors are willing to commit funds to stocks depends on their risk aversion. Investors are risk averse in the sense that, if the risk premium were zero, they would not invest any money in stocks.
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5.5 Time Series Analysis of Past Rates of Return
Time Series versus Scenario Analysis
In a forward-looking scenario analysis, we determine a set of relevant scenarios and associated investment rates of return, assign probabilities to each, and conclude by computing the risk premium (reward) and standard deviation (risk) of the proposed investment.
In contrast, asset return histories come in the form of time series of realized returns that do not explicitly provide investors’ original assessments of the probabilities of those returns; we observe only dates and associated HPRs. We must infer from this limited data the probability distributions from which these returns might have been drawn or, at least, the expected return and standard deviation.
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5.5 Time Series Analysis of Past Rates of Return
Expected Returns and the Arithmetic Average
(5.13)
=Arithmetic Average of Historic Rates of Return
The Geometric (Time-Weighted) Average Return
Column F in Spreadsheet 5.2 shows the investor’s “wealth index” from investing $1 in an S&P 500 index fund at the beginning of the first year.
Wealth in each year increases by the “gross return,” that is, by the multiple (1 + HPR), shown in column E.
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5.5 Time Series Analysis of Past Rates of Return
The Geometric (Time-Weighted) Average Return
(cell F6 in Spreadsheet 5.2) (5.14)
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5.5 Time Series Analysis of Past Rates of Return
The Geometric (Time-Weighted) Average Return
Notice that the geometric average return in Spreadsheet 5.2, .54%, is less than the arithmetic average, 2.1%.
The greater the volatility in rates of return, the greater the discrepancy between arithmetic and geometric averages.
If returns come from a normal distribution, the expected difference is exactly half the variance of the distribution, that is,
. (5.15)
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5.5 Time Series Analysis of Past Rates of Return
Variance and Standard Deviation
When thinking about risk, we are interested in the likelihood of deviations of actual outcomes from the expected return.
. (5.16)
Where replaces to denote that it is an estimate.
The variance estimate from Equation 5.16 is biased downward, however.
The reason is that we have taken deviations from the sample arithmetic average, , instead of the unknown, true expected value, E(r), and so have introduced an estimation error.
Its effect on the estimated variance is sometimes called a degrees of freedom bias. We can eliminate the bias by multiplying the arithmetic average of squared deviations by the factor n/(n − 1).
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5.5 Time Series Analysis of Past Rates of Return
Variance and Standard Deviation
. (5.17)
Mean and Standard Deviation Estimates from Higher-Frequency Observations
Do more frequent observations lead to more accurate estimates? The answer to this question is surprising: Observation frequency has no impact on the accuracy of estimates of expected return. It is the duration of a sample time series (as opposed to the number of observations) that improves accuracy.
Unfortunately, in practice, old data may be less informative. Are return data from the 19th century relevant to estimating expected returns in the 21st century? Quite possibly not, implying that we face severe limits to the accuracy of our estimates of mean returns.
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5.5 Time Series Analysis of Past Rates of Return
Mean and Standard Deviation Estimates from Higher-Frequency Observations
In contrast to the mean, the accuracy of estimates of the standard deviation can be made more precise by increasing the number of observations. This is because the more frequent observations give us more information about the distribution of deviations from the average.
Consequently, standard deviation grows at the rate of ; for example, the standard deviation of annual returns is related to the standard deviation of monthly returns by . While the mean and variance grow in direct proportion to time, SD grows at the rate of square root of time.
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5.5 Time Series Analysis of Past Rates of Return
The Reward-to-Volatility (Sharpe) Ratio
The importance of the trade-off between reward (the risk premium) and risk (as measured by standard deviation or SD) suggests that we measure the attraction of a portfolio by the ratio of its risk premium to the SD of its excess returns.
This reward-to-volatility measure was first used extensively by William Sharpe and hence is commonly known as the Sharpe ratio.
It is widely used to evaluate the performance of investment managers.
Sharpe ratio = Risk premium / SD of excess return (5.18)
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5.6 The Normal Distribution
Figure 5.3 shows a graph of the normal curve with mean of 10% and standard deviation of 20%.
A smaller SD means that possible outcomes cluster more tightly around the mean, while a higher SD implies more diffuse distributions.
The likelihood of realizing any particular outcome when sampling from a normal distribution is fully determined by the number of standard deviations that separate that outcome from the mean.
Put differently, the normal distribution is completely characterized by two parameters, the mean and SD.
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5.7 Deviations from Normality and Alternative Risk Measures
A measure of asymmetry called skew is the ratio of the average cubed deviations from the sample average, called the third moment, to the cubed standard deviation:
(5.19)
Cubing deviations maintains their sign (the cube of a negative number is negative).
When a distribution is “skewed to the right,” as is the dark curve in Figure 5.4A, the extreme positive values, when cubed, dominate the third moment, resulting in a positive skew.
When a distribution is “skewed to the left,” the cubed extreme negative values dominate, and skew will be negative.
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5.7 Deviations from Normality and Alternative Risk Measures
Value at Risk
The value at risk (denoted VaR to distinguish it from Var, the abbreviation for variance) is the loss corresponding to a very low percentile of the entire return distribution, for example, the 5th or 1st percentile return.
Practitioners commonly estimate the 1% VaR, meaning that 99% of returns will exceed the VaR, and 1% of returns will be worse. Therefore, the 1% VaR may be viewed as the cut-off separating the 1% worst-case future scenarios from the rest of the distribution.
Expected Shortfall
This value, unfortunately, has two names: either expected shortfall (ES) or conditional tail expectation (CTE); the latter emphasizes that this expectation is conditioned on being in the left tail of the distribution.
ES is the more commonly used terminology. Using a sample of historical returns, we would estimate the 1% expected shortfall by identifying the worst 1% of all observations and taking their average.
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5.7 Deviations from Normality and Alternative Risk Measures
Another potentially important deviation from normality, kurtosis, concerns the likelihood of extreme values on either side of the mean at the expense of a smaller likelihood of moderate deviations.
Figure 5.4B superimposes a “fat-tailed” distribution on a normal distribution with the same mean and SD.
Although symmetry is still preserved, the SD will underestimate the likelihood of extreme events: large losses as well as large gains.
. (5.20)
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5.7 Deviations from Normality and Alternative Risk Measures
Lower Partial Standard Deviation and the Sortino Ratio
A risk measure that addresses these issues is the lower partial standard deviation (LPSD) of excess returns, which is computed like the usual standard deviation, but using only “bad” returns.
Practitioners who replace standard deviation with this LPSD typically also replace the Sharpe ratio (the ratio of average excess return to standard deviation) with the ratio of average excess returns to LPSD. This variant on the Sharpe ratio is called the Sortino ratio.
Relative Frequency of Large, Negative 3-Sigma Returns
This measure can be quite informative about downside risk but, in practice, is most useful for large samples observed at a high frequency.
Observe from Figure 5.3 that the relative frequency of returns that are 3 standard deviations or more below the mean in a standard normal distribution is only 0.13%, that is, 1.3 observations per 1,000.
Thus in a small sample, it is hard to obtain a representative outcome, one that reflects true statistical expectations of extreme events.
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5.8 Historic returns on risky portfolios
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5.8 Historic Returns on Risky Portfolios
The monthly data series include excess returns on these stocks from July 1926 to June 2016, a sample period spanning 90 years. The annual return series comprise full-year returns from 1927–2015.
Figure 5.5 is a frequency distribution of annual returns on these three portfolios.
The greater volatility of stock returns compared to T-bill or T-bond returns is immediately apparent.
Compared to stock returns, the distribution of T-bond returns is far more concentrated in the middle of the distribution, with far fewer outliers.
The distribution of T-bill returns is even tighter.
More to the point, the spread of the T-bill distribution does not reflect risk but rather changes in the risk-free rate over time.
Anyone buying a T-bill knows exactly what the (nominal) return will be when the bill matures, so variation in the return is not a reflection of risk over that short holding period.
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5.8 Historic Returns on Risky Portfolios
Table 5.3 shows that the standard deviation of the return on stocks over this period, 20.28%, was about double that of T-bonds, 10.02%, and more than 6 times that of T-bills.
Of course, that greater risk brought with it greater reward.
The excess return on stocks (i.e., the return in excess of the T-bill rate) averaged 8.30% per year, providing a generous risk premium to equity investors.
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5.8 Historic Returns on Risky Portfolios
Figure 5.6 plots the standard deviation of the market’s excess return in each year calculated from the 12 most recent monthly returns.
While market risk clearly ebbs and flows, aside from its abnormally high values during the Great Depression(1930s), there does not seem to be any obvious trend in its level.
This gives us more confidence that historical risk estimates provide useful guidance about the future.
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5.8 Historic Returns on Risky Portfolios
Figure 5.7 provides some evidence of this exposure. It shows a frequency distribution of monthly excess returns on the market index since 1926.
The first bar in each set shows the historical frequency of excess returns falling within each range, while the second bar shows the frequencies that we would observe if these returns followed a normal distribution with the same mean and variance as the actual empirical distribution.
You can see here some evidence of a fat-tailed distribution: The actual frequencies of extreme returns, both high and low, are higher than would be predicted by the normal distribution.
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5.8 Historic Returns on Risky Portfolios
Fama-French Model: High (Market Risk), Small (Size Risk), High (Value Risk) are good for excess return.
Following the Fama-French classifications, we drop the medium B/M portfolios and identify firms ranked in the top 30% of B/M ratio as “value firms” and firms ranked in the bottom 30% as “growth firms.”
We split firms into above and below median levels of market capitalization to establish subsamples of small versus large firms. We thus obtain four comparison portfolios: Big/Growth, Big/Value, Small/Growth, and Small/Value.
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5.8 Historic Returns on Risky Portfolios
Table 5.4, Panel A, presents results using monthly data for the full sample period, July 1926–June 2016.
The top two lines show the annualized average excess return and standard deviation of each portfolio.
The broad market index outperformed T-bills by an average of 8.30% per year, with a standard deviation of 18.64%, resulting in a Sharpe ratio (third line) of 8.30/18.64 = .45.
In line with the Fama-French analysis, small/value firms had the highest average excess return and the best risk–return trade-off with a Sharpe ratio of .55.
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5.8 Historic Returns on Risky Portfolios
Skew is generally near zero. No extreme values.
Finally, while the actual 1% VaR of these portfolios are uniformly higher than the 1% VaR that would be predicted from normal distributions with matched means and standard deviations, the differences between the empirical and predicted VaR statistics are not large.
However, there is other evidence suggesting fat tails in the return distributions of these portfolios.
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5.8 Historic Returns on Risky Portfolios
To begin, note that kurtosis (the measure of the “fatness” of both tails of the distribution) is uniformly high.
Unfortunately, these portfolios suggest that the left tail of the return distribution is overrepresented compared to the normal.
If excess returns were normally distributed, then only .13% of them would fall more than 3 standard deviations below the mean.
In fact, the actual incidence of excess returns below that cutoff are at least a few multiples of .13% for each portfolio.
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5.8 Historic Returns on Risky Portfolios
ES in Table 5.4 is the average excess return of those observations that fall in the extreme left tail, specifically, those that fall below the 1% VaR.
By definition, this value must be worse than the VaR, as it averages among all the returns that are below the 1% cutoff.
Because it uses the actual returns of the “worst-case outcomes,” ES is by far a better indicator of exposure to extreme events.
The relevant statistics are given in Panel B of Table 5.4. Perhaps not surprisingly in light of the history of inflation and interest rates, the more recent period is in fact less risky.
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5.8 Historic Returns on Risky Portfolios
Figure 5.8 shows a century-plus-long history (1900–2015) of average excess returns in 20 stock markets.
The mean annual excess return across these countries was 7.40% and the median was 6.50%.
The United States is roughly in the middle of the pack, with a historical risk premium of 7.55%. Similarly, the standard deviation of returns in the U.S. (not shown) was just a shade below the median volatility in these other countries.
So the U.S. performance has been pretty much consistent with international experience.
We might tentatively conclude that the characteristics of historical returns in the U.S. also can serve as a rough indication of the risk–return trade-off in a wider range of countries.
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Assignments
Problem Sets (Paraphrase with your own words.)
1. Explain Determinants of the Level of Interest Rates.
2. Explain the Equilibrium Real Rate of Interest
3. Tax bracket: 20%, nominal return: 10%, Inflation rate: 6%
before-tax real rate?
after-tax real return?
after-tax nominal return?
after-tax real interest rate?
?
?
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Assignments
Problem Sets (Paraphrase with your own words.)
4. Explain Sharpe Ratio.
5. Explain Normal Distribution.
6. Explain Skew and Kurtosis.
7. Explain Table 5.4 as much as you can.
Deadline: 7/3 (Friday)
Submit it via email to [email protected]
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