Portfolio Theory & Practice

Modern Portfolio Theory I: risks, returns, investors’ risk preferences, asset allocation, efficient diversification, & portfolio optimisation.

Disclaimer: Content is summarised from Part 2 (Chapter 5 to 8) of 'Investments' 11th Edition by Zvi Bodie, Alex Kane & Alan J. Marcus

Contents:

5. Risk, Return & Historical Record • Determinants of Level of Interest Rates • Comparing Rates of Return for Different Holding Periods • Bills & Inflation, 1926 to 2015 • Risk & Risk Premiums • Time Series Analysis of Past Rates of Return • Normal Distribution

• Deviations from Normality & Alternative Risk Measures

• Historic Returns on Risky Portfolios

• Normality & Long-Term Investments

• Summary

6. Capital Allocation to Risky Assets • Risk & Risk Aversion • Capital Allocation across Risky & Risk-Free Portfolios • Risk-Free Asset • Portfolios of 1 Risky Asset & 1 Risk-Free Asset • Risk Tolerance & Asset Allocation • Passive Strategies: Capital Market Line

• Summary

5. Risk, Return & Historical Record

5.1 Determinants of Level of Interest Rates

Increased i/r tends to be bad news for longer term fixed-income securities & stock market. Thus, superior technique to forecast i/r would be of immense value to an investor attempting to determine best asset allocation for portfolio. But, forecasting i/r is a notoriously difficult part of applied macro.

Real & Nominal Rates of Interest

I/r: promised rate of return denominated in some unit of account (dollars, yen, euros / purchasing power units) over a time period (month / year / 20 years / longer). When we say i/r is 5%, we must specify unit of account & time period. Even if an i/r is risk-free for 1 unit of account & time period, it will not be risk-free for other units / periods. E.g. i/r that are absolutely safe in dollar terms will be risky in terms of purchasing power due to inflation uncertainty. E.g. Consider a (nominal) risk-free i/r. 1 year ago deposited $1,000 in a 1-year bank deposit guaranteeing a rate of interest of 10%. About to collect $1,100 in cash. What is real return on investment? It depends on what your money can buy today relative to what you could buy a year ago. Consumer Price Index (CPI) measures purchasing power by averaging prices of goods & services in consumption basket of an average urban family of 4. Suppose rate of inflation (% change in CPI; i) is at i = 6%. A loaf of bread that cost $1 last year might cost $1.06 this year. Rate at which your purchasing power has increased is thus 3.8%. Part of interest earnings have been offset by reduction in purchasing power. Nominal i/r: growth rate of your money. Real i/r: growth rate of your purchasing power. Real i/r is nominal i/r reduced by loss of purchasing power due to inflation

Conventional fixed income investments e.g. bank certificates of deposit, promise a nominal i/r. But since future inflation is uncertain, real rate of return that you will earn is risky even if nominal rate is risk-free. Thus you can only infer expected real rate on these investments by adjusting nominal rate for your expectation of rate of inflation.

Equilibrium Real I/r

I/r tend to move together, so economists talk as if there were a single representative rate. 3 basic factors that determine i/r: supply, demand & government actions. Nominal i/r: real rate + expected rate of inflation.

Figure 5.1 shows a downward-sloping demand curve & an upward-sloping supply curve. Horizontal axis: quantity of funds. Vertical axis: real i/r. Supply curve slopes up: higher real i/r, greater supply of household savings. (Assuming at higher real i/r, households will choose to postpone some current consumption & set aside / invest more of disposable income for future use) Demand curve slopes down: lower real i/r, more businesses will want to invest in physical capital. (Assuming businesses rank projects by expected real return on invested capital, firms will undertake more projects for lower real i/r on funds needed to finance those projects) Equilibrium: point of intersection, point E. Government & central bank (Federal Reserve) can shift these curves either to right / left via fiscal & monetary policies. E.g. an increase in government’s budget deficit leads to increased government borrowing demand & shifts demand curve right, causing equilibrium real i/r to rise to point E′. Fed can offset such a rise via an expansionary monetary policy to shift supply curve right.

Equilibrium Nominal I/r

Nominal i/r on an asset is ~ = real i/r + inflation. Higher nominal i/r is necessary to maintain expected real return from on asset especially with high inflation. Irving Fisher (1930) argued that nominal i/r ought to increase 1-for-1 with expected inflation, E(i).

Fisher hypothesis: rnom = rreal + E(i)

This implies that when real rates are stable, changes in nominal rates ought to predict changes in inflation rates. This has been debated & empirically investigated with mixed results. It is difficult to definitively test this as equilibrium real rate also changes unpredictably over time. Although data does not strongly support it, nominal i/r seem to predict inflation as well as alternative methods, in part since we are unable to forecast inflation well with any method.

5.2 Computing Rates of Return for Different Holding Periods

Investor seeks a safe investment in U.S. Treasury securities. We observe zero-coupon Treasury securities with several different maturities. Zero-coupon bonds are sold at discount from par value & provide their entire return from difference between purchase price & ultimate repayment of par value. Risk-free rate given price, P(T) of a Treasury bond with $100 par value & maturity of T years:

E.g. 5.2 Annualised rates of return: longer horizons will provide greater total returns

Comparing returns on investments with differing horizons requires that we express each total return as a rate of return for a common period. Express all investment returns as an effective annual rate (EAR): % increase in funds invested over a 1-year horizon. For 1-year investment: EAR equals total return, rf(1) & gross return (1 + EAR), is terminal value of a $1 investment. For investments less than 1 year, we compound per-period return for a full year. 6-month bill in e.g. 5.2: compound 2.71% half-year returns over 2 semi-annual periods for terminal value of 1 + EAR = (1.0271)^2 = 1.0549, implying EAR = 5.49%.

For investments longer than a year, express EAR as annual rate that would compound to same value as actual investment. 25-year bond in e.g. 5.2 grows by its maturity by a factor of 1 + 3.2918 = 4.2918. EAR is found by solving

Thus

Annual % Rates

Annualised rates on short-term investments (T < 1 year) often are reported using simple rather than compound interest: annual percentage rates (APRs). E.g. APR corresponding to a monthly rate e.g. charged on a credit card is reported as 12 times the monthly rate. If there are 'n' compounding periods per year & per-period rate is rf (T), then APR = n * rf (T). APR of 6-month bond in e.g. 5.2 with a 6-month rate of 2.71% = 2 * 2.71 = 5.42%. For short-term investments of length T, there are n = 1/T compounding periods in a year. Thus, relationship among compounding period, EAR & APR is:

Continuous Compounding (CC)

Difference between APR & EAR grows with frequency of compounding. How far will these 2 rates diverge as compounding frequency grows? What is the limit of [1 + T × APR]^1/ T, as T gets ever smaller? As T approaches zero, we effectively approach continuous compounding (CC), & relation of EAR to APR, denoted by rcc for CC case, is given by exponential function:

Working with such rates like CC can sometimes simplify calculations of expected return & risk. E.g. given a CC rate, total return for any period T, rcc(T), is simply exp(T * rcc ). Total return scales up in proportion to time period, T. This is far simpler than working with exponents that arise using discrete period compounding.

5.3 Bills & Inflation, 1926 to 2015

Table 5.2 summarises history of returns on 1-month U.S. Treasury bills, inflation rate & resultant real rate. Average i/r over more recent portion of history, 1952 to 2015, 4.45%, was higher than earlier due to inflation, main driver of T-bill rates, which also had a higher average value, 3.53%. But, nominal i/r in recent period were still high enough to leave a higher average real rate, 0.90%.

Figure 5.2 shows why we divide sample period at 1952. After that year, inflation is far less volatile & likely as a result, nominal i/r tracks inflation rate with far greater precision, resulting in a far more stable real i/r. This shows up as dramatic reduction in standard deviation of real rate seen at Table 5.2. Lower standard deviation of real rate post-1952 reflects a similar decline in standard deviation of inflation rate. Conclude: Fisher relation appears to work far better when inflation is itself more predictable & investors can more accurately gauge nominal i/r needed to provide an acceptable real rate of return.

5.4 Risk & Risk Premiums

Holding-Period Returns

E.g. investing in a stock-index fund selling for $100 per share. Investment horizon of 1 year, realised rate of return on investment depends on: (a) Price per share at year’s end & (b)Cash dividends collected over the year. If price per share at year’s end is $110 & cash dividends is $4. Realised return; holding-period return (HPR) is 14% (holding period for this is 1 year)

This definition of HPR treats dividend as paid at end of holding period. When dividends are received earlier, HPR should account for reinvestment income between receipt of payment & end of holding period. % return from dividends is dividend yield.

Expected Return & Standard Deviation

You cannot be sure about eventual HPR due to uncertainty of price of share + dividend yield.

Quantify beliefs about state of market & stock-index fund in terms of 4 possible scenarios, with probabilities presented in columns A to E of Spreadsheet 5.1. Characterise probability distributions of rates of return by expected / mean return, E(r) & standard deviation, σ. Expected rate of return is a probability-weighted average of rates of return in each scenario. Calling p(s) probability of each scenario & r(s) the HPR in each scenario, where scenarios are labeled / “indexed” by s, we write the expected return as

Expected rate of return, E(r) = (0.25 * 0.31) + (0.45 * 0.14) + [0.25 * (-0.0675)] + [0.05 * (-0.52)] = 0.0976

Variance of rate of return (σ^2 ) is a measure of volatility. Volatility is reflected in deviations of actual returns from mean return. To prevent positive deviations from canceling out with negative deviations, calculate expected value of squared deviations from expected return. Higher dispersion of outcomes, higher the average value of these squared deviations.

σ^2 = 0.0380. From variance, get back to original units by square rooting for standard deviation, calculated in G14 as 0.1949 = 19.49%. Downside risk of a crash / poor market is what troubles potential investors but standard deviation of rate of return does not distinguish between good / bad surprises; treating both as deviations from mean. Still, if probability distribution is more or less symmetric about mean, σ is a reasonable measure of risk. With a normal probability distribution (bell-shaped curve) E(r) & σ completely characterise the distribution.

Excess Returns & Risk Premiums

To decide on investing in the index fund, first find how much of an expected reward is offered for risk involved.

Risk premium (on common stocks): difference between expected HPR on index stock fund & risk-free rate (rate earned in risk-free assets like T-bills / money market funds / bank) E.g. risk-free rate = 4% per year & expected index fund return = 9.76% thus risk premium on stocks = 5.76% per year.

Excess return: difference between actual rate of return on a risky asset & actual riskfree rate. Thus, risk premium is expected value of excess return & standard deviation of excess return is a measure of its risk. Expected rate on a risky asset = risk-free rate + a risk premium.

Degree to which investors are willing to commit funds to stocks depends on risk aversion. Investors are risk averse in the sense that, if risk premium = 0, they would not invest any money in stocks.

General rule: when evaluating risk premium, maturity of risk-free rate should match the investment horizon. Investors with long maturities view rate on long-term safe bonds as providing benchmark risk-free rate. Thus, for a long-term investment, begin with relevant real, risk-free rate. But in practice excess returns are usually stated relative to 1-month T-bill rates as most discussions refer to short-term investments.

Though scenario analysis in Spreadsheet 5.1 illustrates concepts behind quantification of risk & return, to get a more realistic estimate of E(r) & σ for common stocks & other securities, we look to history. Analysis of historical record of portfolio returns makes use of statistical concepts & tools & so we first turn to a preparatory discussion.

5.5 Time Series Analysis of Past Rates of Return

Time Series vs Scenario Analysis

In a forward-looking scenario analysis, determine a set of relevant scenarios & associated investment rates of return, assign probabilities to each & compute risk premium (reward) & standard deviation (risk) of proposed investment.

Whereas, asset return histories come in form of time series of realised returns that do not explicitly provide investors’ original assessments of probabilities of those returns; only observe dates & associated HPRs. We must infer from this limited data probability distributions or at least expected return & standard deviation.

Expected Returns & Arithmetic Average

When we use historical data, we treat each observation as an equally likely “scenario.”

With n observations, we sub equal probabilities of 1/n for each p(s) in Equation 5.11:

Spreadsheet 5.2 presents a time series of holdingperiod returns for S&P 500 index over a 5-year period. We treat each HPR of n = 5 observations as equally likely annual outcome during sample years & assign equal probability of 1/5. Column B thus uses 0.2 as probabilities & Column C shows annual HPRs. Applying Equation 5.13 to time series shows that adding up products of probability & HPR amounts to taking arithmetic average of HPRs (compare cells C7 & C8). This illustrates logic for wide use of arithmetic average in investments. If time series of historical returns fairly represents true underlying probability distribution, then arithmetic average return from a historical period provides a forecast of investment’s expected future HPR.

Geometric (Time-Weighted) Average Return

Arithmetic average provides unbiased estimate of expected future return. Column Fshows investor’s “wealth index” from investing $1 in S&P 500 index fund at first year. Wealth in each year increases by “gross return”; multiple (1 + HPR), in column E. Wealth index is cumulative value of $1 invested. Value of wealth index at end of fifth year, $1.0275, is terminal value of $1 investment, which implies a 5-year holding-period return of 2.75%. Intuitive measure of performance over sample period is (fixed) annual HPR that would compound over period to same terminal value obtained from sequence of actual returns in the time series. Denote this rate by g, so that

Practitioners call g "time-weighted" average return to emphasise that each past return receives an equal weight in process of averaging. This is important as investment managers often experience significant changes in funds under management as investors purchase / redeem shares. Rates of return obtained during periods when fund is large have a greater impact on final value than rates obtained when fund is small. Geometric average return in Spreadsheet 5.2, 0.54%, is less than arithmetic average, 2.1%. Greater volatility in rates of return, the greater discrepancy between arithmetic & geometric averages. If returns come from a normal distribution, expected difference is half variance of distribution, that is:

When returns are approximately normal, Equation 5.15 is a good approximation. Discrepancy between geometric & arithmetic average arises from asymmetric effect of positive & negative rates of returns on terminal value of portfolio. Example 5.7 illustrates this:

Rate of return of -20% in year 1 & 20% in year 2. Arithmetic average is 0. Yet each dollar invested for two years will grow to only 0.80 * 1.20 = $.96, implying a negative geometric average return. Positive HPR in year 2 is applied to a smaller investment base than negative HPR incurred in year 1. To break even, you needed a rate of return of 25% in year 2. If order of rates of return were switched, (HPRs were 20% in year 1 & -20% in year 2), arithmetic average return is 0, but still end up with $.96 (1.20 * 0.80). Loss in year 2 is applied to a bigger investment base than gain in year 1, resulting in a larger dollar loss. In either case, geometric average is less than arithmetic one.

Variance & Standard Deviation

Likelihood of deviations of actual outcomes from expected return is the concern for risk. Variance is calculated by averaging squared deviations from estimate of expected return, arithmetic average.

Using historical data with n observations, we estimate variance as:

Spreadsheet 5.2 Column D shows squared deviation from arithmetic average & D9 gives variance as 0.0315. This is sum of product of probability of each outcome across 5 observations (as these are historical estimates, we take each outcome as equally likely) times squared deviation of that outcome. Standard deviation given in D10, 0.1774, is square root of variance.

Variance estimate from Equation 5.16 is biased downward, however. As we have taken deviations from sample arithmetic average, instead of unknown, true expected value, E(r), & so have introduced an estimation error. Its effect on estimated variance is aka degrees of freedom bias. We can eliminate bias by multiplying arithmetic average of squared deviations by factor n /(n - 1). Variance & standard deviation become:

D13 shows unbiased estimate of standard deviation, 0.1983, which is higher 0.1774 obtained in D11. For large samples, distinction is usually not important: n/(n - 1) is close to 1 & adjustment for degrees of freedom is trivially small.

Mean & Standard Deviation Estimates from Higher-Frequency Observations

Observation frequency has no impact on accuracy of estimates of expected return. Rather, duration of a sample time series improves accuracy. E.g. longer sample, a 100-year return, provides a more accurate estimate of mean return than a 10-year return, provided probability distribution of returns remains unchanged over 100 years. Thus, use longest sample that you believe comes from same return distribution. But in practice, old data may be less informative. Return data from 19th century may not be relevant, implying we face severe limits to accuracy of our estimates of mean returns.

Accuracy of estimates of standard deviation can be made more precise by increasing number of observations as more frequent observations give us more info about distribution of deviations from average. Estimates of standard deviation begin with variance. When monthly returns are uncorrelated from one month to another, monthly variances simply add up. Thus, when variance is same every month, variance of annual returns is 12 times variance of monthly returns: σsquared (annual) = 12σsquared (monthly) In general, T-month variance is T times 1-month variance. Hence, standard deviation grows at rate of √T; e.g. σ(annual) = √12 σ(monthly). While mean & variance grow in direct proportion to time, SD grows at rate of square root of time.

Reward-to-Volatility (Sharpe) Ratio

Investors price risky assets so that risk premium will be commensurate with risk of that expected excess return & hence it is best to measure risk by standard deviation of excess, not total, returns. Importance of trade-off between reward (risk premium) & risk (measured by SD) suggests we measure attraction of a portfolio by ratio of its risk premium to SD of excess returns (Sharpe Ratio).

Spreadsheet 5.1: scenario analysis for proposed investment resulted in a risk premium of 5.76% & SD of excess returns of 19.49%. Sharpe ratio = 0.30, a value roughly in line with historical performance of stock-index funds. Sharpe ratio is an adequate measure of risk-return trade-off for diversified portfolios, but inadequate when applied to individual assets like shares of stock.

5.6 Normal Distribution

As rates of return are affected by a multiplicity of unanticipated factors, they might be expected to be at least ~ normally distributed as midrange outcomes are far more likely than extremely good / extremely bad outcomes.

Figure 5.3 shows a normal curve with mean of 10% & SD of 20%. Normal distribution is completely characterised by mean & SD. Investment management is far more tractable when rates of return can be well approximated by normal distribution. Normal distribution is symmetric (probability of any positive deviation above mean is = negative deviation of same magnitude). Normal distribution belongs to a special family of distributions characterised as “stable” since assets with normally distributed returns mixed for a portfolio, makes a portfolio return normally distributed too. Scenario analysis is greatly simplified when only mean & SD need to be estimated for probabilities of future scenarios. When constructing portfolios of securities, we must account for statistical dependence of returns across securities. Generally, such dependence is a complex, multilayered relationship but when securities are normally distributed, statistical relation between returns can be summarised with a single correlation coefficient. We must identify criteria to determine the adequacy of the normality assumption for rates of return.

5.7 Deviations from Normality & Alternative Risk Measures

Normality of excess returns hugely simplifies portfolio selection. But, deviations from normality are potentially significant & dangerous to ignore. Statisticians often characterise probability distributions by “moments” & deviations from normality by calculating higher moments of return distributions. nth central moment of a distribution of excess returns, R, is estimated as average value of (R - ¯R)^n . First moment (n = 1) is 0 (average deviation from sample average is 0). Second moment (n = 2) is estimate of variance of returns, σ^2. Measure of asymmetry, "skew" is ratio of average cubed deviations from sample average, the third moment, to cubed standard deviation:

Cubing deviations maintains +/- sign.

When a distribution is “skewed right” as is dark curve in Figure 5.4A, extreme positive values, when cubed, dominate third moment, resulting in a positive skew. When distribution is “skewed left” cubed extreme negative values dominate & skew is negative. Positively skewed distribution's SD overestimates risk, as extreme positive surprises (which do not concern investors) still increase estimate of volatility. Negatively skewed distribution's SD will underestimate risk.

Kurtosis is a deviation from normality that concerns likelihood of extreme values on either side of mean at expense of a smaller likelihood of moderate deviations. When tails of a distribution are “fat” there is more probability at expense of “slender shoulders”: less probability mass near center of distribution.

Figure 5.4B superimposes a “fattailed” distribution. Although symmetry is preserved, SD will underestimate likelihood of extreme events. Kurtosis measures degree of fat tails. Deviations from average raised to fourth moment, scaled by fourth power of SD:

We subtract 3 in as expected value of ratio for a normal distribution is 3. Thus, this formula for kurtosis uses normal distribution as a benchmark: Kurtosis for normal distribution is, in effect, defined as 0, so kurtosis above 0 is a sign of fatter tails. Kurtosis of distribution in Figure 5.4B, is 0.35. Notice that skew & kurtosis are pure numbers. They do not change when annualised from higher frequency observations. Higher frequency of extreme negative returns may result from negative skew &/or kurtosis. Thus, we would like a risk measure that indicates vulnerability to extreme negative returns.

Value at Risk (VaR)

VaR is loss corresponding to a very low percentile of entire return distribution, e.g. 5th or 1st percentile return. VaR is written into regulation of banks & closely watched by risk managers. Aka quantile, q, of a distribution. q of a distribution is value below which lie q% of possible values. Median is q = 50th quantile. Practitioners commonly estimate 1% VaR, meaning that 99% of returns will exceed the VaR, & 1% of returns will be worse. Thus, 1% VaR may be viewed as cut-off separating 1% worst-case future scenarios from rest of distribution. When portfolio returns are normally distributed, VaR is fully determined by mean & SD. -2.33 is first percentile of standard normal distribution (mean = 0 & SD = 1), VaR for a normal distribution is:

VaR(1%, normal) = Mean - 2.33SD

For a sample estimate of VaR, we sort observations from high to low. VaR is return at 1st percentile of sample distribution.

Expected Shortfall

When we assess tail risk by looking at 1% worst-case scenarios, VaR is most optimistic measure of bad-case outcomes as it takes highest return (smallest loss) of all these cases:it tells you investment loss at first percentile of return distribution, but ignores magnitudes of potential losses even further out in the tail. A more informative view of downside exposure would focus on expected loss if in one of the worst-case scenarios. This value has 2 names: expected shortfall (ES) / conditional tail expectation (CTE); CTE emphasises that this expectation is conditioned on being in left tail of distribution. ES is more commonly used terminology. Using a sample of historical returns, we would estimate 1% expected shortfall by identifying worst 1% of all observations & taking their average.

Lower Partial Standard Deviation (LPSD) & Sortino Ratio

Use of SD as measure of risk when return distribution is non-normal presents 2 problems: (1) Asymmetry of distribution suggests we should look at negative outcomes separately & (2) An alternative to a risky portfolio is a risk-free investment, hence we should look at deviations of returns from risk-free rate rather than from sample average, that is, at negative excess returns.

LPSD is a risk measure that addresses these issues of excess returns, which is computed like usual SD, but using only “bad” returns. It uses only negative deviations from risk-free rate (rather than negative deviations from sample average), squares those deviations to obtain an analog to variance & takes square root to obtain a “left-tail SD.” LPSD is thus square root of average squared deviation, conditional on a negative excess return. This measure ignores frequency of negative excess returns. Practitioners who replace SD with LPSD typically also replace Sharpe ratio with ratio of average excess returns to LPSD; "Sortino ratio".

Relative Frequency of Large, Negative 3-Sigma Returns

Concentrate on relative frequency of large, negative returns compared with those frequencies in a normal distribution with same mean & SD. Extreme returns are "jumps". We compare fraction of observations with returns 3 or more SD below mean to relative frequency of negative 3-sigma returns in corresponding normal distribution. 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 relative frequency of returns that are 3 SD or more below mean in a standard normal distribution is only 0.13%. Thus in a small sample, it is hard to obtain a representative outcome, one that reflects true statistical expectations of extreme events.

5.8 Historic Returns on Risky Portfolios

T-bills are considered least risky of all assets. Essentially no risk that U.S. government will fail to honour its commitments to investors & their short maturities means that prices are relatively stable. Long-term U.S. Treasury bonds are certain to be repaid, but prices of these bonds fluctuate as i/r vary, thus imposing meaningful risk. Common stocks are riskiest of 3 groups of securities. Return depends on success / failure of firm. Our stock portfolio is broadest possible U.S. equity portfolio, including all stocks listed on NYSE, AMEX & NASDAQ, denoted as “U.S. market index”. As larger firms play a greater role in economy, this index is a valueweighted portfolio & thus dominated by large-firm corporate sector.  Monthly data series include excess returns on these stocks from July 1926 to June 2016. Annual return series comprise full-year returns from 1927 to 2015.

Figure 5.5 is a frequency distribution of annual returns on the 3 portfolios. Greater volatility of stock returns is immediately apparent. Spread of T-bill distribution does not reflect risk but rather changes in risk-free rate over time. Anyone buying a T-bill knows exactly (nominal) return when bill matures, so variation in return is not a reflection of risk over that short holding period. While frequency distribution is a handy visual representation of investment risk, we also need to quantify that volatility; this is provided by SD of returns.

Table 5.3 shows that SD of return on stocks over this period, 20.28%, was about double that of T-bonds, 10.02% & more than 6 times that of T-bills. That greater risk brought with it greater reward. Excess return on stocks (return in excess of T-bill rate) averaged 8.30% per year, providing a generous risk premium to equity investors. A fairly long sample period is used to estimate average level of risk & reward. While averages may be useful indications of what to expect, we still should expect risk & expected return to fluctuate over time.

Figure 5.6 plots SD of market’s excess return calculated from 12 most recent monthly returns. While market risk clearly ebbs & flows, aside from abnormally high values during Great Depression, there is no obvious trend. This gives us confidence that historical risk estimates provide useful guidance about the future. Unless returns are normally distributed, SD is not sufficient to measure risk. We also need to think about “tail risk”: exposure to unlikely but very large outcomes in left tail.

Figure 5.7 provides frequency distribution of monthly excess returns on market index since 1926. First bar shows historical frequency of excess returns falling within each range, while second bar shows frequencies that we would observe if these returns followed a normal distribution with same mean & variance as actual empirical distribution. Some evidence of a fat-tailed distribution: actual frequencies of extreme returns, high & low, are higher than would be predicted by the normal distribution.

Further evidence on distribution of excess equity returns is given in Table 5.4. We use monthly data on market index & for comparison, several “style” portfolios. Style is defined by: size (large cap / small cap firms) & value vs. growth. Firms with high ratios of market value to book value are “growth firms” (to justify high prices relative to current book values, market must anticipate rapid growth).

Eugene Fama & Kenneth French, extensively documented that firm size & book value-to-market value ratio predict average returns; these patterns have since been corroborated in stock exchanges. High book-to-market (B/M) ratio: value of firm is driven primarily by assets already in place, rather than prospect of high future growth. These are “value” firms. Low book-to-market ratio: typical of firms whose market value derives mostly from ample growth opportunities. Realised average returns, ceteris paribus, historically are higher for value firms than for growth firms & small firms than for large ones. Fama-French database includes returns on portfolios of U.S. stocks sorted by size (Big; Small) & by B/M ratios (High; Medium; Low).

Panel A presents results using monthly data for full sample period. Broad market index outperformed T-bills by an average of 8.30% per year, with a standard deviation of 18.64%, thus Sharpe ratio of 8.30/18.64 = 0.45. In line with Fama-French analysis, small/value firms had highest average excess return & best risk-return trade-off with a Sharpe ratio of 0.55. But, Figure 5.5 warns us that actual returns may have fatter tails than normal distribution, thus consider risk measures beyond just SD.

Some measures do not show meaningful departures from symmetric normal distribution: Skew is generally near zero. LPSD is generally quite close to conventional SD. Actual 1% VaR of these portfolios are uniformly higher than 1% VaR predicted from normal distributions with same means & SDs, differences between empirical & predicted VaR statistics are not large.

But, there is other evidence suggesting fat tails. Kurtosis is uniformly high. But, these portfolios suggest that left tail is over-represented compared to the normal. If excess returns were normally distributed, then only 0.13% of them would fall more than 3 SD below mean. In fact, actual incidence of excess returns below that cutoff are at least a few multiples of 0.13% for each portfolio.

ES estimates show why VaR is only an incomplete measure of downside risk. ES in Table 5.4 is average excess return of those observations that fall in extreme left tail, specifically, those that fall below 1% VaR. By definition, this value must be worse than VaR, as it averages among all returns that are below the 1% cutoff. As it uses actual returns of “worst-case outcomes” ES is by far a better indicator of exposure to extreme events. Figure 5.2 showed that post-war years (after 1951) are far more predictable, for i/r. Thus it may be instructive to examine stock returns in post-1951 period to see if risk & return characteristics for equity investments have changed meaningfully in more recent period. Relevant stats are given in Panel B of Table 5.4. Due to history of inflation & i/r, more recent period are less risky. SD for all 5 portfolios is noticeably lower in recent years & kurtosis drops dramatically. VaR also falls. While number of excess returns that are more than 3 SD below mean changes inconsistently, as SD is lower in this period, those negative returns are also less dramatic: ES generally is lower in latter period. Frequency distribution in Figure 5.5 & stats in Table 5.4 for market index & style portfolios tell a reasonably consistent story. There is some, admittedly inconsistent, evidence of fat tails, so investors should not take normality for granted. Extreme returns are quite uncommon, especially in more recent years. Incidence of returns on market index post-1951 that are worse than 3 SD below mean is 0.66%, compared to a prediction of 0.13% for normal distribution. “Excess” rate of extreme bad outcomes is thus only 0.53%, once in 187 months. It is not unreasonable to accept simplification offered by normality as an acceptable approximation as we think about constructing & evaluating our portfolios.

A Global View of the Historical Record

As financial markets grow & become more transparent, U.S. investors look to improve diversification by investing internationally.

Figure 5.8 shows average excess returns in 20 stock markets. Mean annual excess return across them was 7.40% & median was 6.50%. US is roughly in the middle. SD of returns in U.S. (not shown) was just a shade below median volatility. U.S. performance has been quite consistent with international experience. We have seen that there is tremendous variability in year-by year returns & this makes even long-term average performance a very noisy estimate of future returns. There is an ongoing debate if historical U.S. average riskpremium of large stocks over T-bills of 8.30% is a reasonable forecast for long term: (a) Do economic factors that prevailed over 1926 to 2015 adequately represent those in the future? (b) Is average performance from available history a good estimate for long-term forecasts?

5.9 Normality & Long-Term Investments

From historical experience, you may treat short-term stock returns as ~ symmetric normal distribution. But longterm performance cannot be normal. If r1 & r2 are returns in 2 periods, & have same normal distribution, sum of returns (r1 + r2) would be normal. But 2-period compound return (1 + r1)(1 + r2), is not normal. Shape of distribution changes noticeably as investment horizon extends.

E.g. $100 in a stock with an expected monthly return of 1%. Dispersion around expected value: With equal probability, actual return in any month will exceed mean by 2% / fall short by 2%. These 2 possible monthly returns, -1% / 3%, are thus symmetrically distributed around 1% mean.

Figure 5.9 shows distribution of portfolio value at end of several investment horizons. After 6 months (Panel A), distribution of possible values is beginning to take on shape of familiar bell-shaped curve. After 20 months (Panel B), bell-shaped distribution is even more obvious, but with a hint that extremely good cumulative returns (possible prices extending to $180) are more prevalent than extremely poor ones (worst possible outcome is only $82). After 40 months (Panel C), asymmetry in distribution is pronounced. This pattern emerges from compounding. Upside potential is unlimited. No matter how many months in a row you lose 1%, funds cannot drop below zero, so there is a floor on worst-possible performance: cannot lose more than 100% of investment.

While probability distribution in Figure 5.9 is bell-shaped, it is a distinctly “asymmetric bell” with a positive skew & distribution is clearly not normal. In fact, actual distribution approaches lognormal distribution. “Lognormal”: log of final portfolio value, ln(WT) is normally distributed. Instead of using effective annual returns, use continuously compounded returns.

If continuously compounded returns in 2 months are rcc(1) & rcc(2), invested funds grow by a factor of exp[rcc(1)] in first month & exp[rcc(2)] in second month, where exp( ) is exponential function. Total growth in investment is thus exp[rcc(1)] * exp[rcc(2)] = exp[rcc(1) + rcc(2)]. Total 2-month return expressed as a continuously compounded rate is sum of one-month returns. Thus, if monthly returns are normal, then multi-month returns will also be normal. Thus, by using continuously compounded rates, even long-term returns can be described by normal distribution. If returns have same distribution in each month, expected 2-month return is just twice expected one-month return. If returns are uncorrelated, then variance of 2-month return is also twice one-month return.

We can generalise from this example to arbitrary investment horizons, T. If expected per-period continuously compounded return is E(rcc), expected cumulative return after an investment of T periods is E(rcc)T, & expected final value of portfolio is E(WT) = W0 exp[E(rcc)T]. Variance of cumulative return is proportional to time horizon: Var(rccT) = TVar(rcc). Thus, SD grows in proportion to square root of time horizon:

Short-Run vs Long-Run Risk

Results on risk & return of investments over different time horizons appear to offer a mitigation of investment risk in long run: As expected return increases with horizon at a faster rate than SD, expected return of a long-term risky investment becomes larger relative to its SD. So are investments less risky when horizons are longer? 

E.g. 5.11 investment returns are independent from year to year with an expected continuously compounded rate of return 0.05 & SD of σ = 0.30.

Table 5.5 shows properties of the investment at horizons of 1, 10 & 30 years. Mean cumulative return rises in direct proportion to T. SD of cumulative return rises in proportion to square root of T. Probability that cumulative return will be negative: for 1-year investor to suffer a loss, actual return must be 0.05 below mean (Return is 0.05/0.3 = 0.167 SD below mean return). Assuming normality, probability is 0.434. For 10-year investor, mean cumulative continuously compounded return is 0.500 & SD is 0.949. Thus, investor will suffer losses only if actual 10-year return is 0.500/0.949 = 0.527 SD below expected value. Probability is only 0.299. Probability of a loss after 30 years is 0.181. As horizon expands, mean return increases faster than SD & thus probability of a loss steadily shrinks.

But e.g. 5.11 is misleading: probability of a shortfall is an incomplete measure of investment risk. Size of potential losses is unaccounted. Worst-case scenarios for 30-year investment are far worse than for 1-year investment: Table 5.5 shows that 1% VaR after 1 year entails a continuously compounded cumulative loss of 0.648, implying each dollar invested will fall to e-0.648 = 0.523. This value is “wealth relative” of investment (final value of portfolio as a fraction of initially invested funds). 1% VaR after 30 years implies a wealth relative of 0.098. Thus, while probability of losses falls as investment horizon lengthens, magnitude of potential losses grows.

Better way to quantify risk of a long-term investment: calculate market price of insuring it against a loss. Insurance premium takes into account probability of possible losses & magnitude of losses. Contrary to any intuition that a longer horizon reduces risk, value of portfolio insurance increases dramatically with investment horizon. These policies do not come cheap: for reasonable parameters, a 25-year policy would cost ~ 30% of initial portfolio value. A typical demonstration relies on fact that SD of annualised returns is lower for longer-term horizons. But the demonstration is silent on the range of total returns.

Forecasts for Long Haul

Arithmetic averages forecast future rates of return as they are unbiased estimates of expected rates over equivalent holding periods. Arithmetic average of short-term returns can be misleading when used to forecast long-term cumulative returns as sampling errors in estimate of expected return will have asymmetric impact when compounded over long periods. Positive sampling variation will compound to greater upward errors than negative variation.

Jacquier, Kane & Marcus show that an unbiased forecast of total return over long horizons requires compounding at a weighted average of arithmetic & geometric historical averages. Proper weight applied to geometric average equals ratio of length of forecast horizon to length of estimation period. E.g. to forecast cumulative return for a 25-year horizon from a 90-year history, unbiased estimate would be to compound at a rate of

This correction would take about 0.5% off historical arithmetic average risk premium on large stocks & ~2% off arithmetic average premium of small stocks. A forecast for next 90 years would require compounding at only geometric average & for longer horizons at an even lower number. Forecast horizons that are relevant for current investors would depend on their investment horizons.

5.10 Summary

1. Economy’s equilibrium level of real i/r depends on willingness of households to save, as reflected in supply curve of funds & on expected profitability of business investment in plant, equipment & inventories, as reflected in demand curve for funds. It depends also on government fiscal & monetary policy.

2. Nominal i/r is equilibrium real rate + expected rate of inflation. Generally, we directly observe nominal i/r; from them we infer expected real rates, using inflation forecasts. Assets with guaranteed nominal i/r are risky in real terms as future inflation rate is uncertain.

3. Equilibrium expected rate of return on any security is sum of equilibrium real rate of interest, expected rate of inflation & a security-specific risk premium.

4. Investors face a trade-off between risk & expected return. Historical data confirms that assets with low degrees of risk should provide lower returns on average.

5. Historical rates of return over the last century in other countries suggest U.S. history of stock returns is not an outlier.

6. Historical returns on stocks exhibit somewhat more frequent large negative deviations from mean than that predicted from a normal distribution. Lower partial standard deviation, skew & kurtosis of actual distribution quantify deviation from normality.

7. Widely used measures of tail risk are value at risk & expected shortfall or, equivalently, conditional tail expectations. VaR measures loss that will be exceeded with a specified probability like 1% / 5%. ES measures expected rate of return conditional on portfolio falling below a certain value. Thus, 1% ES is expected value of outcomes that lie in bottom 1% of distribution.

8. Investments in risky portfolios do not become safer in long run in fact, longer a risky investment is held, greater the risk. Basis of argument that stocks are safe in long run is fact that probability of an investment shortfall is smaller. But, probability of shortfall ignores magnitude of possible losses.

6. Capital Allocation to Risky Assets

6.1 Risk & Risk Aversion

Risk, Speculation & Gambling

One definition of speculation: “assumption of considerable investment risk to obtain commensurate gain.” “considerable risk”: risk sufficient to affect decision. “commensurate gain”: positive risk premium, that is, an expected return greater than risk-free alternative. Gamble is “to bet / wager on an uncertain outcome.” Central difference for gambling & speculation is lack of “commensurate gain.”

Gamble: assumption of risk for enjoyment of risk itself. Speculation: undertaken in spite of risk as one perceives a favourable risk-return trade-off. Speculative ventures require adequate risk premium. Risky investment with 0 risk premium, aka fair game, amounts to a gamble as there is no expected gain to compensate for risk.

In some cases, a gamble may appear to be speculation: Paul pays Mary $100 if value of £1 exceeds $1.40, Mary pays Paul if pound is worth less than $1.40. They assign p = 0.5 to each outcome. Expected profit to both is 0. But what is more likely is that they assign different probabilities to outcome. Mary assigns it p > 0.5, Paul’s assessment is p < 0.5. They perceive, subjectively, 2 different prospects. Economists call this “heterogeneous expectations”. Investors on each side of a financial position see themselves as speculating rather than gambling.