where A = investor's aversion to risk as measured by the variance of the portfolio return.

To maximize the function assuming the investor's assets are only in the market portfolio and the riskfree asset,

first let w_m = the fraction of assets in the market portfolio. Then

E(R_p) = r_F + w_m (R_m - r_F)

and

Var(R_p) = w²_m σ²_m.

Then

Z = r_F + w_m [E(R_m) - r_F] - 0.5 A w²_m σ²_m

and

dZ/dw_m = E(R_m) - r_F - A w_m σ²_m = 0

Solving for A,

A = [E(R_m) - r_F] / ( w_m σ²_m)

The risk of an individual security in a well diversified portfolio can be measured by its beta. Such risk is nondiversifiable.

Beta of an individual security with respect to the market is:

β_im = σ_im / σ²_m = Cov(R_i, R_SP500) / Var(R_SP500)

Beta of a risk-free asset with respect to the market = 0.

Betas determined using historical data are subject to estimation error. Merrill Lynch and some other firms adjust this value back towards the mean beta of the market (=1) or industry using

β_adjusted = w β_historical + (1-w) β_"true"

The lower the confidence in β_historical, the lower should be the value chosen for w.

Beta of a portfolio:

β_pm = Σ x_i β_im , where x_i is the weight.

One is willing to accept a lower return on a security or a portfolio having a negative beta since it can reduce the portfolio risk as part of a larger portfolio.

Efficient portfolios lie on the capital market line (CML). This CML is not a part of the CAPM. For this line to be used, there must be perfect correlation between the portfolio in question and the market portfolio. This implies that the line is only for those portfolios that are a combination of the tangential portfolio (usually the market portfolio) and the risk-free rate.

CML: E(R_p) = R_F + [ E(R_m) - R_F ] σ_p / σ_m

If borrowing is not permitted, the rational risk-averse investor will choose a portfolio along the capital market line up to the efficient frontier, and then follow the efficient frontier for levels of higher risk and return.

The variance and expected return of the market portfolio can be obtained by combining any two portfolios that lie on the efficient frontier and solving for the weights in the following expression:

E(R_m) = w₁ E(R₁) + (1-w₁) E(R₂)

The covariance between any two portfolios on the efficient frontier can be found by finding the weights needed to emulate the market portfolio and then solving for σ₁₂ in the following equation:

σ²_p = w²₁ σ²₁ + w²₂ σ²₂ + 2 w₁ w₂ σ₁₂

CAPM

The following is the security market line - any security's expected return will lie on this line. This line applies to all securities, not just efficient portfolios.

E(R_i) = R_F + [ E(R_m) - R_F ] β_im Sharpe-Lintner

E(R_i) = R_z + [ E(R_m) - R_z ] β_im Black

Expected return of a portfolio using CAPM:

E(R_p) = R_F + [ E(R_m) - R_F ] β_pm

If the assumption of equal borrowing and lending rates is relaxed, investors no longer are required to hold the market portfolio; instead, they can hold a range of portfolios along the efficient frontier between the point of tangency of the lending line and the point of tangency of the borrowing line.

CAPM requires the measure of two unknown quantities - market risk premium and beta. However, attempts to estimate expected returns by using historical stock return data have resulted in std errors about double those of CAPM, because for CAPM the better precision in the estimate of the market risk premium more than offsets the additional estimation error in beta.

There have been many difficulties in testing CAPM. Roll argued that the CAPM must always hold for ex post data if the proxy chosen for the market is efficient. He also argued that it is impossible to measure the true market, so the CAPM cannot be tested. However, in 1982 Stambaugh found that adding other risky assets such as corporate bonds, real estate, and consumer durables to the market portfolio did not materially affect the tests.


Single Factor Model (Market Model)

R_it = a_i + β_iR_mt + e_it t = 1, ..., T

where e_it is the distance from the regression line at time t. The mean value of e_it = 0 and the covariance between R_m and e_i = 0. This is a regression model that characterizes the risk of a security over time by measuring its beta over a time interval. β_iis different from the β_im used in the CAPM in that β_im is more of a present-day beta rather than one taken over time. In the traditional approach of testing the CAPM, in the first step one uses this model to measure the beta of all securities (or portfolios). In the second step one estimates the CAPM itself by regressing the security returns on the estimated betas. When testing CAPM in this manner, one must question the validity of tests using ex post data to test the ex ante CAPM. Also, there is measurement error in individual security betas. Using portfolios instead in the first-pass regression helps.

Variance using the single-factor model:

Var(R_i) = β²_i Var(R_m) + Var(e_i)

where R_i, R_m, and e_i are random variables. The variance of the mean return a_i is zero by definition, so this term falls out. In a well-diversified portfolio, Var(e_i) = 0. In this equation, β²_i Var(R_m) is the variance explained by the market. The percent of variance explained by the market then is given by

β²_i Var(R_m) / σ²_i = R²

Note that (1-R²) is the idiosyncratic variance.

These expressions apply to portfolios as well by replacing i with p.

For two portfolios or securities in which their e_i's are uncorrelated, the covariance between them is given by:

σ_ij = β_i β_j σ²_m .

This is derived by finding the covariance between:

R_it = a_i + β_iR_mt + e_it and R_jt = a_i + β_jR_mt + e_jt

Fama and French used a multi-factor model using additional risk factors related to size, price/book, etc.

They concluded that three "risk" factors were sufficient.

Gabriel Hawawini and Donald Keim's paper reports that stock returns depended size, E/P, CF/P, P/B, and prior returns. However, these factors were not due to risk.

The premia related to size and P/B are mainly due to the January effect. It is unlikely that the risk is higher in January. The size & P/B premia are uncorrelated across international markets. This is inconsistent with the notion of well-integrated international markets, in which similar risks should result in similar returns.


Market-Neutral Strategies

Market-neutral strategies balance the market risk by going long on some securities and short on others. Some people propose using the T-bill rate as a benchmark against which to compare the market return of a such a strategy. One can argue that even though the market-neutral strategy is risky, since it has zero beta it does not contribute to the risk of the market portfolio and therefore should not command a premium over the risk-free rate. On the other hand, if the expected returns on the long side are higher than those on the short, the benchmark return should exceed the risk-free rate.


Trading Costs

An important factor in the performance of high-turnover portfolios is the amount of the trading costs, including explicit costs such as commissions, fees, and taxes, the market maker spread, the impact of trading on market price, and the opportunity cost incurred during the delay between the time the decision is made and the time the trade is executed.

Trading costs can be reduced through passive fund management and electronic trading.


Long-Term Investing

The conventional wisdom is that over the long run, stock will generate returns superior to those of bonds. But while the variance of the geometric means of the returns declines as the time horizon increases, the variance of the terminal wealth increases. If a put option were purchased to insure a certain terminal wealth, the cost of that option would increase as the time horizon increases. To the extent that option prices are a measure of risk, the risk of stock investments then increases as the time horizon lengthens.

The optimal asset allocation is a function of the present wealth, target future wealth, risk tolerance, and time horizon. Long-term returns are difficult to analyze statistically because as the historical time horizon increases, the number of possible independent samples of returns decreases. In 1991, Butler and Domian illustrated a procedure that attempts to overcome this difficulty by first listing the monthly returns for the S&P 500 and the long-term bonds over a long historical time horizon. By randomly selecting data, returns over various long-term holding periods can be emulated by multiplying the appropriate number of random samples. An almost limitless number of samples for each holding period can be generated using this method. Performing such an analysis with data taken from the 792 months from 1926-1991 indicates that over a 10-year time period, there is an 11% chance that stocks will underperform bonds; over a 20-year time period this probability reduces to 5%.


Defined-Benefit Pension Plans

In a defined-benefit plan, the plan sponsor (usually an employer) guarantees a level of future benefits to the plan participants, taking responsibility for any shortfall in the investment performance of the plan. FASB 87 requires that any unfunded liability in the present value of the benefits appear on the balance sheet of the employer. One alternative for the plan sponsor is to place the present value of the plan liability into government bonds of the same duration as the liability, in which case there is no chance of shortfall and the liability is fully immunized. Furthermore, because pension plans are not taxed, the incentive to hold equity in order to take advantage of lower taxes on capital gains is diminished. For a given level of risk, tax implications increase the return most for investments such as bonds, which have a large spread between pre-tax and after-tax return. An alternative to bonds is for the pension fund to place its money into riskier assets such as common stocks. Under this latter alternative, there exists both the chance of a shortfall and the chance of a surplus. However, FASB 87 does not permit a surplus to be reported as an asset on the sponsor's balance sheet, and the surplus often gets allocated to the plan participants. Nonetheless, for many reasons it is common for firms to hold equity in their pension funds. The Pension Benefit Guarantee Corporation (a federal agency) guarantees the benefits, and the sponsor's premiums are independent of the risk level of the pension fund's investments. For employers in financial distress, the pension guarantee from PBGC effectively is a put option. Given that put options increase in value as risk increases, there is an incentive for some firms to invest the pension fund in risky assets.

In defined-benefit plans there exists the opportunity for tax arbitrage. The plan sponsor can issue debt in order to buy equity in the pension plan. The pension plan then can invest the funds in bonds. Because of the tax status of the pension fund, the taxes on the pension plan's bond interest will be deferred, and the sponsor will enjoy the interest tax shield from its debt issuance. The sponsor then realizes an arbitrage profit equal to the interest rate multiplied by the corporate tax rate, with no increase in the firm's overall risk.


Arbitrage Pricing Theory

In 1976, Steve Ross presented the arbitrage pricing theory (APT) as an alternative to the CAPM that requires fewer assumptions. The APT is an equilibrium theory, which differs from a factor model in that it specifies relationships between expected returns across securities and attributes that influence those securities. A factor model allows the first term in the model, the expected return, to differ across securities and therefore can represent either and efficient or an inefficient market. Ross assumed that returns have the first term in common, and the other terms depend on several different systematic factors, as opposed to the single market risk premium factor of the CAPM. The model takes the form:

where I_i = value of the i^th factor, β_pi = sensitivity of the return to the i^th factor, k = number of factors, and e_pt equals the idiosyncratic variation in the return. Assuming an efficient market in equilibrium, the first term to the right of the equal sign is the same for all securities and is approximately equal to the risk-free rate. Examples of factors that could be included in the model are monthly industrial production, changes in expected inflation, unexpected inflation, unexpected changes in the risk premium, and unexpected shifts in the term structure of interest rates. Such variables likely affect most or all stocks.


Market-Neutral Strategies Revisited

Given that "alpha" is the return above the market return, by constructing a portfolio long on positive alpha stocks and short on negative alpha stocks, one can cancel the effect of the market. This market-neutral strategy sometimes is referred to as a "double alpha, no beta" strategy. Because such a strategy is uncorrelated with the market, the volatility depends on non-market factors. If the market-neutral portfolio is well-diversified across many types of industries, the volatility can be low. If the portfolio is concentrated in a smaller number of stocks and industries, the volatility can be high. Furthermore, if the portfolio is not balanced among stocks of different size or value/growth measures, there could be higher volatility as a result of these non-market risk factors.


Mutual Funds

Mutual funds charge fees to their investors. Transaction fees called loads sometimes are charged for fund purchases or redemptions. Such fees are deducted directly from the investor's account and represent a charge for the broker's service of providing information and fund selection advice. Operating expenses are fees that are deducted from the fund earnings before distribution to investors, and typically average slightly more than 1% per year. Two components of operating expenses are management fees and 12b-1 fees. The 12b-1 fees represent a reimbursement for the fund's marketing expenses.


Style Analysis

Mutual funds can be characterized according to investment style, such as value or growth. However, the actual fund composition may not correspond closely to its stated investment style, and reports of portfolio holdings may not be very representative since they are only snapshots taken at one point in time. This limitation makes the public's view of the holdings subject to distortions such as "window-dressing," in which the portfolio manager buys stocks that have performed well so that investors will see those stocks in the portfolio holdings (cost basis is not reported) and perceive the manager to be capable of selecting the top performers. Style analysis is a method of characterizing the true style of a fund based on its behavior, not on its stated objectives and holdings.

Style analysis is performed by first selecting a set of indices that correspond to particular styles, such as small-cap value, small-cap growth, large-cap value, large-cap growth, and cash. Using a weighted combination of these indices, one can construct a passive benchmark portfolio that tracks the return of the portfolio being analyzed as closely as possible. Assume that there are five indices available with which to compose the benchmark. The following steps are used to analyze the style:

1. Define the benchmark return for period t to be:

2. Define the tracking error to be:

e_t = R_Fund,t - R_Benchmark,t

3. Solve for the weights by minimizing the standard deviation of the mean tracking error over the entire time period being analyzed under the constraint that the weights sum to one and are each greater than or equal to zero (unless net short positions are permitted in the fund). The standard deviation of the tracking error is given by:

σ(e) = [ ( 1 / T-1 ) Σ_t=1,T (e_t - e_mean)² ]^1/2

Evaluating Fund Performance

When the popular press publishes mutual fund performance rankings, it usually does not consider the risk that the portfolio manager took to achieve that return. Such rankings do not necessarily reflect the skill of the manager. To adjust for risk, one should consider the ratio of excess returns to risk, or consider risk-adjusted differential returns. For the risk, one can use standard deviations or betas.

The "Sharpe Measure": [ E(R_p) - E(R_F) ] / σ_p

The "Treynor Measure": [ E(R_p) - E(R_F) ] / β_p

Std dev. differential measure: [ E(R_p) - E(R_F) ] - [ E(R_B) - E(R_F) ] σ_p / σ_B

"Jensen Measure": [ E(R_p) - E(R_F) ] - [ E(R_B) - E(R_F) ] β_p

The Jensen measure is perhaps the most widely used measure of fund performance.

In the above measures, R_p is the return of the portfolio under test, R_B is the return of a passive benchmark portfolio, R_F is the risk-free rate, and E(R) represents the mean historical returns.

In determining which measure to use, one should consider the purpose of the measurement.

For portfolios that represent a large portion of its investors' assets, a method that uses standard deviation should be used; the Sharpe Measure and the other Std. deviation differential measure are more appropriate.

For ranking fund performance, the ratio of excess return to risk should be measured; the Sharpe Measure or the Treynor Measure are more appropriate.


Market Efficiency

There is some evidence of some autocorrelation in stock prices. Small amounts of both positive autocorrelation, in which stock returns tend to move in the direction of the previous period, and negative autocorrelation, in which returns tend to move in a direction opposite to that of the previous period, have been observed. In situations of positive autocorrelation, momentum investing strategies should be employed, and in situations of negative autocorrelation, contrarian strategies should be used. However, for shorter term trading, any advantage from these techniques is neutralized by trading costs, and for longer terms there is not yet enough data to confirm or deny any net advantage.


Timing the Market

Some investors have attempted to time the market to increase their returns, increasing their stake in equities when they predict an up market and decreasing it when they predict a down market.

QuickMBA's market timing page covers this topic in more detail.


Bonds

Coupon bearing notes and bonds typically make fixed interest payments two times per year. Zero coupon bonds are sold at a discount and pay off their face values at maturity. Zero coupon treasury securities are issued by commercial institutions who separate the interest and principal payments. These zero coupon bonds are known as CAT's, TIGR's, and STRIP's.

Bond prices often are quoted in the format x:y, where x is the integer dollar amount and y is the fractional amount in 32^nd's of a dollar.

The spot rate is the rate that would correspond to a single cash flow at maturity for a bond purchased today, as is the case with a zero coupon bond. A notation used for spot rates is r_n, where n is the number of periods (e.g. years) into the future when a loan made today is to mature. The forward rate is the rate at which a future loan is made today. A notation used for forward rates is f_m,n, where m is the number of periods from the present when the loan is to commence, and n is the number of periods into the future when the loan is to end. Forward rates can be expressed in terms of spot rates:

1 + f_m,n = ( 1 + r_n ) / ( 1 + r_m )

The ask price of a U.S. Treasury bill is calculated from the "asked" rate (not asked yield) as follows:

Ask Price = 10,000 [ 1 - asked rate ( N / 360 ) ]

where N = the number of days until maturity.

The implied rate (spot rate) is ( 10,000 / Ask Price - 1 ). This implied rate does not represent an annualized basis. The annualized rate is found by raising the implied rate to the 365/N power:

Annualized Rate = ( Implied Rate )^{365 / N}

The bond equivalent yield is the yield to maturity y that satisfies the following equation:

P = Σ_n=1,N C_n / ( 1 + y/2 )ⁿ

where P = price, C_n = cash flow at the end of each period, N = number of periods.

For a zero coupon bond there is only one cash flow at maturity.

The value of a coupon bond can be modeled as a portfolio of zero-coupon bonds having face values and maturity dates that correspond to the coupon payments and dates. Summing the prices of the zero-coupon bonds then would give the value of the coupon bond, and any difference would represent an arbitrage opportunity.

Forward rates can be calculated using the prices and returns of bills, notes, or bonds provided they cover the proper time periods. For example, given the six month spot rate r_0.5, one can calculate the one year spot rate r_1.0 by using the data for a one year note the following equation:

Once the spot rates are known, the forward rate can be calculated as already illustrated.

The spot rate is not quoted on an annualized basis. To annualize it:

where x is the number of periods in one year, and y is the number of periods included in the spot rate.

The duration of a bond often is thought of in terms of time until maturity. However, in addition to the payoff of the face value at maturity, there are the coupon payment cash flows that influence effective duration. Two bond with equal yield-to-maturities and maturity dates will have different effective durations if their coupon rates are different. Frederick Macaulay suggested the following method of determining duration:

where T = life of the bond in semiannual periods,

C_t = cash flow at end of t^th semiannual period,

y = yield to maturity, expressed as a bond-equivalent yield.

A zero-coupon bond has no coupon payments and therefore its effective duration always is equal to the time until maturity and does not change as yield-to-maturity changes. Duration essentially measures the sensitivity of a bond's price to movements in interest rates. By this definition, duration is defined as

If one plots the price of a non-callable bond as a function of its yield, the plot will be concave up (convex down) rather than linear. This curvature is called convexity, and in this case, positive convexity. Convexity is due to the fact that effective duration increases as interest rates decrease.

Because of the effectively shorter duration, the coupon bond yield curve will be below that of the zero coupon bond when forward rates are rising with time, and above it when they are dropping. Zero coupon rates often are more useful for capital budgeting purposes.

Research has found that diversified portfolios of junk bonds have lower variance than those of high-grade bonds. There are several contributing factors to this initially surprising result. First, while individual junk bonds are risky, much of this risk can be diversified in a portfolio. Second, because of the higher coupon rate, junk bonds effectively have a shorter duration than do higher grade bonds and therefore a lower sensitivity to interest rate movements. Third, junk bonds are more likely to be called than are higher-grade bonds, since there is a strong incentive to refinance at lower rates if the issuer's credit improves. This characteristic reduces the effective duration resulting in less volatility.

Recommended Reading

David F. Swensen, Pioneering Portfolio Management : An Unconventional Approach to Institutional Investment