Giddy/NYU Foundations of Finance Course

Professor Ian Giddy's Foundations of Finance

Risk and Return: Highlights

Risk and Return

A security's return is often measured by its holding-period return: the change in price plus any income received, expressed as a percentage of the original price. A better measure would take into account the timing of dividends or other payments, and the rates at which they are reinvested.
The total return on an investment has two components: the expected return and the unexpected return. The unexpected return comes about because of unanticipated events. The risk from investing stems from the possibility of an unanticipated event.
The total risk of a security refers to the extent to which realized returns may deviate from the expected return. A common measure is standard deviation, although for many investors the downside risk is more important than the sheer dispersion of returns. For funds managers, the risk of underperforming a benchmark may be the most relevant risk.
If one assumes returns are nomally distributed, then variance (or its square root, standard deviation) is a reasonable measure of risk, since a normal distribution is symmetrical and fully described by its expected value and variance. There is evidence that stock-price returns are more leptokurtic (fat-tailed) than would be predicted by the standard normal distribution.
Systematic risks (also called market risks) are unanticipated events that affect almost all assets to some degree because the effects are economywide. Unsystematic risks are unanticipated events that affect single assets or small groups of assets. Unsystematic risks are also called unique or asset-specific risks.
Investors face a trade-off between risk and expected return. Historical data confirm our intuition that assets with low degrees of risk provide lower returns on average than do those of higher risk.
Shifting funds from the risky portfolio to the risk-free asset is the simplest way to reduce risk. Another method involves diversification of the risky portfolio.
U.S. T-bills provide a perfectly risk-free asset in nominal terms only. Nevertheless, the standard deviation of real rates on short-term T-bills is small compared to that of assets such as long-term bonds and common stocks, so for the purpose of our analysis, we consider T-bills the risk-free asset. Besides T-bills, money market funds hold short-term safe obligations such as commercial paper and CDs. These entail some default risk but relatively little compared to most other risky assets. For convenience, we often refer to money market funds as risk-free assets.
A risky investment portfolio (referred to here as the risky asset) can be characterized by its reward-to-variability ratio. This ratio is the slope of the capital allocation line (CAL), the line that goes from the risk-free asset through the risky asset. All combinations of the risky and risk-free assets lie on this line. Investors would prefer a steeper sloping CAL, because that means higher expected returns for any level of risk. If the borrowing rate is greater than the lending rate, the CAL will be "kinked" at the point corresponding to investment of 100% of the complete portfolio in the risky asset.
An investor's preferred choice among the portfolios on the capital allocation line will depend on risk aversion. Risk-averse investors will weight their complete portfolios more heavily toward Treasury bills. Risk-tolerant investors will hold higher proportions of their complete portfolios in the risky asset.
The capital market line is the capital allocation line that results from using a passive investment strategy that treats a market index portfolio such as the Standard & Poor's 500 as the risky asset. Passive strategies are low-cost ways of obtaining well-diversified portfolios with performance close to that of the market as a whole.

Interest Rate Risk

Even default-free bonds such as Treasury issues are subject to interest rate risk. Longer term bonds generally are more sensitive to interest rate shifts than short-term bonds. A measure of the average life of a bond is Macaulay's duration, defined as the weighted average of the times until each payment made by the security, with weights proportional to the present value of the payment.
Macauley's duration measures the time horizon when a bond's yield will be realized. During that time, losses (gains) from price change will be offset by gains (losses) from reinvestment of coupon interest.
Modified Duration is a direct measure of the sensitivity of a bond's price to a change in its yield. Modified Duration is equal to Macauley's Duration/(1+yield).
Duration is only an approximation of the percentage price change of a bond for a 1% change in yield. It assumes parallel changes in a flat yield curve, and only works for small changes (such as 10 basis points) in yield.
The longer the maturity, the lower the yield, and the smaller and less frequent the bond's coupon, the greater is the duration. The Macauley's duration of a zero-coupon bond is equal to its maturity.
Convexity measures the degree to which duration changes as the yield to maturity changes.
Positive convexity, which characterizes most straight (plain, non-callable) bonds, refers to the fact that price sensitivity, as measured by duration, declines as the yield increases, and rises as the yield decreases. Positive convexity is regarded as a desirable feature of a bond, particularly when yields are volatile. Callable bonds such as US mortgage-backed securities have negative convexity over some yield range.
Duration is additive, so the duration of a portfolio of bonds is the weighted sum of the duration of the individual bonds. Because duration and convexity measure price risk, they can be helpful in bond portfolio management.
Immunization strategies are characteristic of passive fixed-income portfolio management. Such strategies attempt to render the individual or firm immune from movements in interest rates. This may take the form of immunizing net worth or, instead, immunizing the future accumulated value of a fixed-income portfolio. Immunization of a fully funded plan is accomplished by matching the durations of assets and liabilities. To maintain an immunized position as time passes and interest rates change, the portfolio must be periodically rebalanced.
A more direct form of immunization is dedication or cash flow matching. If a portfolio is perfectly matched in cash flow with projected liabilities, rebalancing will be unnecessary.

Quantifying Credit Risk

For many years, academics and financial insitutions have sought to predict losses from credit risk. The best-known methodogy is based on Altman's Z-score, which seeks to predicts defaults using company financial data.
The newer CreditMetrics approach estimates volatility from upgrades, downgrades, and defaults. Historical data are used to attribute a likelihood of possible credit events, including upgrades and downgrades, not just defaults.
For example, CreditMetrics calculates the probability that a bond’s current rating will shift to any other rating within a given time. Each shift results in an estimated change in value (derived from historical credit spread data or recovery rates in default). Each value out-come is weighted by its likelihood to create a distribution of value across each credit state, from which each asset’s expected value and volatility of value is computed.
To compute the volatility of portfolio value from the volatility of individual asset values requires estimates of correlation in credit quality changes. Since these cannot be directly observed from historical data, one approach is to infer these from historical asset correlation data derived from equity price series. Several different approaches, including a simple constant correlation, can be used.

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