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In this chapter we will discuss the historical performance of the major asset classes. We will use a risk free asset as a benchmark to evaluate that performance. The risk free asset is the Treasury Bill or TBill because it is regarded as the safest asset, the main reason for this is that the US government issues these bills and maintains its credit worthiness via the tax payers money. We will start with a review of the determinants of the risk free rate, the rate available on Tbills and we will focus on the important distinction between real and nominal returns. Second we will discuss the measurement of the expected returns and volatility of risky assets, and show how historical data can be used to construct such estimates. The purpose of this is that we will construct on optimal investment portfolio and in order to construct it we need some idea how risk can be measured. Finally we will review the historical record of several portfolios of interest to provide some insight how different portfolios have performed over time.
Forecasting interest rates is very difficult, But we do however have a good understanding of the fundamental determinants of the level of interest rates:
1. The supply of finds from savers, primary households
2. The demand for funds from businesses to be used to finance investments in plant, equipment and inventories.
3. The government’s net supply of or demand for funds as modified by actions of the Federal Reserve Bank.
Now we will focus on the important distinction between real and nominal returns. The real rate of interest is the nominal rate of interest minus the expected rate of inflation. In general, we can observe only the nominal interest rates. From these nominal interest rates we can derive expected real rates using inflation forecasts. The equilibrium expected rate of return on any security is the sum of the equilibrium real rate of interest, the expected rate of inflation and a securityspecific risk premium.
Real versus nominal interest rates an example:
General
Fisher effect: Approximation
nominal rate = real rate + inflation premium
R = r + i or r = R – i
Example
r = 3%, i = 6%
R = 9% = 3% + 6% or 3% = 9%  6%
Fisher effect: Exact
r = (R  i) / (1 + i)
2.83% = (9%6%) / (1.06)
The empirical relationship is that inflation and interest rates move closely together.
HPR = (P1  P0 D1) : P0
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period
This holding period return is always uncertain. The expected rate of return is a probability weighted average of the rates of return in each scenario. P(s) is the probability of each scenario, and r(s) is the HPR in each scenario:
See formula sheet, formula 1
To quantify the volatility of this HPR, the risk, the standard deviation (square root of the variance) is used as a measure:
See formula sheet, formula 2
An investment decision first of all depends on the expected reward, which is the difference between the expected HPR (E(r)) and the risk free rate, which is the rate of return on a risk free asset, such as Treasury bills. This difference is called the risk premium.
Excess return is the actual difference between the risk free rate of return and the actual rate of return of a risky asset. The risk premium is therefore the expected value of the excess return.
Investors are said to be risk averse, which means that they always want to be compensated by a premium for taking risk.
A single period example:
Ending Price = 48
Beginning Price = 40
Dividend = 2
HPR = (48  40 + 2 )/ (40) = 25%
Time series
The probability distributions of these rates of return must be inferred from the data at hand, historical data. The average rate of return can be calculated in two ways:
1. The Arithmetic average of rates of return, giving an unbiased estimate of the expected rate of return:
See formula sheet, formula 3
2. The geometric average (g) is a timeweighted average return:
See formula sheet, formula 4
An estimate of the variance is usually based on the estimate of the expected return, the arithmetic average (). Using historical data the estimated variance looks like this:
See formula sheet, formula 5
This estimate needs to be compensated for the degrees of freedom bias, which is the result of the estimation error resulting from using . It is resolved by multiplying it with n/(n1):
See formula sheet, formula 6
Formula 7 is the standard deviation.
The trade off between reward and risk is important and represented by the sharpe ratio:
Sharpe ratio = risk premium / SD of excess return
Assuming that expectations are rational, the actual rates of return should be normally distributed around the expectations. Because the normal distribution is symmetric, stable and binary, this assumption is very practical in investment and portfolio management.
Deviations from normality are so common, that we cannot leave it undiscussed. Skew is a measure of symmetry:
See formula sheet, formula 8
(cubing these deviations ensures that the sign is maintained)
Kurtosis is a measure of fat tails:
See formula sheet, formula 9
Historical returns on stock have more frequent large negative deviations from the mean than would be predicted from a normal distribution.
The lower partial standard deviation (LPSD) of the actual distribution quantify the deviation from normality. The LPSD, instead of the standard deviation, is sometimes used by professionals as a measurement of risk. A more widely used measurement of risk is value at risk (VaR). VaR measures the loss that will be exceeded with a specified probability such as 5%. The VaR does not add new information when returns are normally distributed. When negative deviations from the average are larger and more frequent than the normal distribution, the 5% VaR will be more than 1.65 standard deviations below the average return.
Historical rates of return over the twentieth century in developed capital markets suggest the US history of stock returns is not an outlier compared to other countries. The arithmetic average of the risk premiums on stocks over the period 19262002 is arguably too optimistic as a forecast for the long term as we can see on page 155 of BKM figures 5.8 and 5.9. Some evidence suggests returns over the later half of the twentieth century were unexpected high, and hence the fullcentury average is upward biased. Another argument is that the arithmetic average returns may five upward biased estimates of longterm cumulative return. Longterm forecasts require compounding at an average of the geometric and arithmetic historical means, which reduces the forecast.
A compounding portfolio with a terminal value has a strong positive skew. It converges to a lognormal rather than a normal distribution. In a lognormal distribution the logarithms of a variable are normally distributed. For example, if an investment has low rates of returnmm the expected rate of return of the continuously compounded investment is close to the normal rate: Rcc = ln (1+r) = r. This changes however if it concerns longer periods or hinger r’s.
In this chapter we will discuss three themes in portfolio theory, all of them centering around risk. The first theme is that investors avoid risk and demand a reward for engaging in a risky investment. The reward is taken as a risk premium, the difference between the expected rate of return and that rate of return on a risk free investment. The second theme allows us to quantify investor’s personal tradeoffs between portfolio risk and expected return. To do this we introduce the utility function which assumes that investors can assign a welfare/benefit or “utility” score to any investment portfolio depending on its risk and return. Finally, the third theme is that we cannot evaluate the risk of an asset separate from the portfolio of which it is a part; that is the proper way to measure the risk of an individual asset is to assess its impact on the volatility of the entire portfolio of investments. Taking this approach, we find that seemingly risky securities may be portfolio stabilizers and actually low risk assets. In appendix A of this chapter we will describe the theory and practice of measuring portfolio risk by variance or standard deviations of returns. In Appendix B we will discuss the classical theory of risk aversion.
One definition of speculation is: the assumption of considerable business risk obtaining commensurate gain. With commensurate gain we mean a positive risk premium, that is, an expected profit greater than the riskfree alternative.
By considerable risk we mean that risk is sufficient to affect the decision. Gambling is to bet with an uncertain outcome. If you compare this definition to that of speculation, you will see that the central difference is the lack of commensurate gain. Economically speaking, a gamble is the assumption of risk for no purpose but enjoyment of risk itself, whereas speculation is undertaken in spite of risk involved because one perceives a favourable risk return trade off. Hence, risk aversion and speculation are not necessarily inconsistent.
A prospect that has a zero risk premium is called a fair game. Investors who are risk averse reject investment portfolios that are fair games or worse. Risk averse investors are willing to consider only risk free or speculative prospects with a positive risk premium. In a certain way riskaverse investors penalizes the expected rate of return of a risky portfolio to account for the risk involved. We can formalize the notion of a risk penalty system. In order to do so, we will assume that each investor can assign a welfare or utility, score to competing investment portfolios based on the expected return and risk of those portfolios.
We can formulate this concept into a formula:
Utility Function
See formula sheet, formula 10
A is the insexof the investor’s risk aversion
Investors can have three different views of risk:
The utility function weighs the return and the risk, taking the risk aversion of the investor into account. Based on comparing the utility values of different scenarios an investor can make a profound decision. Simply said, portfolio A dominated B if its expected return is higher and if the risk is lower. Using this utility function, indifference curves can be drawn, comparing all possible portfolios.
Because we can compare utility values to the rate offered on riskfree investments when choosing between a risky portfolio and a safe one, we may interpret a portfolio’s utility value as its “certainty equivalent ”rate of return to an investor. That is, the certainty equivalent rate of a portfolio is the rate that riskfree investments would need to offer with certainty to be considered equally attractive as the risky portfolio.
Figure 6.1. on page 193 describes the tradeoff between risk and return of a potential investment portfolio. We can see expected return E(r) on the yaxis and risk represented as variance on the xaxis). We say that this is the mean variance criterion. When we plot different mean variance combinations we can draw a line which results into the indifference curve as graphed in figure 6.2. on page 173.
Shifting funds from the risky portfolio to the riskfree asset is the simplest way to reduce risk. Other methods involve diversification of the risky portfolio and hedging. In allocating capital across risky and risk free portfolios we consider Tbills the risk free asset and stocks as the risky asset. Issues we need to examine are risk versus return tradeoff. We will demonstrate how different degrees of risk aversion affect allocation between risk free and risky assets.
Tbills provide a perfectly risk free asset in nominal terms only. Nevertheless, the standard deviation of real rates on shortterm Tbills is small compared to that of other assets such as longterm bonds and common stocks, so for the purpose of our analysis we consider Tbills as the riskfree asset. Money market funds hold, in addition to Tbills, relatively safe obligations such as CP and CDs. These entail some default risk, but again, the additional risk is small relative to most other risky assets. For convenience, we often refer to money market funds as riskfree assets.
So investors compose complete portfolios containing both risky investments and lowrisk, even riskfree, assets. Only the government can issue defaultfree bonds. Although such treasury bills are in fact not entirely riskfree, it is common sense to use the rates of return of treasury bills as the riskfree rate.
The rate of return of a complete portfolio is calculated as follows:
See formula sheet, formula 11
You can take expectations of this portfolio rate of return.
It can be rearranged to:
See formula sheet, formula 12
Concerning the volatility, only the standard deviation of the risky asset is relevant, and its weight:
See formula sheet, formula 13
Using that y is the division of both standard deviations, the relationship between the expected rate of return of the complete portfolio and the risk is as follows:
See formula sheet, formula 14
S is the slope of the capital asset allocation line (CAL). (see page 200, figure 6.4) This line represents all the riskreturn combinations available to the investor. The slope (S) is called the rewardtovolatility ratio. Other things equal, an investor would prefer a steepersloping CAL, because that means higher expected return for any level of risk. If the borrowing rate is greater than the lending rate, the CAL will be 'kinked' at the point of the risky asset.
The investor's degree of risk aversion is characterized by the slope of his or her indifference curve. Indifference curves show, at any level of expected return and risk, the required risk premium for taking on one additional percentage of standard deviation. More riskaverse investors have steeper indifference curves; that is, they require a greater risk premium for taking on more risk.
The optimal position, y*, in the risky asset, is proportional to the risk premium and inversely proportional to the variance and degree of risk aversion:
See formula sheet, formula 15
In other words, when we look at the optimal position of the risky asset shown as y* we can see that this is also strongly depended on A or in other words the level of risk averseness of the investor. Recall that A = 0 means a risk neutral investor and A > 0 are risk averse investors. The next step is to look for the optimal portfolio for a given level of risk aversion. To fully understand the level of risk aversion we need to construct a indifference curve to graphically plot the indifference curve for the level of risk aversion. An example of how an indifference curve can be plotted can be seen on page 208 in table 7.2. and figure 7.5, where we can see several indifference curves for a given several given levels of risk aversion. The next step is to find the optimal complete portfolio by using indifference curves. This can be graphically seen in figure 7.6 on page 209 where the tangent line of the indifference curve and the CAL depicts the optimal complete portfolio, this is also shown in table 7.3.
A passive investment strategy disregards security analysis, targeting instead the riskfree asset and a broad portfolio of risky assets such as the S&P 500 stock portfolio. We call it a passive strategy because it describes a portfolio decision that avoids any direct or indirect security This can also be conceptually be drawn, this line is comprised by a 1month Tbills and a broad index of common stocks and is called the capital market line (CML). There are several reasons why an investor would choose a passive strategy. The first reason is that an active strategy is not free, in involves for example research costs, analysing the different securities and buying or selling these securities, transaction costs. The second reason is the freerider benefit. If markets work perfectly there is no need to try to outperform the market, better to hold a well diversified portfolio that represents the market, in other words if you can beat them, join them. The box on page 213 describes that passive strategies outperform active strategies.
In this chapter we explain how to construct that optimal risky portfolio. We begin with a discussion of how diversification can reduce the variability of portfolio returns. After establishing this basic point we examine efficient diversification strategies at the asset allocation and security selection levels. We start a simple example of asset allocation that excludes the riskfree asset. To that effect we use two risky mutual funds: a longterm bond fund and a stock fund. With this example we investigate the relationship between investment proportions and the resulting portfolio expected return and standard deviation. We then add a riskfree asset to the menu and determine the optimal asset allocation. We do so by combining the principals of optimal allocation between risky assets and riskfree assets with the risky portfolio construction methodology. Moving from asset allocation to security selection, we first generalize asset allocation to a universe of many risky securities. We show how the best attainable capital allocation line emerges from the efficient portfolio algorithm, so that portfolio optimalization can be conducted in two stages, asset allocation and security selection. We examine in two appendixes common fallacies relating the power of diversification to the insurance principal and to investing for the long run.
The reduction of risk to very low levels in the case of independent risk sources is sometimes called the insurance principal, because of the notion that an insurance company depends on the risk reduction achieved through diversification when it writes insurance policies insuring against many independent sources of risk, each policy being a small part of the company’s overall portfolio.
The risk that remains even after extensive diversification is called market risk, risk that is attributable to marketwide risk sources. Such risk is also called systematic risk, or nondiversifiable risk. In contrast, the risk that can be eliminated by diversification is called unique risk, firmspecific risk, nonsystemic risk or diversifiable risk.
We will now move on and study efficient diversification, whereby we construct risky portfolios to provide for the lowest possible risk for any given level of expected return. We will start with considering a portfolio of two risky assets because they are relatively easy to analyze and illustrate the principal and considerations that apply to portfolios of many assets.
Return
We will consider a portfolio of two risky assets. We can formalize this as:
Rp = Wd Rd + We Re: this is the return on such a portfolio.
Wd = Proportion of funds in Security 1
We = Proportion of funds in Security 2
rd = Expected return on Security 1
re = Expected return on Security
and the weights need to add up to 1: See formula sheet, formula 16
Risk
The risk of the portfolio with two risky assets can also be formalized as:
See formula sheet, formula 17
Cov (Rd, Rg) = Covariance of returns for Security 1 and Security 2
This covariance is calculated by using the correlation between security 1 and 2:
See formula sheet, formula 18
The correlation between the two risky assets is important. Because it tells us in what way these two risky assets move together. When the securities are positively correlated they will move together. When they are negatively correlated they will move the opposite way of each other.
Portfolios of less than perfectly correlated assets always offer better riskreturn opportunities than the components on their own. This is because the standard deviation is less than the weighted average of both components, as the expected return actually is. Here the diversification opportunities arise. A minimumvariance portfolio has a standard deviation that is smaller than that of either of the individual component assets. This is the effect of diversification. . In the extreme case of perfect negative correlation, there is a perfect hedging opportunity and it would be possible to construct a zerovariance portfolio.
Portfolios with different correlations:
We use assets with different correlations to reduce the risk of the overall portfolio. The correlation effects can b summarized as:
The relationship depends on correlation coefficient.
1.0 r +1.0
The smaller the correlation, the greater the risk reduction potential.
If r = +1.0, no risk reduction is possible
Range of values for r 1,2
+ 1.0 > r > 1.0
If r= 1.0, the securities would be perfectly positively correlated
If r=  1.0, the securities would be perfectly negatively correlated
The relationship of expected return and standard deviation in relation to different levels of correlation can be seen in the graph below. This graph is also called the portfolio opportunity set. Here we can clearly see the different levels of expected return associated with different levels of correlation between the two risky assets. A correlation of r = 1 achieves a lower expected return than a r = 1.
Figure 1: portfolio opportunity set ( See formula sheet)
This is our next step in our refinement process of understanding portfolio selection. In the previous section we have looked at the simplest asset allocation decision. That involves the choice of how much of the portfolio to leave in riskfree money market securities versus in a risky portfolio. We will now take it a bit further by specifying the risky portfolio as comprised of a stock and bond fund. We will investigate this more refined selection of the risky part of the portfolio in this section.
Optimal risky portfolio with two risky assets and a riskfree asset
We will start our analysis with plotting two different capital allocation lines for two different portfolios. This can be seen in figure 7.6 on page 234 where the opportunity set of the debt and equity funds and two feasible CALs are shown. By calculating the different sharp ratios we can see which Cal dominates the other, when we calculate this we can see that portfolio B dominates portfolio A.
The next step is to think why do we stop here? We can continue to create the optimal portfolio. We do this by plotting the optimal CAL which is tangent to the opportunity set of risky assets, which is shown in figure 7.7 on page 236. P in this graph is the optimal portfolio. In practice, the process of creating an optimal risky portfolio is done with more than two risky assets we do this in a spread sheet or another computer program.
A numerical example of creating a optimal complete portfolio can be seen in examples 7.2 and 7.3 on pages 236237 of the book. In general, we take the following steps to arrive at the complete portfolio:
1. Specify the return characteristics of all securities (expected returns, variances, covariances)
2. Establish the risk portfolio:
a. Calculate the optimal risky portfolio using the following formula:
W1 = σ1^2  Cov(r_{1}r_{2}) / σ1^2 + σ2^2 – 2COV(r1,r2)
W2 = 1W1
b. Calculate the properties of Portfolio P using the weights we have calculated.
3. Allocate funds between the risky portfolio and the riskfree asset:
a. Calculate the fraction of the complete portfolio allocated to Portfolio P (the risky portfolio) and to Tbills (the riskfree asset)
b. Calculate the share of the complete portfolio invested in each asset and in tbills.
Security selection
In general, we can divide the security selection problem in three phases. First, we identify the riskreturn combinations available from the set of risky assets. Second, we identify the optimal portfolio of risky assets by finding the portfolio weights that result in the steepest CAL. And third and finally, we choose an appropriate complete portfolio by mixing the riskfree asset with the optimal risky portfolio.
Harry Markowitz (1952), published a formal model of portfolio selection embodying diversification principals. For his work he received the nobel prize in 1990. His model is precisely step one of portfolio management: identification of the efficient set of portfolios or as we have seen and named the efficient frontier of risky assets. The critical idea behind the frontier set of risky portfolios is that, for any risk level, we are interested only in that portfolio with the highest expected return. An alternative we can view the frontier as asset of portfolios that minimize the variance of any target expected return.
In more detail, the first step is to determine the riskreturn opportunities available to the investor. These are summarized by the minimumvariance frontier of risky assets. This is a graph of the lowest possible variance that can be attained for a given portfolio expected return. With data about expected returns, variances, and covariances we can calculate the minimumvariance portfolio for any targeted expected return. The plot of the minimum variance frontier of risky assets can be seen in the graph below (fugure 2).
Figure 2: The minimum variance frontier of risky assets, see formula sheet
All the portfolios that lie on the minimum variance frontier from the global minimum variance portfolio are candidates for the optimal portfolio. The part of the frontier that lies above the global minimumvariance portfolio is therefore called the efficient frontier of risky assets. The second step of the optimization plan involves the riskfree asset. As we can see in figure 7.11 on page 240 we can see that the portfolio P is tangent with the efficient frontier. Portfolio P is clearly the optimal portfolio. The final step, is where the individual investor chooses the optimal mix between the risky portfolio P and Tbills as we can see in figure 7.8 on page 238.
In this section we briefly describe how we can create an optimal portfolio with a spread sheet program like MS excel. This we can do with a data set as shown in table 7.4 and the section describes how we can plot an efficient frontier as shown in figure 7.13 with excel with the dataset.
Capital allocation and the separation property
Let us assume that we now have established the efficient frontier with excel. The next step is to introduce the riskfree asset. Whatever the preference of the client, the client will always choose portfolio P, because it is the optimal risky portfolio. The assumption with this conclusion is that the riskfree asset is available and that the input lists are identical for every investor. This result is called a separation property.
The separation property tells us that the portfolio choice problem may be separated into two tasks.
The critical point is that the optimal portfolio P that the manager offers is the same for all clients. This result makes professional management more efficient and hence less costly. In practice however the differentiating factor of great portfolio managers and the rest is the quality of security analysis in other words the input list analysis as the universal rule also applies here garbage in is garbage out. This is the factor that makes great versus poor portfolio managers.
We have seen that the that the theories of asset allocation and security selection are identical. So the next logical question is: Why do we make a distinction between asset allocation and security selection? There are three reasons:
Risk pooling is the collection of uncorrelated assets in one portfolio. This is widely seen as the insurance principle, but it is based on the misunderstanding that adding several bets would reduce the risk. Despite the fact that risk pooling benefit from uncorrelatedness, it does not reduce risk by itself. Risk only increases less than proportionally to the number of securities. The probability of loss however does diminish.
Risk sharing is selling shares in an attractive risky portfolio to limit risk an yet maintain the profitability of the resultant position. Risk sharing combined with risk pooling is the key to the insurance industry. Adding insurance policies increases the sharpe ratio, or the profitability, and steadily reduces the risk to each shareholder.
This chapter introduces index models that simplify estimation of the covariances and greatly enhances analysis if risk premiums. Risk is explicitly decomposed in systematic and unsystematic risk. This simplifies the analysis because positive covariances among security returns arise from common economic forces that affect the fortunes of most firms, for example business cycles, interest rates or natural resources. Covariances and correlations are more easily estimated now.
We assume in the single factor security market that just one variable drives the normally distributed returns. Statistical implications of this normality assumption give the following model:
See formula sheet, formula 19
Whereas,
Ri is rate of return on security i,
E (Ri) is the expected rate of return on security i,
Ei is the unexpected component of the rate of return on security i,
m is a parameter measuring macroeconomic components, unanticipated macro surprises,
βi is the sensitivity coefficient of the specific security i (relating to a specific firm, some respond differently to m than others).
This beta coefficient gives the sign of the systemic risk. Cyclical firms for example respond greater to the market, resulting in a higher beta.
Total risk of security i is given by a composition of beta and standard deviations:
See formula sheet, formula 20
Also the covariance between a pair of securities is determined by its beta’s:
See formula sheet, formula 21
The single index model uses the market index to estimate the common macroeconomic factor. Since the model is linear, the beta coefficient of a security can be estimated using singlevariable linear regression. The basis is regressing the excess return of a security i on the excess return of the market index using historical data (from for example the S&P500):
See formula sheet, formula 22
Whereas,
Ri is the excess return of a security i
Ai is the expected excess return when the market excess return is zero,
βi is the security’s sensitivity to the market index,
Rm is the excess return of the market index,
Ei is the residual, the estimation error, with .
Taking expectations of this model results in the following:
See formula sheet, formula 23
The first term (alpha) on the right hand side is the nonmarket risk premium. The second part, derived from the market risk premium, is the systemic risk premium.
This model simplifies the analysis because less separate coefficients need to be estimated. This model also enables specialization of effort in the analysis. The model however simplifies the World of risks, dividing them into a Sharp dichotomy: market versus firmspecific risk.
Page 283 until 289 describe the simplified estimation process of the model using six large corporations. The regressions of the rate of returns of these (six in total) corporations describe the security characteristic line (SCL), drawn through a scatter diagram. (basic econometrics)
This regression is complemented by an analysis of the variance (ANOVA), the estimate of the alpha, the estimate of the beta, and the covariance and correlation matrix.
In the context of portfolio Construction the alpha term is crucial. The beta’s are widely known and standardized. A sound estimation of the alpha however tells the manager if the security is good or bad. Intuitively: a positive alpha provides a premium on top of the premium that would result from following macroeconomic movements.
To optimize a portfolio, the goal is to maximize the Sharpe ratio. The procedure is summarized as follows:
See formula sheet, formula 24
Practical aspects
The full Markowitz model would be a better model in principle, since all necessary estimations would have to be made. However the great number of possible estimation errors cumulatively could account for a major failure. The singleindex framework has a clear practical advantage.
Another practical issue is the estimation of betas. betas seem to drift toward 1 over time, meaning that estimating a beta based on past betas is usually not the best option. Forecasting models have been developed that use regression to estimate the beta’s from various variables, such as variance of earnings, market capitalization, and dividend yield or debttoasset ratio.
Beta capture is the procedure of constructing a tracking portfolio which has the same beta as the portfolio of interest. This tracking portfolio captures the systemic risk. Buying this tracking portfolio short combined with the portfolio of interest long, the systemic risk is cancelled out. This is characteristic for many hedge funds.
The capital asset pricing model (CAPM) is the core of modern finance. It provides a prediction of the relationship between risk and expected return that should be observed. As such it is a benchmark and it provides ground for educated guesses on nontraded securities.
The model
The model is based on six assumptions:
Although these assumptions are often not realistic, they provide insight in many realworld complexities.
These assumptions will result in the following equilibrium:
The mutual fund theorem is the result that the passive strategy of investing in a market index portfolio is efficient. If this is true, it would imply that attempts to beat the passive strategy only generates trading and research costs with no offsetting benefits. However, in the real world, investors do choose different portfolios from M.
The market price of risk is the extra return that investors demand to bear portfolio risk. It is represented by the following ratio:
See formula sheet, formula 28
A basic principle of equilibrium is that all investments should offer the same rewardtorisk ratio, otherwise trading a rearranging would be beneficial. That is why the rewardtorisk ratio of an individual security needs to equal the market price of risk.
See formula sheet, formula 29
Rearranging terms results in the expected returnbeta relationship:
See formula sheet, formula 30
In which is the ratio that measures the contribution of this individual security to the variance of the market portfolio as a fraction of the total variance of the market portfolio:
See formula sheet, formula 31
If this holds for every i, this would have to hold for the entire portfolio.
This expected returnbeta relationship can graphically be represented by the security market line (SML), see figure 9.2 on page 317. The SML graphs the individual asset risk premiums as a function of asset risk. Relevant in this case is the contribution of the asset to the portfolio variance, measures by beta. The capital market line (CML) in contrast graphs the risk premiums of efficient portfolios as a function of portfolio standard deviation.
Testing the implication of CAPM is difficult. First of all because all traded risky assets would need to be considered, which is immense. Second, the CAPM implies relationships among expected returns and such expected values are never actually observed.
A model, consisting of assumptions, logical/mathematical manipulation of these assumptions, and predictions, can be tested normatively and positively. Normative tests test the assumptions, positive tests test the predictions. Few models can pass the normative test. In case of the CAPM the positive test implies testing the efficiency of the market portfolio and the accurateness of security market line. The principle problem with testing these, is that the market portfolio M is unobservable.
This leaves us with empirical tests of the expected returnbeta relationship, but the CAPM miserably fails these tests.
Despite its empirical shortcomings, the CAPM is the accepted norm in the US and other developed countries. The first reason is that the theoretical decomposition of systematic risk from firmspecific risk is compelling. Second, there is impressive evidence that the central conclusion of CAPM, which is the efficiency of M, may be close to truth.
Greater accuracy could be gained by adding complexity to the model.
Zerobeta model
Efficient frontier portfolios have some interesting implications:
Labor income
Two important assets are not traded, human capital and privately held businesses. Such capital is less portable across time and thus may be more difficult to hedge with using traded securities. Such facts may put pressure on security prices and results in departures from CAPM.
Multiperiod model
Robert C. Merton has relaxed the assumption of myopic investors. He envisions investors who optimize a lifetime consumption or investment plan and who adapt their decisions to changes. His model however predicts the same expected returnbeta relationship when the only source of risk is the uncertainty about portfolio returns. Other sources of risk could be changes in the parameters such ask the riskfree rate or expected returns. Another possibility would be the prices of consumption goods, inflation risk for example. All these risks would require hedging activities, highly complicating the model.
Consumption based CAPM
It might be useful to center the model on consumption, assuming that an investor would try to optimally smooth maximum consumption. In a lifetime consumption plan, he must in each period balance the allocation of current wealth between today’s consumption and the savings and investment that will support future consumption. When optimized, the utility value from an additional dollar of consumption today must be equal to the utility value of the expected future consumption that can be financed by that additional dollar of wealth.
Investors will value additional income more highly during difficult economic times. An asset will therefore be viewed as riskier in terms of consumption if it has positive covariance with consumption growth. Equilibrium risk premiums will be greater for assets that exhibit higher covariance with consumption growth.
The liquidity of an asset is the ease and speed with which it can be sold at fair market value. Illiquidity is measures by the discount that a seller must accept if the asset is to be sold quickly. Liquidity is increasingly seen as an important determinant of prices. Investors are likely to act on expected liquidity constraints or changes in such constraints. Liquidity premises might change unexpectedly as well. Therefore, investors may demand compensation for their exposure to liquidity risk.
The efficient market hypothesis (EMH) is the notion that stocks already reflect all available information. As market participants try to anticipate on all available information, in principle stock prices should contain all information that could possibly be used to predict them. Consequently stock prices that respond to information must move unpredictably. Prices should follow a random walk.
The weakform of the EMH asserts that stock prices already reflect all information that can be derived by examining market trading data. The semi strongform states that all publicly available information regarding the prospects of a firm must be reflected already in the stock price. The strongform of the hypothesis, the most extreme position, claims that all information relevant to a firm is reflected.
The EMH implies that technical analysis is of no merit. Technical analysis is the search for recurrent and predictable patterns in stock prices. Wellknown concepts of such technical analysis are resistance levels and support levels, respectively probable upper and lower levels of stock prices.
Fundamental analysis uses earnings and dividend prospects of the firm, expectations of future interest rates, and risk evaluation of the firm to determine proper stock prices. Just as technical analysis, most fundamental analysis is useless following the reasoning of EMH. The trick is to identify firms that are better than everyone else’s estimate.
Proponents of the EMH are advocates of passive investment strategy, because the costs of active management is unlikely to be compensated by benefits. Such passive management simply aims for a welldiversified portfolio of securities. One common strategy is creating an index fund, designed to replicate the performance of a broadbased index.
Even in an efficient market there is a role for portfolio management. Optimal positions depend on such things as tax, risk aversion and employment. The role of the portfolio manager is to adjust the portfolio to these factors, not to beat the market.
An event study is empirical financial research to assess the impact of a certain event on a firm’s stock price. The general approach commences with an estimate of the stock price if the event would not have occurred. The abnormal return is then the difference between the actual return and this benchmark.
Index models are widely used to estimate these abnormal returns. Rewriting the mathematical formula for stock return ( formula 34) gives the following equation to estimate: formula 35
Whereas
Et is the part of a security’s return resulting from firmspecific events,
Rt is the actual rate of return of this stock,
Rmt is the return on the market portfolio,
b is the sensitivity of this particular stock to market return,
a is the average rate of return the stock would realize in a period with a zero market return.
This model can be easily upgraded to include all kinds of sophisticated factors. Estimation of the parameters a and b is a delicate issue. The standard estimate of abnormal return is easily complicated by leakage of information. The cumulative abnormal return is a better measure in such a case. Event studies are widely used nowadays.
EMH has never been widely accepted on Wall Street. Three important issues are important: the magnitude issue, the selection bias issue and the lucky event issue. With these in mind, we can discuss the empirical test of EMH.
Weakform tests
One way of finding trends in stock prices is measuring the serial correlation of returns. This reflects the tendency of returns to be related to past returns. Broad market indexes only reflect very weak serial correlation. There seems to be a stronger relationship across specific sectors.
Some studies have shown the predictive power of particular easily collected variables. On the one hand this could imply the violation of the EMH. On the other hand, such variables probably account for variation in market return.
Semi strongform
Fundamental analysis is always in line with the semi strongform of the hypothesis, since fundamental analysis uses publicly available information to clarify and predict stock prices. Examples of such fundamental analysis and its findings are the smallfirm effect, the neglectedfirm effect, the booktomarket effects, and the postearningsannouncement price drift.
Strongform
It is very common sense that insiders are able to make superior profits trading in their own stock. This practice is regulated and limited. This however implies that the strongform of the hypothesis is not very likely to appear.
Anomalies
Lots of literature has been produced on the anomalies of the financial markets. Are these markets just inefficient? Some argue that the anomaly effect named above are actually in line with efficiency and just reflect manifestations of risk premiums. The opposite interpretation is also provided, claiming that these effects are proof of inefficiency.
Prices can also lose their grounding in reality. Such bubbles show prices that depart from any semblance of intrinsic value. These bubbles are usually only acknowledged in retrospect.
Can market professionals outperform the passive index funds? This provides a short discussion of the professionals, stock market analysts and mutual fund managers.
Stock market analysts recommend investment positions based on their analysis. Only the relative performance of these analysts is really of interest. Literature suggests that they add some value, but ambiguity remains. The same accounts for mutual fund managers, who actually manage portfolios.
Behavioral finance is a relatively new school in finance, arguing that the literature on finance strategies has overlooked the most important point: the correctness of security prices. Conventional financial theory ignores how people really make decisions. Behavioral finance starts with the assumption that investors might not be rational. Irrationalities fall into two categories: people do not always process information correctly, and people make often inconsistent decisions.
1. Information processing: this leads to misestimating of probabilities. Four important biases have been identified.
2. Behavioral biases
3. The above biases would not lead to inefficient markets if some rational arbitrageurs would operate. For the following reasons, such arbitrage is limited:
Behavioral finance is not uncontroversial yet. It does make important points on the limits of rationality. It however does not provide investment opportunities based on its insights, for example. Some believe that the behavioral critique is too unstructured.
Technical analysis attempts to exploit recurring and predictable patterns in stock prices to generate superior investment performance. This section discusses the relation between technical analysis and behavioral finance. Technical analysis reflects all kinds of behavioral biases.
Technical analysis mostly uncovers trends. Dow theory is one of the oldest of trend analysis. According to Dow, three major trends influence stock prices:
This model is built on the notion of predictability. EHM argues however that in this case investors would exploit these possibilities, affecting the prices and resulting in a selfdestructing strategy. These trends are thus only observed after the fact.
Two other measures are (1) the moving average, and (2) the breadth, a measure of the extent to which movements in the market index are reflected widely in the price movements of all the stocks in the market.
There are three indicators of the investor’s sentiment to be named here:
We will start with analysing debt securities. A debt security is a claim on a specified periodic stream of income. Debt securities are often called fixedincome securities, because they promise a (fixed) stream of income that is determined according to a specified formula. These securities have the advantage of being relatively easy to understand because the payment formulas are specified in advance. Risk considerations are minimal as long as the issuer of the security is sufficiently creditworthy.
The bond is the basic debt security. We will start with an overview of the universe of bond markets. This includes Treasury, corporate, and international bonds. We next turn to bond pricing, showing how bond prices are set in accordance with market interest rates and why bond prices change with those rates. Given this background we compare the different measures of bond returns such as yield to maturity, yield to call, holdingperiod return or realized compound yield to maturity. We show how bond prices evolve over time. Finally we consider the impact of default or credit risk on bond pricing and look at the determinants of credit risk and the default premium built into bond yields.
A bond is a security that is issued in connection with a borrowing arrangement. The borrower issues (i.e., sells) a bond to the lender for some amount of cash; the bond is a “IOU” of the borrower. The arrangement obligates the issues to make specified payments to the bondholder on specified dates. A typical coupon bond obligates the issuer to make semiannual payments of interest to the bondholder for the life of the bond. These are called coupon payments. When the bond matures, the issuer repays the debt by paying the bondholder the bond’s par value another word for par value is face value. The coupon rate of the bond serves to determine the interest payment: the annual payment is the coupon rate times the bond’s par value. The coupon rate, maturity date, and par value of the bond are part of the bond indenture, which is the contract between the issuer and the bondholder.
Bonds are usually issued with coupon rates set high enough to induce investors to pay par value to buy the bond. Sometimes, however, zerocoupon bonds are issued that make no coupon payments. In this case, investors receive par value at the maturity date but receive no interest payment until then. The bond has a coupon of zero. This type of bond is issued at priced considerably below par value, and the investor’s return comes solely from the difference between issued price and the payment of par value at maturity.
Figure 14.1 on page 468 is an excerpt from the listing of treasury issues of the Wall Street Journal. Aside from the differing initial maturities, the only major distinction between Tnotes and Tbonds is that in the past, some Tbonds were callable for a given period. The US treasury no longer issues callable bonds, but some previously issued callable bonds still outstanding. Page 468 explains how to read a bond price quote in the Wall Street Journal (WSJ).
Accrued interest and quoted bond prices. The bond prices that you see quoted in the financial pages are not the actual prices that investors pay for the bond. This is because the quoted price does not include the interest that accrues between coupon payment dates.
In general, the formula for the amount of accrued interest between two dates is:
Accrued interest = (Annual coupon payment / 2 ) × (days since last coupon payment/ daysseparating coupon payments)
Corporate bonds. These are bonds issued by corporations. An example is given in figure 14.2 on page 451 here you can a sample from the WSJ.
Preferred Stock. Although preferred stock strictly speaking is considered to be equity, it often is included in the fixedincome universe. This is because like bonds preferred stock promises to pay a specified stream of cash in the form of dividends.
Other Issuers. There are also other issuers of bonds. For example local governments and federal agencies as discussed in chapter 2.
International Bonds. International bonds can be divided into two categories: foreign bonds and Eurobonds. Foreign bonds are issued by a borrower from a country other than the one in which the bond is sold. The bond is denominated in the currency of the country in which it is market. Foreign bonds sold in the US are called Yankee bonds, for example German BMW issues a bond in dollars in the US are called Yankee bonds. Foreign bonds in Japan are called Samurai bonds and foreign bonds in the UK are called bulldog bonds.
Innovation in the bond market. Below we will describe some innovations in the bond market:
A bond’s coupon and principal payment all occur months or years in the future, the price an investor would be willing to pay for a claim to those payment depends on the value of dollars to be received in the future compared to dollars in hand today. This sort of present value calculation depends in turn on market interest rates. The nominal riskfree interest rate equals: 1) a real riskfree rate of return 2) a premium above the real rate to compensate for expected inflation. In addition a premium reflects bond specific characteristics such as default risk, liquidity, tax attributes, call risk etc. To value a security we discount its expected cash flows by the appropriate discount rate.
Bond value = Present value of coupons + Present value of par value
See formula sheet, formula 36
PB = price of the bond
Ct = interest or coupon payments
T = number of periods to maturity
y = semiannual discount rate or the semiannual yield to maturity
An example:
What is the price of the bond?
We know:
10 year bond
Face value or Par value 1000
8% Coupon
Calculations: formula 37
Ct = 40 (SA)
P = 1000
T = 20 periods
r = 3% (SA)
An important insight is that with a higher interest rate, the present value of the payments to be received by the bondholder is lower. Therefore, the price of the bond will fall as market interest rates rise. This illustrates a crucial general rule in bond valuation. When interest rates rise, bond prices must fall because the present value of the bond’s payments are obtained by discounting at a higher interest rate.
Figure 14.3 on page 476 explains the inverse relationship between bond prices and yields. The price of an 8% coupon bond with 30year maturity making semiannual payments. An important insight from this figure is that the shape of the curve implies that an increase in the interest rate results in a price decline that is smaller than the price gain resulting from a decrease of equal magnitude in the interest rate. This property of bond prices is called convexity because of the convex shape of the bond price curve. This curvature reflects the fact that progressive increases in the interest rate result in progressively smaller reductions in the pond price. Therefore, the price curve becomes flatter with higher interest rates. Prices and Yields (required rates of return) have an inverse relationship
We can say that when yields get very high the value of the bond will be very low. When yields approach zero, the value of the bond approaches the sum of the cash flows.
A general rule in evaluating bond price risk is that, keeping all other factors the same, the longer the maturity of the bond, the greater the sensitivity of price to fluctuations in the interest rate. This is also the reason why shortterm Treasury securities such as Tbills are considered to be the safest. They are free not only of default risk but also largely of price risk attributable to interest rate volatility.
We would like a measure to rate of return that accounts for both current income and the price increase or decrease over the bond’s life. The yield to maturity is the standard measure of total rate of return. The yield to maturity is defined as the interest rate that makes the present value of a bond’s payments equal to its price.
In Formula:
See formula sheet, formula 38
An example: formula 39
10 yr Maturity Coupon Rate = 7%
Price = $950
Solve for r = semiannual rate
Answer: r = 3.8635%
The financial press reports on an annualized basis, and annualizes the bond’s semiannual yield using simple interest techniques, resulting in an annual percentage rate, or APR. Yields annualized using simple interest are also called “bond equivalent yields”. Therefore, the semiannual yield would be doubled and reported in the newspaper as a bond equivalent of 6%. The effective annual yield of the bond, however, accounts for compound interest. If one earns 3% interest every 6 months, then after 1 year, each dollar invested grows with interest to
$1 X (1,03)^2 = $ 1,0609, and the effective annual interest rate on the bond is 6,06%.
Yield to maturity is different from the current yield of a bond, which is the bond’s annual coupon payment divided by the bond price.
A general rule is that for premium bonds (bonds selling above par value), coupon rate is greater than current yield, which in turn is greater than yield to maturity. For discount bonds (bonds selling below par value), these relationships are reversed. Some numeral examples:
Bond Equivalent Yield
7.72% = 3.86% x 2
Effective Annual Yield
(1.0386)2  1 = 7.88%
Current Yield
Annual Interest / Market Price
$70 / $950 = 7.37 %
A bond sell at par value when its coupon rate equals the market interest rate. We shall now discuss the Yield to maturity versus holdingperiod return. The difference between yield to maturity and holding period return is that yield to maturity depends only on the bond’s coupon, current price and par value at maturity. All of these values can de observed today, so it is relatively easy to calculate. In other words we can see the yield to maturity as a measure of the average rate of return if we hold the bond until the bonds maturity. The holdingperiod return is the rate of return over a particular investment period and depends on changes in rates affects returns, reinvestment of coupon payments and change in price of the bond.
Zero coupon bonds. Original issue discount bonds are less common than coupon bonds issued at par. There are bonds that are issued intentionally with low coupon rates that cause the bond to sell at a discount from par value. An extreme example are zerocoupon bonds, which carries no coupons and provides all its return in the form of price appreciation. Zeros provide only one cash flow to their owners, on the maturity date of the bond.
Although bonds generally promise a fixed flow of income, that income stream is not risk less unless the investor can be sure the issuer will not default on obligation. Bond default risk, is usually called credit risk, this risk is measured by rating agencies such as Moody’s, Fitch and Standard and Poor. They give a bond a rating such as in figure 1.8 on page 472.
To determine the safety of a bond we can use some ratios to analyze. We will discuss five of them briefly:
1. Coverage ratio. Ratios of company earnings and fixed costs.
2. Leverage ratios. Debtto equity ratios
3. Liquidity ratio. Two common liquidity ratios are:
4. Profitability ratios. Measures of rates of return on assets or equity.
5. Cash flow to debt ratio. This is the ratio of cash flow to outstanding debt.
A bond is issued with an indenture, which is the contract between the issuer and the bondholder. Part of the indenture is a set of restrictions that protect the rights of bondholders. To make sure the bond issuer does not come into a cash flow problem the firm agrees to establish a sinking fund to spread the payment burden over several years. The sinking fund may operate in one of two ways:
1. The firm may repurchase a fraction of outstanding bonds in the open market each year.
2. The firm may purchase a fraction of the outstanding bonds at a special call price associated with the sinking fund provision.
The firm has an option to purchase the bonds at either the market price or the sinking fund price, whichever is lower. To allocate the burden of the sinking fund call fairly among bondholders, the bonds chosen for the call are selected at random based on serial number. Other issues are subordination of further debt in case of liquidation, dividend restrictions and collateral.
Bonds with a relatively high risk of default yield lower prices and consequently its rate of return will rise. Following the same reasoning implies that collateralized bonds usually yield a lower rate of return, because the risk of losses in case of default is smaller. The default premium is the compensation that corporate bonds offer for the possibility of default. The development over time of these default premiums on these risky bonds is sometimes called the structure of interest rates.
A credit default swap (CDS) is in short an insurance on the default risk of an investment. By using such a CDS a highly risky bond can be repackaged as a very safe investment. These CDSs have widely been used to speculate, resulting in the credit boom that led to the financial crisis of 2008.
Another example of such a financial product dealing with risk mitigation is the Collateralized Debt Obligation (CDO). A separate legal financial institution would first raise funds, collect different kinds of debt obligations, pool them togehter, and resell the total in small ‘slices’ or ‘tranches’ in different priorityscales varying in the risk they would entail.
Until now we have assumed for the sake of simplicity that the same constant interest rate is used to discount cash flows of any maturity. In the real world this is rarely the case. In this chapter we explore the pattern of interest rates for differentterm assets. We will try to identify the factors that account for that pattern and determine what information may be derived from an analysis of the so called term structure of interest rates, the structure of interest rates for discounting cash flows of different maturities. We will show how traders use the term structure to compute forward rates that represent interest rates on “forward” or deferred loans, and consider the relationship between forward rates and future interest rates. Finally, we give an overview of some issues involved in measuring the term structure.
We could conclude that longerterm bonds usually offer higher yields of maturity because longerterm bonds are riskier and that the higher yields are evidence of a risk premium that compensates for interest rate risk. Another reason is that at these times investors expect interest rates to rise and that the higher average yields on longterm bonds reflect the anticipation of high interest rates in the latter years of the bond’s life.
Bond pricing.
The interest rate for a given time interval is called the short interest rate for that period. Table 15.1 on page 509 shows the Interest rates on 1year bonds in coming years. Expected onerear rates in coming Years:
Year Interest Rate
0 (today) 8%
1 10%
2 11%
3 11%
The interest rates are the expected interest rates in the future from today. We can price a bond using these expected interest rates with the following formula:
See formula sheet, formula 40
PVn = Present Value of $1 in n periods
r1 = Oneyear rate for period 1
r2 = Oneyear rate for period 2
rn = Oneyear rate for period n
We use this table to calculate the prices and yields of zero coupon bonds
Time to Maturity Price of Zero* Yield to Maturity
1 $925.93 8.00%
2 841.75 8.995
3 758.33 9.660
4 683.18 9.993
* $1,000 Par value zero
An important note is that the yield to maturity on zerocoupon bonds is sometimes called the spot rate that prevails today for a period corresponding to the maturity of the zero.
The forward interest rate is the interest rate that is inferred from the growth rate of the observed interest rates of the years before. Consequently, and since future interest rates are uncertain, this forward interest rate does not need to equal the interest rates that will actually prevail. With the following formula we can calculate the forward rates from the observed rates.
See formula sheet, formula 41
fn = oneyear forward rate for period n
yn = yield for a security with a maturity of n
See formula sheet, formula 42
An example as explained in the BKM: How to calculate a forward?
4 yr = 9.993 3yr = 9.660 fn = ?
(1.0993)^4 = (1.0966)^3 (1+fn)
(1.46373) / (1.31870) = (1+fn)
fn = .10998 or 11%
Note: this is expected rate that was used in the prior example.
Downward Sloping Spot Yield Curve
ZeroCoupon Rates Bond Maturity
12% 1
11.75% 2
11.25% 3
10.00% 4
9.25% 5
1yr Forward Rates downward sloping yield curve
1yr [(1.1175)2 / 1.12]  1 = 0.115006
2yrs [(1.1125)3 / (1.1175)2]  1 = 0.102567
3yrs [(1.1)4 / (1.1125)3]  1 = 0.063336
4yrs [(1.0925)5 / (1.1)4]  1 = 0.063008
In general there are three theories concerning term structure:
We will explain each theory briefly:
1. Expectations theory
This is the simplest theory of the term structure. A common version of this hypothesis states that the forward rate equals the market consensus expectation of the future short interest rate. The assumptions of this theory are:
2. Liquidity preference:
This theory states that the forward rate exceeds expected future interest rates. It assumes that 1) Longterm bonds are more risky. 2) Investors will demand a premium for the risk associated with longterm bonds. 3) The yield curve has an upward bias built into the longterm rates because of the risk premium. 4) Forward rates contain a liquidity premium and are not equal to expected future shortterm rates.
3. Market segmentation theory / preferred habit theory:
This is the theory that long and shortmaturity bonds are traded in essentially distinct or segmented markets and that prices in one market do not affect those in the other. It assumes that 1) Short and longterm bonds are traded in distinct markets. 2) Trading in the distinct segments determines the various rates. 3) Observed rates are not directly influenced by expectations. 4) Investors will switch out of preferred maturity segments if premiums are adequate.
A common version of the expectations hypothesis holds that forward interest rates are unbiased estimates of expected future interest rates. However, there are good reasons to believe that forward rates differ from expected short rates because of a risk premium know as a liquidity premium. A liquidity premium can cause the yield curve to slope upward even if no increase in short rates is anticipated.
The existence of liquidity premiums makes it very difficult to infer expected future interest rates from the yield curve. Such an inference would be made easier if we could assume the liquidity premium remains reasonable stable over time. However, both empirical and theoretic; considerations cast doubt on the constancy of that premium.
A pure yield curve could be plotted easily from a complete set of zerocoupon bonds. In practice, however, most bonds carry coupons, payable at different future times, so that yieldcurve estimates are often inferred from prices of coupon bonds. Measurement of the term structure is complicated by tax issues such as tax timing options and the different tax brackets of different investors.
Forward rates are market interest rates in the important sense that commitments to forward (deferred) borrowing or lending arrangements can be made at these rates. Even though the forward rates eventually won’t equal the realized interest rates in the future.
In this chapter we will discuss several strategies bond portfolio managers can pursue. We make the following distinction between these strategies. Active and passive bond strategies. Active strategies are strategies that trade on interest rate predictions and trade on market inefficiencies. In contrast, passive strategies focus on control risk and balance risk and return. We will start with discussing interest rate risk and the important concept of duration. Second, we will move to convexity. Third, passive bond management. Fourth, active bond management and finally interest rate SWAPS.
We have seen in the previous chapter that there is an inverse relationship between bond prices and yields, and we know that interest rate fluctuate. As we can imagine the sensitivity of bond prices to changes in market interest rates is obviously of great concern to investors. Six propositions underlie this sensitivity:
We need a measurement as guide to the sensitivity of a bond to interest rate changes, because the price sensitivity tends to increase with time to maturity. This measurement is called duration.
Duration is the effective measure of the duration of a bond. Duration is shorter than maturity for all bonds except zero coupon bonds. Duration is equal to maturity for zero coupon bonds.
For three reasons duration is a usefull concept for fixedincome portfolio management. First it is a simple summary statistic of the effective average maturity of the portfolio. Second, it is a useful tool to immunize portfolios from interest rate risk. Third, it is a measure of interest rate sensitivity.
In formula duration:
See formula sheet, formula 43
An example to calculate duration
8% bond  time years  Payment  PV of CF(10%)  Weight  C1*C4 

 0,5  40  38.095  0.395  0.0197 

 1  40  36.281  0.0376  0.0376 

 1,5  40  34.553  0.357  0.537 

 2  1040  855.611  0.8871  17.742 

Sum 

 964.540  1  18.852  Duration 







CPrice change is proportional to duration and not to maturity.
^P/P = D x [^(1+y) / (1+y)
D* = modified duration
D* = D / (1+y)
^P/P =  D* x ^y
Note the convexity of this function. The priceyield relationship is a convex relationship. Convexity is the rate of change of the slope of the curve as a fraction of the bond price:
Rule 1: the duration of a zerocoupon bond equals its time to maturity.
Rule 2: holding maturity constant, a bond’s duration is higher when the coupon rate is lower.
Rule 3: holding the coupon rate constant, a bond’s duration generally increases with its time to maturity.
Rule 4: holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower.
Rules 5: the duration of a level perpetuity is equal to: (1+ y) : y
Different sort of durations:
Duration. A measure of the average life of a bond, defined as the weighted average of the times until each payment is made, with weights proportional to the present value of the payment.
Macauly’s duration. Effective maturity of bond, equal to weighted average of the times until each payment, with weights proportional to the present value of the payment.
Modified duration. Macauly’s duration divided by 1 + yield to maturity. Measures the sensitivity of the bond.
Effective duration. Percentage change in bond price per change in the level of market interest rates.
As a measure of interest rate sensitivity, duration is a critical tool in fixedincome portfolio management. But the duration rule for the impact of interest rates on bonds is only an approximation. The duration rule is a good approximation for small changes in bond yield, but it is less accurate for large changes. This point is illustrated in figure 16.4 on page 532. The true priceyield relationship is a curvature. Curves with shapes such as the priceyield relationship are said to be convex, and the curvature of the priceyield curve is called the convexity of the bond. As figure 16.4 shows we want to compensate for the convex curvature of the bond. We do this with the following formula:
See formula sheet, formula 44
If we correct the formula for convexity we get:
See formula sheet, formula 45
Investors think that convexity is a desirable characteristic of a bond. The reason is that bonds with greater curvature gain more in price when yields fall than they lose when yields rise. Although convexity is desirable it is not available for free, investors have to pay more and accept lower yields on bonds with greater convexity.
Passive fixedincome portfolio management has two broad categories, indexing and immunization strategies.
Bond indexing basically composes a portfolio that mirrors the broad market and is similar to stock market indexing. Some differences exist however. It is for example much more complicated to keep track of the owners of bonds, and the bonds available at the market change continuously. A cellular approach is used to solve such practical problems.
Immunization strategies attempt to render the individual of firm immune from movements in interest rates. This may take the form of immunizing net worth or instead immunize the future accumulated value if a fixed income portfolio. We can accomplish immunization by matching the durations of assets and liabilities. If we want to maintain an immunized position we need to rebalance the portfolio over time, the reason is that as time passes interest rates pass as well.
This classical approach to immunization also depends on parallel shifts in a flat yield curve. Given that this assumption is unrealistic, immunization generally will be less than complete. To solve this problem, multifactor duration models can be used to allow for variation in the shape of the yield curve. A more direct form of immunization is dedication or cash flow matching. If the portfolio is perfectly matched in cash flow with projected liabilities, rebalancing will not be needed.
Active bond management could be divided in two broad categories. First there is interest rate forecasting, when managers use techniques to adjust their portfiolios to movements across the markets. An example of such a technique is horizon analysis, adjusting its strategies based on a particular holding period.
The second categorie is searching for relative mispricing within the fixedincome market. Interest rate swaps are common techniques of active bond management:
Interest rate swaps are major recent developments in the fixed income market. In these deals parties trade the cash flows of different securities without actually exchanging any security directly. This is a useful tool to manage the interestrate exposure of a portfolio. Five categories of swaps can be identified:
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