Bond Portfolio Management: A Conceptual Review

Worapot Ongkrutaraksa*
Spring 1997
 

Abstract

Three issues in the area of bond portfolio management are discussed in this review essay namely, first, the factors that influence the level of interest rates, second, the theory and application of the term structure of interest rate or yield curve, and third, the theory and application of duration and bond portfolio immunization.
 
 
Introduction
 
The equilibrium pricing models such as the capital asset pricing model (CAPM) and the arbitrage pricing theory (APT) attribute the expected security returns as the function of term premium of risk-free security and risk premium of risky market portfolio. This two-fund separation, in effect, divides capital markets into two parts: one for riskless assets such as government securities and less-risky corporate debentures and the other for more risky assets such as common stocks. Therefore, the study of term premium is no less important than the study of market risk premium in the determination of stock or portfolio returns.
 
Factors Influencing Interest Rate Level Back to Top
 
Changes in the level of interest rate is has lent significant impact on bond and stock portfolios management because interest rate serves as a foundation for the expected returns on those portfolios. This time-varying variance or volatility of interest rate level is basically the result of differences in term to maturity of the underlying fixed-income assets. From the standpoint of an economic agent who maximizes her expected utility level based on wealth and income subject to her risk and time preferences, the real interest rate compensates such individual for deferring present consumption by investing in the risk-free security. Time preference serves to differentiate between the agent's transaction demand (consumption) and speculative demand (investment) for money. Not only are speculative money demanders compensated by the real interest rate, they also earn inflation premium for the loss in their purchasing power of the returns they expect to receive from risk-free security. The sum of the real interest rate and the inflation premium, called nominal interest rate, therefore links the level of interest rate to the level of price of goods and services in the economy which, in turn, influence the time preference of all economic agents. There are three factors that can influence interest rate level: 1) disequilibria in the money demand-supply system, 2) interplays among the macroeconomic factors, and 3) government intervention through active fiscal and monetary policies.
 
Factor 1: Disequilibria in the Money Demand-Supply System
 
In an open economy, the equilibrium level of interest rate is determined by the interaction between the aggregate demand for and aggregate supply of money by all economic agents. When there is a disequilibrium between the demand for and supply of money, interest rates will fluctuate and adjust until they reach the equilibrium levels. Whenever the demand for money, either transaction, speculative or both, exceeds the money supply, there are incentives for the investors to liquidate their current asset holdings and reinvest in other risk-free securities that pay higher real interest rates. These higher levels of interest rates will lower the excess demand until it equals supply. The opposite is true whenever there is excess money supply in the market.
 
Factor 2: Interplays among Macroeconomic Factors
 
Beside the disequilibria in money demand and supply system, the level of interest rates is also indirectly affected by prime macroeconomic factors including gross national product (GNP), aggregate private-sector investment, and the level of domestic savings. GNP measures the level of national income in each year which affect the money demand and supply system. An increase in GNP will increase demand for money which puts an upward pressure on the level of interest rate. Larger national income tends to increase consumption and/or saving levels of the households and investment expenditures in the private sector. Households' consumption will induce more domestic production which requires higher investment by the firms. Firms raise their capital for long-term investment through issuing corporate bonds and common stocks. As demand for production and investment rises, the firms are pressured to pay higher interest rates on their securities to attract funds. They can also borrow funds from financial intermediaries which attract domestic savings through increased interest rates. Therefore, the interplays among these three macroeconomic factors can influence the changes in the level of interest rate quite considerably.
 
Factor 3: Intervention by Government Through Active Fiscal and Monetary Policies
 
Government can influence the level of interest rates directly through the implementation of fiscal (taxation, subsidy, and spending) and monetary (bank reserves, discount window, open market operation, and exchange rate regime) policies which alter the money supply in the economy. There are three effects occur as a result of such policies: 1) liquidity or substitution effect, 2) income effect, and 3) price effect. As money supply expands due to tax rate reduction or higher government spendings, the excess money supply will change the liquidity preference of the individuals by substituting investment for consumption through increased savings and investment in securities which put downward pressure on the interest rates. However, tax cut also increases households' disposable income and firms' profit. This income effect will increase consumption by households will stimulate investment which puts upward pressure on the interest rates. The increased income may also put an inflationary pressure on price which drives up the nominal interest rates through higher inflation premium. Therefore, while liquidity effect reduces real interest rates level, income effect raises it. The price effect will result in the nominal interest rates level depending on whether or not the income effect is inflationary. When money supply contracts, liquidity, income, and price effects work in the opposite directions.
 
Theory and Application of The Term Structure of Interest Rate Back to Top
 
Cox, Ingersoll, and Ross (1985) define the term structure of interest rates as the measure of the relationship among the yields on risk-free securities that differ only in their term to maturity. The yield is a rate at which the present value of all future payments of interest and principal is equated to the market price of the security. When the security is held until its maturity, the resultant yield is called yield to maturity (YTM). A plot of yields as a function of time to maturity is a yield curve. The yield curve is positively sloped, which means that the yield of long-maturity securities are higher than the yields of short-maturity securities. It contains the market's anticipations of future events which allows financial economists to predict how changes in the underlying economic variables will affect the yield curve.
 
There are four hypotheses regarding the term structure: 1) expectations hypothesis, 2) liquidity preference hypothesis, 3) market segmentation hypothesis, and 4) preferred habitat hypothesis.
 
Expectations Hypothesis
 
Fisher (1896) states that bonds are priced so that the implied forward rates are equal to the expected spot rates or holding-period returns. Its underlying assumption is that the return on holding a long-term bond to maturity is equal to the expected return on repeated investment in a series of the short-term bonds. Alternatively, the expected rate of return over the next holding period is the same for bonds of all maturities. Thus, it implies that the YTM on an n-year bond is equal to the average of the YTM on one-year bond for the next n years. In effect, the risk-neutral investors would be indifferent holding either bonds. This condition will not hold when investors are risk-averse between holding short- and long-maturity bonds.
 
Liquidity Preference Hypothesis
 
Hicks (1939) argues that when investor preference about risk is included in the assumption of expectations hypothesis, the implied forward rates will differ from the holding-period returns. Risk aversion will cause forward rates to be systematically higher than expected spot rates by an amount increasing with time to maturity. This term premium is required by the investors as compensation for taking more liquidity risk in holding the long-term bonds instead of a series of shorter-term bonds. The impact of term premium on the term structure is that it would cause the yield curve to be more upward sloping when the interest rates are expected to rise and less downward sloping when the interest rates are expected to fall.
 
Market Segmentation Hypothesis
 
Culbertson et al. (1957) offer a different explanation of the term premium by postulating that 1) individuals have strong liquidity preferences and risk aversion and 2) bonds of different maturities are traded in separate and distinct markets. The demand for and supply of bonds of a particular maturity are mildly affected by the prices of bonds of neighboring maturities. Based on these postulates, there should be no reason for the term premia to be positive or increasing functions of maturity. This implies that bonds with close maturities will not be the close substitutes for each other. Under this hypothesis, the yield curve is not derived by the expectations of spot rates or by the term premia, but by the attempts of the investors to match their assets maturity with the maturity of their liabilities in different markets.
 
Preferred Habitat Hypothesis
 
Based on the market segmentation hypothesis which assumes that investors in short-term bond market are reluctant to invest in the long-term bond market, Modigliani and Sutch (1966) pose some limitations that individuals can have specific preferences about timing of their consumption and the risk preference for the characteristics of other investment alternatives in other markets, i.e., preferred habitat. This hypothesis offers a more plausible rationale for term premium to co-exist with market segmentation rather than being a necessary causal explanation for term structure of interest rate.
 
Using The Term Structure to Forecast Future Interest Rates
 
In order to earn the highest holding-period returns from bond investment, the investors seek the accurate forecasts of the future interest rates given the current market prices of bonds with different maturities. There are two methods to forecast future interest rates: one is to calculate the forward return and the other is to calculate the implied forward rate. The first method deals with the comparison between the future values of two alternative investments, for example, a one-year bond yielding 9.92% and two 6-month bonds yielding 9.54%. The calculation of the future values of these two investment strategies is straightforward:
 
A one-year bond FV = (1+0.0992/2)+(1+0.0992/2) = 1.100
Two 6-month bonds FV = (1+0.0954/2)+(1+0.0954/2) = 1.098
 
If the investors expect that the second 6-month bond would yield 11%, then
 
Two 6-month bonds FV = (1+0.0954/2)+(1+0.1100/2) = 1.105
 
Therefore, the investors will be earn higher holding-period return by investing in the one-year bond if the expected 6-month rate six month from now remains unchanged. However, if such rate were expected to rise, they will be better off investing in two 6-month bonds. This first method depends very much on the subjective belief of the investors about the future short-term interest rate.
 
The second method provides the forecasts of future interest rates based on the implied forward rate, i.e., the rate at which two alternative investments are equally profitable that would make the investors indifferent to maturity when choosing the bond(s). By equating the future value of the one-year bond to that of the two 6-month bonds:
 
Two 6-month bonds FV = A one-year bond FV
(1+0.0992/2)(1+0.099./2) = (1+0.0954/2)(1+r/2)
 
Solving for r gives the implied 6-month forward rate:
 
r = 0.103 or 10.3%
 
In general, the implied forward rate formula is given by:
 
rt2-t1,t1 = n[{(1+rt2,0/n)nt2(1+rt1,0/n)-nt1}1/n(t2-t1) - 1]
 
where
rt2-t1,t1 = implied forward rate from t1 until time t2-t1
rt2,0 = rate from today until time t2
rt1,0 = rate from today until time t1
n = number of reinvestment periods per year
 
The implied forward rates for all maturities at the same future date can be calculated to give the implied forward yield curve. Its formula is given by:
 
ym,0 = 2[{P2mi=0 (1+ri/2,0/2)}1/2m - 1]
 
where
ym,0 = yield of an m-year bond today
ri/2,0 = spot rate of maturity 1/n for period i/2
 
The implications from the implied forward yield curve can be summarized as follows: 
  1. A flat yield curve implies unchanged rates in the future.
  2. A positively sloped yield curve implies higher rates in the future.
  3. A negatively sloped yield curve implies lower rates in the future.
Theory and Application of Bond Duration Back to Top
 
The valuation of bonds and other fixed-income securities not only requires the knowledge of yield curve which provide the investors with an information about the term structure and term premium of the interest rate, but also the degree of bond price's sensitivity to the change in interest rates. Duration is a used as a measure of interest rate risk of bond and other the fixed-income securities which causes the volatility in their market prices. Bond valuation and bond volatility are linked together through the yield curve. In order to establish the foundation for duration concept, it is imperative that the generic bond valuation model be presented.
 
B = c/y + (F-c/y)(1+y/2)-n
 
where
B = market value of bond
c = annual coupon rate
y = yield to maturity
F = face value of bond
n = time to maturity
 
The relationships between B, c, y, and F are the following: 
  1. If y = c, then B = F; this implies that the bond price is equal to its face value (par).
  2. If y < c, then B > F; this implies that the bond price is traded at premium.
  3. If y > c, then B < F; this implies that the bond price is traded at discount.
The reason why the yield curve does not provide a good measure of bond volatility is because it ignores the timing and magnitude of interest rates fluctuation between the period before maturity. Thus, the inherent risks associated with bond trading beside term risk are: 1) bond price risk (duration), 2) interest rate risk (modified durations), and 3) yield risk (convexity).
 
Duration
 
The price risk of bond or dollar duration is the measure of the magnitude of change in bond price per unit change in yield. In other words, it is measures the rate of change, or speed, in bond price with respect to a change in yield. The formula is given by:
RB = -dB/dy
where
RB = price risk of bond
dB = instantaneous change in bond price
dy = instantaneous change in yield to maturity
 
The negative sign indicates that the relationship between bond price and yield is always inverse. Bond price risk is normally expressed as the change in price per 1% change in YTM.
 
Modified Durations
 
Macaulay (1938) suggests the modification to dollar duration in order to quantify the interest rate risk. Macaulay duration measures the magnitude of the percentage change in bond price including accrued interest with respect to a change in its yield. Where dollar duration measures price movement as a percentage of face value, modified duration measures price movement as a percentage of dollars invested. The formula to derive Macaulay duration is:
 
DM = - (100/B)(dB/dy) = 1/B[c/y2{1-(1+y/2)-n} + n/2(F-c/y)(1+y/2)-n+1]
 
where
DM = Macaulay duration
B = market value of bond
F = face value of bond
c = annual coupon rate
y = yield to maturity
n = time to maturity
 
Alternatively, Macaulay duration can be stated in the following fashion:
 
DM = 1/B[C1(1+y)-1] + 2/B[C2(1+y)-2] + ... + n/B[(Cn+F)(1+y)-n]
 
where
Ci = annual coupon payment at year i, i = 1,2,...,n
 
Later, Fisher and Weil (1971) derive a more general modified duration formula which allows for future interest rates to change:
DF-W = 1/B[C1/(1+r1)] + 2/B[C2/{(1+r1)(1+r2)}] + ... + n/B[(Cn+F)/{(1+r1)(1+r2)×××(1+rn)}]
 
where
DF-W = Fisher-Weil duration
ri = estimated future interest rate at year i, i = 1,2,...,n
 
Finally, Cox, Ingersoll, and Ross (1979) allow for more complicated changes in the level of interest rates by assuming that the interest rate on the shortest-term bond changes over time in the following manner:
 
rt - rt-1 = E[rt-rt-1] + szr½t-1
 
where
rt-rt-1 = change in interest rate from t-1 to t
E[×] = expected change in interest rate
s = standard deviation of interest rate
z = random shock
 
The specification of Cox-Ingersoll-Ross duration is the most complicated among the three measures of modified duration. Evidently, Macaulay duration is the simplest but as effective as either of the other two. These modified durations are used to immunize the bond portfolio returns from the volatility in the future interest rates.
 
Convexity
 
Convexity is the rate of change in the slope of the price-yield relationship and always positive. In other words, convexity measures the acceleration of bond price change with respect the change in yield. Because convexity tends to be a small number, its standard definition includes multiplication by a scale factor of 100. Convexity formula is given by:
 
CX = (1000/B)(d2B/dy2)
CX = B-1[n(n+1)(F-c/y)/4(1+y/2)n+2 - cn/y2(1+y/2)-(n+1) - 2c/y3{(1+y/2)-n-1}]
 
Mathematically, convexity is the second derivative of the duration and/or modified durations. Therefore, it can be said that convexity measures the change in duration with respect to the change in yield. If duration were constant at all level of yield, convexity would not exist. The change in yield thus creates the duration, modified durations, and convexity effects.
 
Using Duration to Immunize Bond Portfolio Returns
 
The concept of duration as a measure of bond price volatility is widely applied in the bond portfolio management apart from the use of implied forward yield curve concept to maximize the bond portfolio returns. A prominent application of duration in hedging the bond portfolio returns against interest rate risk is called the interest immunization. The objective of interest immunization is to match the value of one bond portfolio with the value of another debt portfolio (i.e., present value of the liabilities). Duration matching can be used to immunize the fluctuation in interest rates by structuring the bond portfolio so as to match the duration and the performance of a debt portfolio. If both portfolios have the same average duration, then equal amount invested in and borrowed from either portfolios will produce the same returns for a given change in the interest rate. When interest rates change, durations of both bond and debt portfolios have to be rematched because the present and maturity values of both portfolios are affected. Duration matching differs from cash matching because there is a variety of bond combinations that can create the duration match.
 
Douglas (1990) identifies the advantages of duration matching as follows. First, duration matching is cheaper and more flexible to implement than cash matching. The flexibility is in the selection of individual bond to satisfy the interest immunization. The immunized bond portfolio can be structured to take advantage of undervalued issues in the bond market. Second, the immunized bond portfolio experiences market fluctuations identical to those if the debt portfolio. Duration matching protects the asset-liability structure from general movements in interest rates. There are also some limitations for duration matching. Often, the immunized bond portfolio requires periodic rebalancings to maintain the duration matching. Additional rebalancing is necessary if the debt portfolio is altered. Sometimes, the long-duration immunized bond portfolio can be adversely affected by non-parallel interest rate shifts such as the frequent changes in yield curve shapes, coupon spread and quality spread relationships. Moreover, the immunized bond portfolio is exposed to a modest degree of market risk and reinvestment risk. And finally, the duration matching is difficult to implement if the debt portfolio also has a long duration.
 
Conclusion Back to Top
 
Since bond and fixed-income portfolios are held by most institutional investors in addition to stock portfolios, their relative importance in terms of total portfolio returns necessitate a closer analyses of bond valuation and bond volatility. The underlying determinants of bond value are the nominal interest rates (i.e., real interest rate plus inflation premium) and their term premia. Term premium results from the hypotheses that investors have different time preferences when trading bonds with different maturities in different markets. Both nominal interest rate and term premium combine to form the bond yield thereby determining the market value of bond. The yield curve - a relationship between the bond yield and its maturity - represents the term structure of the interest rate. A positively sloping yield curve implies that the future interest rates will be higher, and vice versa. This term structure helps the bond investors to form their expectations correctly in order to maximize their bond portfolio returns.
 
The change in yield curve affects the bond price through duration (interest rate risk) and convexity. Duration gives the bond investors how sensitive the bond prices are to the changes in bond yields or the interest rates. As yield increases, duration decreases and vice versa. The rate of change in duration is measured by convexity which is used in bond duration matching. Bond portfolio can be immunized from the effect of yield fluctuation through the use of both duration and convexity. We can relate bond price, coupon, yield, duration, and time to maturity by the following conclusions: 1) price is inversely related to yield but directly related to duration, 2) a decrease in yield raises price more than the same increase in yield lowers price, and 3) if yield equals coupon, then duration and time to maturity are directly related.

 

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* Worapot Ongkrutaraksa is a lecturer in Finance and Strategic Management at Maejo University's Faculty of Agricultural Business, Chiang Mai, Thailand. He used to conduct his post-graduate research in financial economics at Kent State University and international political economy at Harvard University through the Fulbright sponsorship between 1995 and 1998.

E-mail: worapot@iname.com

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