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- Introduction
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- 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.
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- Factors Influencing Interest Rate Level Back to Top
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- 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.
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- Factor 1: Disequilibria in the Money Demand-Supply System
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- 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.
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- Factor 2: Interplays among Macroeconomic Factors
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- 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.
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- Factor 3: Intervention by Government Through Active Fiscal and
Monetary Policies
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- 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.
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- Theory and Application of The Term Structure of Interest Rate Back to Top
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- 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.
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- There are four hypotheses regarding the term structure: 1) expectations
hypothesis, 2) liquidity preference hypothesis, 3) market segmentation hypothesis, and 4)
preferred habitat hypothesis.
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- Expectations Hypothesis
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- 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.
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- Liquidity Preference Hypothesis
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- 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.
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- Market Segmentation Hypothesis
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- 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.
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- Preferred Habitat Hypothesis
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- 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.
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- Using The Term Structure to Forecast Future Interest Rates
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- 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:
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- 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
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- If the investors expect that the second 6-month bond would yield 11%,
then
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- Two 6-month bonds FV = (1+0.0954/2)+(1+0.1100/2) = 1.105
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- 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.
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- 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:
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- 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)
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- Solving for r gives the implied 6-month forward rate:
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- r = 0.103 or 10.3%
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- In general, the implied forward rate formula is given by:
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- rt2-t1,t1 = n[{(1+rt2,0/n)nt2(1+rt1,0/n)-nt1}1/n(t2-t1)
- 1]
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- 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
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- 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:
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- ym,0 = 2[{P2mi=0 (1+ri/2,0/2)}1/2m - 1]
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- where
- ym,0 = yield of an m-year bond today
- ri/2,0 = spot rate of maturity 1/n for period i/2
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- The implications from the implied forward yield curve can be summarized
as follows:
- A flat yield curve implies unchanged rates in the future.
- A positively sloped yield curve implies higher rates in the future.
- A negatively sloped yield curve implies lower rates in the future.
- Theory and Application of Bond Duration Back to Top
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- 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.
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- B = c/y + (F-c/y)(1+y/2)-n
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- where
- B = market value of bond
- c = annual coupon rate
- y = yield to maturity
- F = face value of bond
- n = time to maturity
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- The relationships between B, c, y, and F are the following:
- If y = c, then B = F; this implies that the bond price is equal to its
face value (par).
- If y < c, then B > F; this implies that the bond price is traded
at premium.
- 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).
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- Duration
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- 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
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- 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.
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- Modified Durations
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- 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:
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- DM = - (100/B)(dB/dy) = 1/B[c/y2{1-(1+y/2)-n}
+ n/2(F-c/y)(1+y/2)-n+1]
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- 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
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- Alternatively, Macaulay duration can be stated in the following fashion:
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- DM = 1/B[C1(1+y)-1] + 2/B[C2(1+y)-2]
+ ... + n/B[(Cn+F)(1+y)-n]
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- where
- Ci = annual coupon payment at year i, i = 1,2,...,n
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- 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)}]
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- where
- DF-W = Fisher-Weil duration
- ri = estimated future interest rate at year i, i = 1,2,...,n
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- 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:
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- rt - rt-1 = E[rt-rt-1] + szr½t-1
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- 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
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- 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.
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- Convexity
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- 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:
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- 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}]
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- 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.
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- Using Duration to Immunize Bond Portfolio Returns
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- 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.
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- 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.
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- Conclusion Back
to Top
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- 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.
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- 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.
Back to Top
* 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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