CHAPTER 16 MANAGING BOND PORTFOLIOS
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Outline of the Chapter
• Bond pricing and sensitivity of bond pricing to interest rate changes • Duration analysis
– What is duration? – What determines duration?
• Convexity • Passive bond management
– Immunization
• Active bond management
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Interest Rate Risk
• There is an inverse relationship between interest rates (yields) and price of the bonds. • The changes in interest rates cause capital gains or losses. • This makes fixed-income investments risky.
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Interest Rate Risk (Continued)
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Interest Rate Risk (Continued)
• What factors affect the sensitivity of the bonds to interest rate fluctuations? • Malkiel’s (1962) bond-pricing relationships
– Bond prices and yields are
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Convexity (Continued)
• The duration rule is a good approximation for small changes in bond yields. • The duration approximation always understates the value of the bond.
• It underestimates the increase in price when yields fall. • It overestimates the decline in prices when yields rise. •Due to the curvature of the true price-yield relationshipconvexity
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Convexity (Continued)
• Convexity is the rate of change of the slope of the price-yield curve, expressed as a fraction of the bond price.
– Higher convexity refers to higher curvature in the price-yield relationship. – The convexity of noncallable bonds are usually positive. – The slope of the cuve that shows the price-yield relation increases at higher yields.
Convexity
1 P (1 y ) 2
n t 1
CFt (t 2 t ) (1 y )t
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Convexity (Continued)
• We can improve the duration approximation for bond price changes by taking into account for convexity. • The new equation becomes:
P P
D
y
1 [Convexity ( y ) 2 ] 2
• The convexity becomes more important when potential interest rate changes are larger.
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Convexity (Continued)
• Why convexity is important? • In the figure bond A is more convex than bond B. •The price increases are more in A when interest rates fall. •The price decreases are less in A when interest rates rise.
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•
Inflation erodes the purchasing power of a bond 's future cash flows. A rise in inflation will cause investors to demand higher yields to compensate for inflation rate risk. Also, prices will tend to drop because the bond will be paying interest with less purchasing power.
2. Jordan, Bradford D., Susan D. Jordan, and David R. Kuipers. "The Mispricing of Callable U.S. Treasury Bonds: A Closer Look." Journal of Futures Markets 18.1 (1998): 35-51. Web.
The yield to maturity on a 15-year bond is a true estimate of the cost of 30-year bond
∆P/P = –D*(∆y) D* = D/(1 + y) = 7/1.073 = 6.52 ∆P/P = –D*(∆y) = –6.52(–0.09%) = .59% New price = $1,073(1.0059) = $1,079.33 Learning Objective: 11-02 Compute the duration of bonds; and use duration to measure interest rate sensitivity.
• Shorter terms have less risk of default and inflation because the near future is easier to predict. Broadly speaking, if interest rates increase, then investment decreases due to the higher cost of borrowing (all else being equal).
10) The term structure of interest rates is a graphical presentation of the relationship between the
There is a large number of approaches to evolve real-world yield curves including non-, semi- or parametric ones based on historical data as the primary source of real-world model calibration. In general, their common feature is that they preserve historical quote patterns and statistical properties including the following important ones:
The liquidity preference theory states that bond holders are risk averse and wish to be compensated for holding the long term security by a liquidity premium. The normal yield curve to explain this should be up sloping indicating preference for liquidity by investors and lower risk of shorter term
The 3-month T-bill rates and Dow Jones index are really close to the whole economic environment; the 3-month T-Bill rates are the preeminent default-risk-free rates in the US money market that is often used by
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The most popular way in which duration is employed by the investors is to measure the volatility of the bond or any other financial security. As time progresses, stock prices and yields fluctuate and in most cases, this fluctuation was not foreseen. Due to this reason, investors usually select a specified period to measure the performance of securities, which is significant identifying the level of risk associated with their
Concave curve: Shows the increase in opportunity cost, which means when resources are better to produce bread as compare to house so the opportunity cost of house will be higher than the opportunity cost of bread. (we need to sacrifice more unit of bread to produce less no of house)
As discussed in the lecture notes, duration is nothing but a metric to quantify bond price risk (price sensitivity to a change in interest rate). Let me give you a naïve example, but it really helps understand the duration concept. Supposed that there are two securities: A and B. A gives you $1 next year and $999 in 100 years whereas B gives you $999 next year and $1 in 100 years. Both A and B have the “SAME” maturity of 100 years.
Suppose there is diseconomies of scale or decreasing returns to scale. Under full utilization of resources, the production possibility frontier is concave towards the origin. This shape is basically due to the differences in optimal factor intensities between industries. If a country faces increasing opportunity costs or marginal rate of transformation (MRT) in producing more units of a commodity, then this is shown by a PPF that is concave. The country will produce where the MRT is equal to the equilibrium relative commodity price.
It can be seen that if the interest rate rises then the price of such fixed return bonds fall, making bonds less attractive proposition to investors than money. When the price of such fixed return bond rises, then they are more attractive for an investor than money. Some investors will expect the rate of interest to rise, while others will expect it to fall. When the rate of interest is realised to be unduly high by individuals, most will assume that the next move is in a downward direction. When the rate of interest falls, the price of bonds increases and so there are capital gains to be made. When the rate of interest is high there will be a substantial demand for bonds and hence a low speculative demand to hold money. If, however, the rate of interest is perceived to be unduly low then most individuals will assume that the next move is upwards, resulting in a fall in bond prices and, therefore, a capital loss for those who own bonds. The speculative demand for money to hold will be high when people owing bonds are looking to sell them before the price falls.