Theories of the term structure
Comparison" Suppose 1 year interest rate is expected to be the same this year and next year : When SHORT investors dominate, term structures are sloping : When LONG investors dominate, term structures are sloping Observed fact: term structure is upward sloping : Conclusion: investors dominate This is called the Liquidity Preference Theory
Liquidity
Observed fact: TS is mostly upward sloping : So SHORT investors appear to dominate Buyers of long-term bonds want to be compensated : for "tying up" money for a long time : for having risk if they need to sell before maturity Issuers of bonds are willing to pay a higher interest rate on long-term bonds because : they can lock in an interest rate for many years The associated risk premium is called the liquidity premium
Interest rate sensitivity and Duration
We've seen that when interest rates go up, bond prices fall Exact example: 2 year annual payment bond, coupon rate 10%, face value $1000 What is price when YTM=10%? How about when YTM increases to 11%? Almost linear Would like to estimate how much prices fall : Natural to think of taking derivatives .What is the sign of dP/dy?
The duration (D) of a bond is defined as minus the elasticity of its price (P) with respect to (1 plus) its yield (y): .Why such a funny definition?
d P
1
And if you actually do the differentiation, you find
PV(cashflow at t at YTM)
D
wtt, where wt :
P
Weights sum to 1. Why? Since duration is "weighted times", it's measured in years Natural to think of it as weighted-average-maturity: longer the life, more the sensitivity
Example
What is the duration of a $1000 face value, 4-year coupon bond with an annual coupon of 8% and a YTM of 10% ?
Using duration
Can quickly calculate change in price given a change in yield
d P I P
AP P
D
1+y modified duration
Example
What is the duration of a 4-year coupon bond with an annual coupon of 8% and a YTM of 10% ? If the YTM changes to 10.1%, what would be the (relative) change in price ? If the YTM changes to 11%, what would be the (relative) change in p