Chapter 15 - Term Structure of Interest Rates Flashcards
Why are interest rates more difficult to model compared to share prices?
Interest rates depend on current time t, and the term of the investment.
Share Prices are only dependent on time.
What is the price of a zero-coupon bond with constant interest rate R between time t and T?
P(t,T) = 1/(1+R(t,T))^(T-t)
Define the spot rate R(t,T).
The spot rate is the average effective interest rate for time t to T, pa.
What is the formula to calculate the spot rate R(t,T) if we have the price of a zero-coupon Bond P(t,T).
R(t,T) = 1/P(t,T)^(1/(T-t)) -1
What is the formula for the price of a zero-coupon bond at current time t, maturing at time T, with force of interest r(t,T)
P(t,T) = exp(-r(t,T)*(T-t))
What is the formula for the force of interest, r(t,T), using the price of a zero-coupon bond at current time t, maturing at time T.
r(t,T) = -ln(P(t,T))/(T-t)
What is r(t,T)?
The continuously compounded spot rate over time t to T - it is the average force of interest over the period from time t to time T.
What is R(t,T)?
The annual effective spot rate that represents the average effective rate of interest over the period of time t to time T.
Define the Expectation Theory
The expectations theory argues that the long-term rate is determined purely by current and future expected short-term rates, so that the expected final value of investing in a sequence of short-term bonds equals the final value of wealth from investing in long-term bonds.
Define the Market Segmentation Theory
The market segmentation theory argues that different agents in the market have different objectives: pension funds determine longer-term rates, market makers determine short-term rates, and businesses determine medium-term rates, which are all determined by the supply and demand of debt for these different market segments.
Define the Liquidity Preference Theory
The liquidity preference theory argues that lenders want to lend short term while borrowers wish to borrow long term, and so forward rates are higher than expected future zero rates (and yield curves are upward sloping).
What is the short rate r(t)?
The short or instantaneous rate, r(t), is force of interest applying at time t for a very short period (ie overnight).
What is the Forward Rate F(0,t,T)?
F(0,t,T) defines the annual effective forward rate of interest over the time period t to time T by market prices at time 0.
What is the formula for F(0,t,T)?
(1+F(0,t,T)^(T-t)=P(0,T)/P(t,T)
F(0,t,T) = ]P(0,T)/P(t,T)]^(1/(T-t)) -1
What is the Forward Rate f(0,t,T)?
F(0,t,T) defines the continuously compounded forward rate of interest over the time period t to time T by market prices at time 0.
What is the formula for f(0,t,T)?
exp(f(0,t,T)*(T-t)) = P(0,T)/P(0,t)
f(0,t,T) = 1/(T-t) * ln(P(0,T)/P(0,t))
What is f(0,T)
The instantaneous rate as lim t->T f(0,t,T) = f(0,T)
Prove f(0,T) = -d[ln(P(0,T)]/dT
What is the relationship between P(t,T) and f(0,T)?
P(t,T) = -exp(int[t,T]f(0,T) dt)
What desirable features should a term structure model?
- The model should be arbitrage free
- Interest rates should exhibit some form of mean-reverting behaviour
- Interest rates should ideally be positive
- Bonds and derivative contracts should be easy to price.
- It should produce realistic interest rate dynamics.
- It should fit historical interest rate data adequately.
- It should be easy to calibrate to current market data.
- It should be flexible enough to cope with a range of derivatives
Why are they called one factor models?
Only one factor of randomness, i.e. the short rate r(t)
What is the risk neutral approach
What is the formula for the Vasicek model?
drt = alpha(mu-rt)dt + sigmadWt
What is the Cox-Ingersoll-Ross model?
drt = alpha(mu-rt)dt + sigmasqrt(rt)*dWt
What is the hull-white model
drt = alpha(mu(t)-rt)dt + sigmadWt
Discuss the Vasicek Model.
Allows for negative interest rates - which is not realistic
Constant sigma - imperial evidence to show volatility is not constant especially in times of economical uncertainty where volatility is higher
Arbitrage free
Mean reverting - good as empirical evidence to suggest interest rates are mean reverting
Mathematically tractable
Positive correlation between yields on bond of diff duration - not realistic
“Static” interest rates when high or when low - model doesnt allow for that
Discuss the CIR Model
Does not allow for negative interest rates (because of the sqrt) - realistic
Constant sigma - imperial evidence to show volatility is not constant especially in times of economical uncertainty where volatility is higher
Arbitrage free
Mean reverting - good as empirical evidence to suggest interest rates are mean reverting
Not very mathematically tractable because of the square root
Positive correlation between yields on bond of diff duration - not realistic
“Static” interest rates when high or when low - model doesn’t allow for that
