SYLLABUS Previous: 5.1 Discound bonds
Up: 5.1 Discound bonds
Next: 5.1.2 Parameters illustrated with
5.1.1 Term structure models for dummies
[ SLIDE
hedging 
experiments 
forecasting models 
VIDEO
login]
Imagine a portfolio with two identical discount bonds, except that
the first
expires some time before the second
.
What is the effect of a market fluctuation, which suddenly rises the
spot rate at a time
before the first bond reaches maturity?
The bonds are correlated and both will loose some of their original
value; since there is more time left for another fluctuation to step
back in the opposite direction, it is reasonable to assume that the
second bond with a longer time to maturity will be less affected.
Taking advantage of this correlation, Vasicek creates a portfolio with a
positive holding in the first bond and a negative holding in the second.
By choosing exactly the right balance, this deltahedging cancels out
the uncertain effect from fluctuations and leaves only a deterministic
change in the portfolio value. This is then used to calculate the fair
price of a bond.
The normalized value of the discount function is of course known at the
maturity
and the calculation is carried out with a forecast
of the interest rates backward in time to predict the fair value
for an increasing lifetime
.
The VMARKET applet below illustrates the
procedure for a bond lifetime with up to RunTime=10 years.
VMARKET applet: press Start/Stop
to simulate the price of a zerocoupon bond backward in time, for a
market with a volatile spot rate paying a reward for the associated
risk.
The plots show the value of the discount function as a function of the
spot rate (P[r] in black) for an increasing time to maturity
t (Time on the top of the window, in years). Directly
derived from that using (2.2.2#eq.1),
two plots show the evolution of the yield curve (Y[r] in blue,
for a fixed Time) and the term structure of the interest rates
(Y[t] in grey, for a fixed SpotRate).
The latter acquires a finite value and sweeps across the plot window
over the time span of one simulation [0; RunTime] and is best
viewed after rescaling with Display.

For a given value of the spot rate
(horizontal axis, chosen to
reflect the current market conditions), the discount function
is decreasing backward in time
. Indeed, investors expect a return
from their investment, which shows up as a growth of the discount
function when the time runs forward so as to reach exactly one at
maturity. The reward can be measured using (2.2.2#eq.1) as a yield
and differs from the spot rate
because of
the uncertain evolution of the future rates.
Due to the cyclic nature of the economy and the changes in the central
bank interest rates, economists generally forecast what may be the
future evolution of spot rates
with
.
This opinion consists of a drift (``the spot rate will fall'') and a
volatility (``the spot rate will fluctuate'') that can be estimated
from historical values (exercise 1.05).
