Philip H. Dybvig
Washington University in Saint Louis
spot rate r_t: quoted at t-1 for borrowing/lending from t-1 to t
forward rate f(s,t): quoted at s for borrowing/lending from t-1 to t
discount factor D(s,t): price at s of receiving 1 at a future date t
zero-coupon rate z(s,t): yield at s for a zero-coupon bond maturing at t
par coupon rate c(s,t): coupon rate (= yield) quoted at s for a coupon bond maturing at t and trading at par
present value PV: value today of a series of future cash flows
present value NPV: value today of a series of future cash flows, less the initial price
r_t = f(t-1,t)
1
D(s,t) = -----------------
t
Prod (1+f(0,s))
s=1
D(s,t-1)
f(s,t) = -------- - 1
D(s,t)
1
D(s,t) = -------------
t-s
(1+z(s,t))
-1/(t-s)
z(s,t) = D(S,t) - 1
1 - D(s,T)
c(s,T) = --------------
T
Sum D(s,t)
t=s+1
T
PV = Sum D(0,s)c_s
s=1
NPV = PV - P
t
= Sum D(0,s)c_s
s=1
Suppose the price is $212 for a 2-year coupon bond with face of $200 and an annual coupon (first one one year from now) of $40. Suppose also that the price is $150 for a 1-year coupon bond with face of $150 and an annual coupon (one remaining one year from now) of $15.
Remaining pension benefits in a plan having two more years to go are $95,000 one year from now and $60,000 two years from now. What replicating portfolio of the two coupon bonds covers the pension liabilities exactly? What is the price of the replicating portfolio?
/ T \ 1/T 1 T
z(0,T) = | Prod (1+f(0,s)) | - 1 ~ - Sum f(0,s).
\ s=1 / T s=1
The zero-coupon rate is an average of forward rates up to that
maturity.
T
c(0,T) = Sum w(0,s)f(0,s)
s=1
where
/ T
w(0,s) = D(0,s) / Sum D(0,t)
/ t=1
The par-coupon rate is a weighted average of forward rates up to
that date, with more weight on the earlier maturities.
In-class exercise: formulas connecting rates
Suppose the spot rate is 5% and the forward rate one year out is
6%. What are the one- and two-year zero-coupon rates? What are
the one- and two-year par-coupon rates?
Duration Formulas
traditional (Macauley) duration:
T c_t D(0,t)
duration = Sum ----------------- t
t=1 T
Sum c_s D(0,s)
s=1
The discount factors D(0,t) are usually computed using either the bond's
yield (i.e. D(0,t) = 1/(1+y)^t) or using the actual discount factors.
Macauley duration assumes that random shocks impact forward rates equally at all maturities.
effective duration (sens = short for sensitivity):
/ T \ / / T \
sens(duration) = | Sum sens(s) c_s D(0,s) | / | Sum c_s D(0,s) |
\ s=1 / / \ s=1 /
For effective duration, shocks affect different forward rates differently, so the
amount of interest rate exposure is no longer proportional to time-to-maturity,
even for a discount bond.
the particular effective duration measure we have used:
sens(duration) = exp(-.125 * duration)
duration(sens) = -log(1 - .125 * sens)/.125
In-class exercise: duration and effective duration
Suppose the yield curve today is flat at 5%. Compute the duration and effective
duration of a portfolio paying $100, 10 years from now, and $100, 20 years from now.
Recompute the duration and effective duration assuming a flat yield curve at 10%.
Basic option pricing formulas
single-period:
-1
Value = R (pi_U V_U + pi_D V_D)
Risk-neutral probabilities could be computed from the payoffs of some asset,
but more commonly we make assumptions about them directly.
multi-period:
* 1 1 1 1
Value = E [--- --- --- ... --- C_T]
R_1 R_2 R_3 R_T
This formula is especially useful for simulations but can also be used in simple
binomial cases without American or conversion features.
In-class exercise: binomial model
The spot interest rate is 5%. Each year it goes up by 5% (e.g. from 5% to 10%)
with risk-neutral probability 1/3 or down by 2% (e.g. from 5% to 3%) with
risk-neutral probability 2/3. What is the price of a 2-year interest-rate cap
with a cutoff rate of 5%?
In-class exercise: binomial model
The spot interest rate is 5%. Each year it goes up by 5% (e.g. from 5% to 10%)
with risk-neutral probability 1/3 or down by 2% (e.g. from 5% to 3%) with
risk-neutral probability 2/3. What is the price of a 2-year interest-rate cap
with a capped rate of 5%?
Mean reversion and fudge factors
For mean reversion
E[r - r ] = k(rbar - r ).
t+1 t t
in the binomial model with equal changes delta or -delta in rates, set
k(rbar - r )
1 t
pi = - + ------------- .
U 2 2 delta
Without mean reversion, k=0 and pi_U=1/2.
fudge factors: To match actual discount factors D(0,t), modify the original
model -- om -- as follows:
om D(0,s-1)/D(0,s)
R = R -------------------
s s om om
D (0,s-1)/D (0,s)
or approximately
om om
r = r + f(0,s) - f (0,s)
s s
In-class exercise: capstone problem with fudge factors and mean reversion
Consider a two-year binomial model. Start with an original model in which
the short riskless interest rate starts at 5% and moves up or
down by 2.5% each period (i.e., up to 7.5% or down to
2.5% at the first change). The artificial probability of each
of the two states at any node is determined by whatever makes mean reversion
k equal to 20% per year with a long-term mean of 5%.
What is the price of a one-year discount bond in this original model? the
two-year discount bond?
Suppose the actual one-year discount rate in the economy is 6% and the
actual two-year discount rate is 6.5%. Compute the fudge factors and
draw the tree for the adjusted interest rate process.