Forward Rates
FNCE 4070
Financial Markets and Institutions
Market Data
• T-Bills
– 26W 0.130%
– 52W 0.175%
• T-Note
– 2 year
• Coupon rate 0.25%
• Semi-Annual Yield 0.273%
T-NotePricing
• As there are 4 cashflows in a 2-year T-Note we
need 4 discount factors to price the note
– These are the 6M, 1Y, 18M and 2Y
• We could simply use the s.a. yield to determine a
PV but we do not get any term structure
information from this.
– For example, short term rates are lower than longer
term rates. The T-Note price gives information about
the relationship but we cannot determine that
information from the s.a. yield directly.
Goal
• In order to view the yield curve we need to
look at consistent rates.
– The natural choice for rates is the YTM for a
discount bond for a given maturity
– An alternate view would be to look at Expected
6M T-Bill prices for 6M, 1Y and 18M.
• We need enough rates to value a 2 year TNote
– The rates we need are the 6M, 1Y, 18M and 2Y
rates
What we already know
• Given the T-Bills we can easily compute the
6M and 1Y YTM.
– We can also easily compute the discount factor for
a cashflow received on these dates.
df = 1- DiscountRate ´
æ1ö
YieldToMaturity = ç ÷
è df ø
days
360
365
days
-1
What we need to figure out
• To create the yield curve we are trying to
– Come up with the yield to maturity for 18M and
2Y
– Come up with forward T-Bill rates (the first is
straightforward)
• Start date 6M, 1Y, 18M
• End date 12M, 18M, 2Y
– These are equivalent representations
Forward T-Bill rates
• To figure out Forward T-Bill rates we need
– Forward discount factors
• Represent the value of 1 dollar paid at the end date as
of the start date
dft1,t2 =
df0,t2
df0,t1
DiscountRate = (1- df ) 360
days
Forward T-Bill Rate
• The expected Discount Rate for the 6M T-Bill
starting in 6M time is straightforward
182
df0,6 M = 1- 0.13%
= 0.99934378
365
364
df0,12 M = 1- 0.175%
= 0.99823056
365
df
df6 M ,12 M = 0,12 M = 0.998887046
df0,6 M
DiscountRate = (1- df6 M ,12 M )
360
= 0.2201%
182
The Problem
• We still need to find two rates
– Expected T-Bill starting in 12M and maturing in
18M
– Expected T-Bill starting in 18M and maturing in
24M
• Alternatively
– Expected YTM for 18M
– Expected YTM for 24M
The Problem cont
• We have a single equation that we have not
used.
• Or alternatively – we have three pieces of
market data and 4 unknowns necessary for
pricing our T-Bond
Assumption
• We will assume that the 6M T-Bill starting in
12M time will have an expected discount rate
that is the average of the 6M T-Bill starting in
6M time and the 6M T-Bill starting in 18M
time
Pricing the T-Note
• If you price using discount factors you find
PV = 0.125%´ df0,6 M + 0.125%´ df0,12 M + 0.125%´ df0,18M +100.125%´ df0,24M
• Otherwise you use a standard s.a. yield
calculation
Discount Factors
• The missing discount factors can be derived
from:
df0,18M = df0,12 M ´ df12 M,18M
df0,24M = df0,18M ´ df18M,24M
• The missing discount factors from these
equations can be derived from the T-Bill
discount ratediscount factor formula
Final Solutions
• We then use the solver from excel to price the
T-Note using discount factors and using s.a.
yield calculation and iterate on the T-Bill
prices until they match.