Forward Rates FNCE 4070 Financial Markets and Institutions

Forward Rates FNCE 4070 Financial Markets and Institutions

Market Data • T-Bills – 26 W 0. 130% – 52 W 0. 175% • T-Note – 2 year • Coupon rate 0. 25% • Semi-Annual Yield 0. 273%

T-Note. Pricing • As there are 4 cashflows in a 2 -year T-Note we need 4 discount factors to price the note – These are the 6 M, 1 Y, 18 M and 2 Y • 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 6 M T-Bill prices for 6 M, 1 Y and 18 M. • We need enough rates to value a 2 year TNote – The rates we need are the 6 M, 1 Y, 18 M and 2 Y rates

What we already know • Given the T-Bills we can easily compute the 6 M and 1 Y YTM. – We can also easily compute the discount factor for a cashflow received on these dates.

What we need to figure out • To create the yield curve we are trying to – Come up with the yield to maturity for 18 M and 2 Y – Come up with forward T-Bill rates (the first is straightforward) • Start date 6 M, 1 Y, 18 M • End date 12 M, 18 M, 2 Y – 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

Forward T-Bill Rate • The expected Discount Rate for the 6 M T-Bill starting in 6 M time is straightforward

The Problem • We still need to find two rates – Expected T-Bill starting in 12 M and maturing in 18 M – Expected T-Bill starting in 18 M and maturing in 24 M • Alternatively – Expected YTM for 18 M – Expected YTM for 24 M

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 6 M T-Bill starting in 12 M time will have an expected discount rate that is the average of the 6 M T-Bill starting in 6 M time and the 6 M T-Bill starting in 18 M time

Pricing the T-Note • If you price using discount factors you find • Otherwise you use a standard s. a. yield calculation

Discount Factors • The missing discount factors can be derived from: • 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.