Treasury Markets


Kevin Crotty
BUSI 448: Investments

Where are we?

Last time:

  • Empirical facts about equities

Today:

  • Treasury market basics
  • Term structure
  • Spot rates

Treasury Securities

Bills

  • Bills
    • Maturity of 1 year or less (1, 3, 6, 12 months)
    • Usually issued as discount securities
    • Taxes – exempt from state and local income taxes
    • Small denomination – can purchase in $100 increments from Treasury Direct

Bonds and Notes

  • Notes
    • Maturity between 2 years and 10 years (2, 3, 5, 7, 10 years)
    • Coupon securities (semiannual)
  • Bonds
    • Maturity greater than 10 years (20, 30 years)
    • Coupon securities

TIPS and STRIPS

  • Treasury inflation protection securities (TIPS)
    • Principal is indexed to consumer price index
    • Maturities of 5, 10, 30 years
  • STRIPS (Separate Trading of Registered Interest and Principal Securities)
    • Allows individual component of Treasuries to be traded
    • Improves liquidity for zero-coupon Treasury markets

Historical yields

import pandas as pd
from pandas_datareader import DataReader as pdr
y3mo = pdr("TB3MS", "fred", start="1929-12-01")

Treasury Curve

Term structure of rates

  • Interest rates (yields) of different maturity bonds are generally different
    • For instance, 10-year bond may have a different yield than a 2-year note
  • The yield curve is the plot of yields as a function of time to maturity
  • The term structure of rates is the relation between yields and maturity

Key aspects of the term structure

  1. Level
  2. Slope
  3. Curvature

Historical Yield Curves

Time-series of yields

  • What do you notice prior to the shaded recessions?

Some fixed income empirical facts

Size of the market

Stocks, bonds, and gold returns

Spot rate curve

Spot rates

  • Spot rates are the discount rates associated with CFs of a particular maturity.

Two methods to get them:

  • Use zero-coupon bonds (i.e., Tbills or STRIPS)
  • Bootstrap them from coupon bonds

Bond pricing revisited

If \(z_1\), \(z_2\), …, \(z_T\) are maturity-specific riskless spot rates, then the bond price is:

\[ P(\mathbf{z}) = \frac{C/m}{(1+z_1)} + \frac{C/m}{(1+z_2)^2} + ... + \frac{C+FACE}{(1+z_T)^T} \]

\[ P(\mathbf{z}) = \sum_{t=1}^T\frac{C/m}{(1+z_t)^t} + \frac{FACE}{(1+z_T)^T} \]

where

  • \(C/m\) is the periodic coupon payment
  • \(m\) is the compounding periods per year
  • \(T\) is the total number of payments (# years \(\cdot m\))

Spot rates from zero-coupon bonds

  • A zero-coupon bond pays no coupons \[ P(z_t) =\frac{FACE}{(1+z_t)^t}\]
  • Using traded prices, we can solve for \(z_t\) \[z_t = \left(\frac{Face}{P(z_t)}\right)^{1/t}-1\]

Spot rates from coupon bonds

  • Bootstrapping: method of extracting spot rates from coupon bond prices.

  • Iterative procedure: 1st solve for \(z_1\), then \(z_2\) using \(z_1\)…

  • To get spot rate \(z_t\), we must know \(z_1\),\(z_2\), …, \(z_{t-1}\): \[z_t = \left(\frac{CF_t}{PV(CF_t)}\right)^{1/t}-1\]

  • \(PV(CF_t) = P_t - \sum_{i=1}^{t-1} \frac{CF_i}{(1+z_i)^i}\)

  • \(P_t\) is the price of the coupon bond maturing at time \(t\).

Example

Bond Price Coupon Rate Maturity Face Value
A 97.5 0% 0.5 100
B 95 0% 1.0 100
C 955 2.5% 1.5 1,000
D 1,000 5.75% 2 1,000

Assume semiannual coupon payments and no credit risk.

  1. Determine the spot rates for the four periods
  2. What is the fair price of a 2-year 10% coupon bond with a face value of $1,000 if it pays annual coupons?

For next time: Arbitrage





BUSI 448