Case Objective

To understand the concept of duration and how the bond immunization theorem is applied.

Key Concepts

Duration; interest rate risk; bond immunization theorem

Case description

There are 4 fixed-income securities. You trade during the first day of year 1 only.  At the end of this trading day time flashes by and your position is market to the realized end of year yield curve.  This affects the end of year marked value because each security has cash flows up to four years from the beginning of year 1. The cash flow from each bond at the end of each period is shown below.  You start with a negative (short) position in either the first or second security, but you can only trade the two zero coupon bonds (securities three and four).

# Realization Of A Yield Curve

In this economy trading is restricted to the first year only.  The realized spot rate for year 1 is 25% and the spot rates for years 2‑4 are stochastic.  At the beginning of year 1 assume that the spot yield curve is a flat 25% and that by the end of year 1 the realized yield curve remains flat.  However, the realized yield to maturities are very volatile and can exhibit a shift upto plus or minus 20%.

To understand what is implied by the previous assumptions consider the following example.  Suppose at the end of year 1 the realized yield curve shifts by -10%.  That is,  the end of year yield to maturities are 15%, 15%, 15% and all forward rates are 15% for years 2 to 4.  This means that all cash flows are discounted at 15% for years 2 to 4 for purposes of marking to market.  The three year zero-coupon bond is marked at \$100/(1.15^2) and the two year zero-coupon bond is marked at \$100/(1.15).

Case Data

The cash flows from the securities are:

 Payout at end of Period 1 Payout at end of Period 2 Payout at end of Period 3 Payout at end of Period 4 Sec 1 0 160 200 250 Sec 2 30 100 47 0 2YrZer 0 100 0 0 3YrZer 0 0 100 0

Initial Position

There are two types of traders with the following initial positions:

There are two trader types in this market.  Type I has an initial short position (i.e., liability) in non-tradable security 2, a long position in security 4, and cash.  Type II has an initial short position (i.e., liability) in the non-tradable security 1, a long position in security 3, and cash.  Your end of year position is marked-to-market cash using the realized yield curve.  This market value in turn determines your grade cash as described below.

The object is to earn as much grade cash as possible by managing your position's exposure to changes in the  yield curve.

In each market year securities are traded using market cash.  One market year is defined as one trial.  If at the end of any trial you have a closing balance of market cash equal to \$9,999 or higher, you will earn \$10 grade cash.  If you have a closing balance of \$5,000 market cash or lower, you will earn zero grade cash.  Any other amount of market cash will determine your grade cash for the trial as follows:

Trading is conducted over a number of independent trials and a record of your cumulative grade cash is maintained.

Your aim is to hedge the risk of your position to parallel shifts in the yield curve.