Fixed Income Risk Management
You are asked in this assignment to insure the value of a bond portfolio during the (in hindsight) turbulent 8-month (or 245-day) period from 1 March to 1 November 1988. You'll have to place yourself back in that period, using only information available at the time.
The portfolio consists of 100 bonds (nominal $25,000) each of the U.S. Treasury bonds:
1. 12-5/8s May 1995;
2. 13-3/8s August 2001;
3. 7-7/8s November 2002-2008; and
4. 13-1/4s May 2009-2014.
Refer to the attached Table 1 for monthly mid-prices taken from The Wall Street Journal in the period 1 March to 1 November 1988, assumed to be for settlement on the first calendar day of the month. These bonds pay coupons on the 15th day of the May/November or August/February cycle.
Callable bonds, with a range of maturity years, may be assumed to be called at the earliest date when priced at or above par, and at the latest date otherwise.
You must insure this portfolio for a gross yield of 4.5% APR, i.e. not counting the implied insurance premium. You may derive the hedge requirements with the Black-Scholes formula. The volatility or variance rate of this portfolio at the time was estimated at σ = 0:12 per year; and the financing rate were 6% APR.
The required insurance programme must be instituted by means of a rolling delta hedge with T-bond futures (nominal $100,000). You may limit position adjustments, for simplicity's sake, to the first day of every month. Hedging is done with the next maturing contract, provided it has at least one-month of life left. Contracts are settled on the 25^{th} day of December, March, June and September, requiring positions to be rolled over at the start of those months.
Futures prices are also listed in the table under Ft ; the two prices for settlement months are for the contracts expiring in 25 days and three months later, respectively. The CTD-bond during this period is assumed to be the 14s Nov 2006, prices for which are also given in the table. Conversion factors at the time were based on a yield of 8%.
The following costs and rules apply:
- Transactions cost $50 per futures contract (roundtrip in-and-out), also to be applied to roll-overs.
- Fixed management costs of $5,000 per month, to be paid in advance.
- Margin cash deposit $2,500 per contract, earning no interest.
- Futures marking-to-market only at month's end.
- Cash balances (including coupon interest receipts) are invested at 5% APR when positive, or borrowed at 7% APR when negative, using the 30/360 convention.
- There is no trading in bonds during this period.
- Cash balances are part of the overall portfolio value to be insured, even though the duration of cash balances may be assumed to be zero.
It is advisable to carry out most of the required calculations with the portfolio value scaled down by 25,000; only to use the full portfolio value in the final accounting.
Reporting
You should provide for each monthly period a complete accounting of all futures trades, costs, marking-to-market cash flows, and interest income, using the combined position value of the bond portfolio and the cash balance (which may be negative) at the end of a month as the basis on which to compute the hedge for the next month.
The computations are best carried out in an Excel spreadsheet and using the financial functions from a suitable add-on ToolPak. These computations may be carried out in groups with at most three (3) members. The names of all group members must be listed on Excel printouts.
Each student must write an individual report on the assignment, with an insightful discussion of this hedge method in general and of the attained hedge results in particular.