CHAPTER%208%20Interest%20Rate%20Futures%20Refinements

1
CHAPTER 8Interest Rate Futures Refinements

• In this chapter, we extend the discussion of
  interest rates futures. This chapter is organized
  into the following sections
• The T-Bond Futures Contract
• Sellers Options for T-Bond Futures
• Interest Rate Futures Market Efficiency
• Hedging with T-Bond Futures

2
T-Bond Futures Contract

• In this section, the discussion of T-bond futures
  is extended by analyzing the cheapest-to-deliver
  bond.
• Recall that a number of candidate bonds can be
  delivered against a T-bond future contract.
  Recall further that short traders choose when to
  deliver and which combination of bonds to
  deliver.
• Some bonds are cheaper to obtain than others. In
  this section, we learn techniques to identify the
  cheapest-to-deliver bond, including
• Cheapest-to-deliver bond with no intervening
  coupons.
• Cheapest-to-deliver bond with intervening
  coupons.
• Cheapest-to-deliver and the implied repo rate.

3
Cheapest-to-Deliver with No Intervening Coupons

• Assume today, September 14, 2004, a trader
  observes that the SEP 04 T-bond futures
  settlement price is 107-16 and thus decides to
  deliver immediately. That is, the trader selects
  today, September 14, as her Position Day.
  Therefore, she will have to deliver on September
  16.
• The short is considering the following bonds with
  100,000 face value each for delivery. The short
  wishes to determine if delivering one or the
  other bond will produce a larger profit for her.
• How much should the short receive?
• Which bond should the short deliver?

To answer these two questions, we need to
determine the invoice amount and then which bond
is cheapest-to-deliver.
4
Cheapest-to-Deliver with No Intervening Coupons

• Recall that the total price of a bond depends
  upon the quoted price plus the accrued interest
  (AI).

Where DSP decimal settlement price
the decimal equivalent of the quoted price CF
conversion factor the conversion factor as
provided by the CBOT AI accrued interest
the Interest that has accrued since the last
coupon payment on the bond Pi cash
market price
5
Cheapest-to-Deliver with No Intervening Coupons

• The accrued interest (AI) is computed as follows

The days in half-year can be obtained from Table
8.1.
6
Cheapest-to-Deliver with No Intervening Coupons

• Step 1 compute the cash price and invoice price.
• 5.25 Bond
• AI (122/184) (0.5) (0.0525) (100,000)
  1,740.49
• Invoice Amount 1.0750 (100,000) (0.9052)
  1,740.49
• Invoice Amount 99,049.49
• 8.00 Bond
• AI (122/184) (0.5) (0.08) (100,000)
  2,652.17
• Invoice Amount 1.0750 (100,000) (1.2113)
  2,652.17
• Invoice Amount 132,866.92
• The 8 bond has an invoice amount 34 greater
  than the 5.25 bond.

7
Cheapest-to-Deliver with No Intervening Coupons

• Sept 2 compute the cheapest-to-deliver bond.
• The bond that is most profitable to deliver is
  the cheapest-to-deliver bond. The shorts profit
  is the difference between the invoice amount and
  the cash market price.
• For a particular bond I, the profit pi is
• pi Invoice Amount - (Pi AIi)
• Recall that the invoice amount is

Substituting the formula for the invoice amount
into the profit equation gives pi (DFPi)
(100,000) (CFi) AIi - (Pi AIi) And
simplifying pi DFPi (100,000) (CFi) - Pi
8
Cheapest-to-Deliver with No Intervening Coupons

• The cheapest-to-deliver is
• 5.25 Bond
• p 1.0750 (100,000) (0.9052) - 93,468.75
  3,840.25
• 8.00 Bond
• p 1.0750 (100,000) (1.2113) - 127,093.75
  3,121.00
• Thus, in this case the cheapest-to-deliver bond
  is the 5.25 bond.

9
Cheapest-to-Deliver with No Intervening Coupons

• General rules based on interest rates
• When interest rates are below 6, there is an
  incentive to deliver short maturity/high coupon
  bonds.
• When interest rates exceed 6, there is an
  incentive to deliver long maturity/low coupon
  bonds.
• General rules based on duration
• A trader should deliver low duration bonds when
  interest rates are below 6.
• A trader should deliver high duration bonds when
  interest rates are above 6.

10
Cheapest-to-Deliver with Intervening Coupons

• This section examines, cheapest-to-deliver bonds
  when a bond pays a coupon between the beginning
  of the cash-and-carry holding period and the
  futures expiration.
• To find the cheapest-to-deliver bond before
  expiration, the cash-and-carry strategy is used.
• The bond with the greatest profit at delivery
  from following the cash-and-carry strategy will
  be the cheapest-to-deliver bond.
• For this analysis Assume that
• A trader buys a bond a today and carries it
  until delivery.
• Interest rates and futures prices remain
  constant.
• Consider the estimated invoice amount plus the
  estimate of the cash flows associated with
  carrying the bond to delivery.

11
Cheapest-to-Deliver with Intervening Coupons

• The estimated invoice amount depends on three
  factors
• Today's quoted futures price.
• The conversion factor for the bond we plan to
  deliver.
• The accrued interest on the bond at the
  expiration date.
• Acquiring and carrying a bond to delivery
  involves three cash flows as well
• The amount paid today to purchase the bond.
• The finance cost associated with obtaining
  money today to buy a bond in the future.
• The receipt and reinvesting of coupon payment.
• Figure 8.1 brings all these factors together.

12
Cheapest to Delivery and Bond Yield

• Insert Figure 8.1 here

13
Cheapest-to-Deliver with Intervening Coupons

• Estimated Invoice Amount
• DFP0 100,000 (CF) AI2
• Estimated Future Value of the Delivered Bond
• (P0 AI0)(1 C0,2) - COUP1(1 C1,2)
• For bond I, the expected profit from delivery is
  the estimated invoice amount minus the estimated
  value of what will be delivered
• p DFP0 (100,000) (CF) AI2 - (P0 AI0)(1
  C0,2) - COUP1(1 C1,2)
• where
• P0 quoted price of the bond today, t 0AI0
  accrued interest as of today, t 0C0,2
  interest factor for t 0 to expiration at t
  2COUP1 coupon to be received before delivery
  at t 1C1,2 interest factor from t 1 to t
  2DFP0 decimal futures price today, t
  0CF conversion factor for a particular bond
  and the specified futures
  expirationAI2 accrued interest at t 2

14
Cheapest-to-Deliver with Intervening Coupons

• To illustrate these computation consider the
  following situation.
• Suppose that today is Sept 14, 2004, and you want
  to find the cheapest-to-deliver bond for the DEC
  04 futures expiration. The bond has a 100,000
  face value and a target delivery date of Dec 31,
  2004. The futures contract is the DEC 04. The
  T-bond contract had a settlement price of 106-23
  today. The coupon invested repo rates is 7.
• Summary
• Today Sept 14Bond face value
  100,000Target delivery date Dec 31Futures
  contract DEC 04 T-bondCoupon invested repo
  rate 7Settlement price Sept 14 106-23
• You are considering two bonds for delivery. The
  bonds are as follows

15
Cheapest-to-Deliver with Intervening Coupons

• Step 1 estimate the value of AI.
• 5.25 Bond
• P0 AI0 93,468.75 1,740.49 95,209.24
• 8 Bond
• P0 AI0 127,093.75 2,652.17 129,745.92
• Step 2 estimate the accrued interest that will
  accumulate from the next coupon
  date, Nov 15, 2004 if the planned
  delivery date is Dec 31, 2004 (46 days).
• 5.25 Bond
• AI2 (46/181) (0.5) (0.0525) (100,000)
  667.13
• 8 Bond
• AI2 (46/181) (0.5) (0.08) (100,000) 1,016.57

16
Cheapest-to-Deliver with Intervening Coupons

• Step 3 compute the estimated invoice amounts.
• 5.25 Bond
• 1.0671875 (100,000) (0.9056) 667.13
  97,311.63
• 8 Bond
• 1.0671875 (100,000) (1.2094) 1,016.57
  130,082.23
• Step 4 compute financing rates.
• Period Sept 15 until Dec 31 (108 days)
• C0,2 0.07 (108/360) 0.0210
• Period Nov 15 until Dec 31 (46 days)
• C1,2 0.07 (46/360) 0.008944
• Table 8.2 summarizes these calculations.

17
Cheapest-to-Deliver with Intervening Coupons

• Step 5 Compute expected profit for each bond.
• 5.25 Bond
• p (1.06718750) (100,000) (0.9056) 667.13-
  ( 93,468.75 1,740.49) (1.0210) -
  (2,625) (1.008944)
• p 2,751.48
• 8 Bond
• p (1.06718750) (100,000) (1.2094) 1,016.57-
  (127,093.75 2,652.17) (1.0210) - 4,000
  (1.008944)
• p 1,647.43
• The profit from the 5.25 bond is higher, so it
  is the cheapest- to-deliver.

18
Cheapest-to-Deliver Bond and The Implied Repo Rate

• We can analyze the same situation using the
  implied repo rate. The implied repo rate for a
  given period equals the net cash flow at delivery
  divided by the net cash flow when the carry
  starts.
• Repo Rate General Rules
• A cash-and-carry arbitrage nets a zero profit
  if the actual borrowing cost equals the
  implied repo rate.
• If the effective borrowing rate is less than
  the implied repo rate, one can earn an
  arbitrage profit by using cash-and-carry
  arbitrage (i.e., buy a cash bond and sell a
  futures).
• If the effective borrowing rate exceeds the
  implied repo rate and if one can sell bonds
  short, then one can earn an arbitrage profit
  by using a reverse cash-and-carry arbitrage
  ( i.e., sell a bond short, buy the futures, and
  cover the short position at the expiration of
  the futures).

19
Cheapest-to-Deliver Bond and The Implied Repo Rate

• The numerator consists of cash inflows of the
  Invoice Amount, plus the future value of the
  coupons at the time of delivery, less the cost of
  acquiring the bond initially.
• The denominator consists of the cost of buying
  the bond. Thus, the Implied repo rates is

For the 5.25 bond, we have
Implied Repo Rate 0.0499 Annualized, the
implied repo rate is 0.0499(360/180) 16.63
20
Cheapest-to-Deliver Bond and The Implied Repo Rate

• For the 8 bond, we have

Implied Repo Rate 0.0337 Annualized, the
implied repo rate is 0.0337(360/180)
13.23 The cheapest-to-deliver bon has the
highest repo rate in a cash-and-carry arbitrage,
so we should deliver the 5.25 bond.
21
Cheapest-to-Deliver Bond and Implied Repo Rate

• Table 8.3 shows how financing a cash-and-carry
  arbitrage at the implied repo rate yields a zero
  profit.

22
T-Bond Risk Arbitrage

• Arbitrage in the T-bond futures market is really
  risk arbitrage. Risks stem from three sources
• Intervening coupon payments that must face
  reinvestment.
• The use of conversion factors.
• The seller options.

23
T-Bond Risk Arbitrage

• A closer examination of Table 8.3 shows some
  potentially risky elements of the cash-and-carry
  arbitrage. Notice that
• The debt was financed at a constant rate
  throughout the 108-day carry period.
• The trader was able to invest the coupon at the
  reinvestment rate of 7.
• The futures price did not change over the horizon.

24
T-Bond Risk Arbitrage Cash-and-Carry Strategy

• Changes in the futures price can affect the cash
  flow from the cash-and-carry strategy, as Table
  8.4 illustrates. In this case, the futures price
  drops from 106-23 to 104-23 over the life of the
  contract.

Thus, the cash-and-carry strategy now produces a
negative profit.
25
T-Bond Risk ArbitrageReserve Cash-and Carry
Here we examine an attempt to earn a profit using
a reserve cash-and-carry strategy. Recall that
the trades used in a reverse cash-and-carry are
as follows
26
T-Bond Risk ArbitrageReserve Cash-and Carry

• Utilizing the same information from Table 8.3, we
  have

As expected ,there is no arbitrage profit in this
case.
27
T-Bond Futures Sellers Options

• The structure of T-bond futures contract gives
  sellers timing and quality options.
• Timing option
• The sellers right to choose the time of
  delivery.
• 2. Quality option
• The sellers right to select which bond to
  deliver.
• These two main seller's options become entangled
  in the actual T-bond futures contract. The timing
  and quality options are commonly present in what
  the futures markets refers to as
• The wildcard option.
• The end-of-the-month option.

28
Wildcard Option

• The settlement price is determined at 200 PM.
  However, the short seller has until 800 PM to
  notify the exchange of his/her intent to deliver.
  Thus, the seller can observe what happens between
  200 PM and 800 PM before making his/her
  decision.
• If interest rates jump between 200 PM and 800
  PM, the short trader notifies the exchange
  his/her intent to deliver at the 200 PM price.
• If interest rates stay the same or go down, the
  short seller waits for the next day to notify the
  exchange of an intent to deliver.

29
The End-of-the-Month Option

• Recall that the last trading day for T-bond
  futures is the 8th of the month.
• The settlement price established on the final
  trading day is the settlement price used in all
  invoice calculations for all deliveries in the
  month. Thus, the seller can still make two
  choices
• The seller can choose the delivery date.
• The seller can choose the bond to deliver.
• Assuming that interest rates are stable, then the
  seller may apply the following general rules
• If the coupon yield on the bond exceeds the
  financing rate to hold the bond, the seller
  should deliver on the last day.
• If the financing rate exceeds the coupon yield,
  the seller should deliver immediately.

30
Value of The Sellers Options

• Recall that under perfect market conditions, the
  Cost-of-Carry Model concludes that the futures
  price is equal to
• F S (1 C)
• If the seller's options have value, then market
  equilibrium requires that the following equation
  holds
• F SO S (1 C)
• where
• SO value of seller's options
• This implies that
• F S ( 1 C ) - SO
• This implies that the futures price observed in
  the market should be below the cost of carry by
  an amount equal to the sellers options.

31
Interest Rate Futures Market Efficiency

• There are three commonly distinguished forms of
  the market efficiency hypothesis
• The weak form.
• The semi-strong form.
• The strong form.
• While many studies neglect the full magnitude of
  transaction charges, more recent studies find
  potential for arbitrage even after transaction
  costs.
• Pure Arbitrage
• For a pure arbitrage, the yield discrepancy must
  be large enough to cover all transaction costs
  faced by a market outsider.
• Quasi-Arbitrage
• Occurs when a trader with an initial portfolio
  can successfully engage in an arbitrage. For
  quasi-arbitrage, the trader faces less than full
  transaction costs.

32
Pure Arbitrage

• Table 8.9 is from a famous study on the
  efficiency of the T-bill futures market conducted
  by Elton, Gruber and Rentzler. They found large
  arbitrage profits exist, many with single
  contract profits in excess of 800.

33
Pure Arbitrage in T-Bond Futures

• Kolb, Gay and Jordan conducted a study on T-bond
  futures. They investigated the possibility of a
  pure arbitrage for all T-bond contracts from
  December 1977 through June 1981. Figure 8.4 shows
  the profitability of deliverable bond for 15
  contracts maturities.

• Insert Figure 8.4 here

34
Alternative Risk Management Strategy

• In this section, alternative risk management
  strategies using short-term interest rate futures
  are explored, including
• Changing the Maturity of an Investment
• Shortening the maturity of a T-bill investment
• Lengthening the maturity
• Fixed and Floating Loan Rates
• Strip and Stack Hedges
• Tailing Hedge

35
Changing The Maturity of an InvestmentShortening
the Maturity

• Many investors find themselves holding a
  portfolio with undesirable maturity
  characteristics.
• Spot market transaction costs are relatively
  high, and many investors prefer to alter the
  maturities of investment by trading futures.
• Consider a firm that has invested in a T-bill
  with a 1,000,000 face value. Today, March 20,
  the T-bill has a maturity of 180 days. The firms
  manager learns that the company will need cash in
  90 days. Assume that the short-term yield is flat
  with all rates at 10 and a 360-day year.

36
Changing The Maturity of an InvestmentShortening
the Maturity
Table 8.10 illustrate the process of shortening
the maturity.

• The price of a bill is given by
• P FV - DY(FV)(DTM)/360
• P 1,000,000- (.10)(10,000,000(180)/360
• P 1,000,000 - 50,000 9,500,000,000
• By making the above trades, the firm has
  effectively shortened the maturity from 6 months
  to three months.

37
Changing The Maturity of an InvestmentLengthening
the Maturity

• On August 21, an investor holds a T-bill with a
  100 million face value. The T-bill matures in 30
  days (September 20). The investor plans to
  reinvest for another 3 months after the T-bill
  matures. The investor fears that interest rates
  might fall. The investor finds the current SEP
  T-Bill futures yield of 9.8 attractive and would
  like to lengthen the maturity of the T-bill
  investment. The transaction necessary to do so
  are presented in Table 8.11.

These transactions locked in a 9.8 rate over the
four months (Aug-Dec). Thereby, lengthening the
maturity of the individuals investments.
38
Fixed and Floating Loan Rates

• This section examines
• Converting a Floating Rate to a Fixed Rate Loan
• How a borrower holding a floating rate loan can
  effectively convert this loan into a fixed rate
  loan.
• Converting a Fixed Rate to a Floating Rate Loan
• How a lender who feels compelled to offer fixed
  rate loans can use the futures markets to make
  the investment perform like a floating rate loan.

39
Converting a Floating Rate to a Fixed Rate Loan

• Converting a floating rate loan to a fixed rate
  loan, also known as creating a synthetic fixed
  rate loan, occurs when you start with a floating
  rate loan and transact to fix the interest rate.
• Today is Sept 20th, assume that a construction
  company has planned a project which will take 6
  months to complete. The cost of the project is
  100,000,000. The firms bank offers the
  following conditions on a loan.
• Rates
• First 3 months LIBOR 200 basis point Last 3
  months DEC 20 LIBOR 200 basis point
• The bank insists that the second 3-month rate be
  based on the LIBOR prevailing 3 months from
  today. This is a risky preposition for the
  construction company.

40
Converting a Floating Rate to a Fixed Rate Loan

• The construction company wishes to lock in a
  fixed rate loan for the entire period. The
  company has accumulated the following
  information
• LIBOR Loan
• Sept 20 7 9.0DEC Eurodollar
  futures 7.3. 9.3
• These rates give the following cash flows on the
  loan
• Sept 20 Receive principal 100,000,000Dec
  20 Pay interest - 2,250,000Mar 20 Pay
  interest and principal - 102,325,000
• The cash flows for September and December are
  certain but the cash flow for March is unknown.
• Using the above information, construct a
  synthetic fixed rate loan.

41
Converting a Floating Rate to a Fixed Rate Loan
To convert to a variable rate loan to a fixed
rate loan, the following transaction are
completed.
By engaging in the above transactions, the
company knows with certainty the interest expense
that it will pay over the life of the loan. As
such, it has created a fixed rate loan.
42
Converting a Fixed Rate to a Floating Rate Loan

• From the banks perspective, it can grant the
  fixed rate loan. However, doing so exposes the
  bank to risks.
• The bank expects to obtain money to make to the
  loan by borrowing at LIBOR 7 today and 7.3 for
  the next quarter, for an average of 7.15. The
  bank makes a fixed rate loan at 9.15. The bank
  sources of funds are as follows
• BANK
• Sep 20 Borrow principal 100,000,000 Make
  loan to - 100,000,000Dec 20 Pay interest -
  1,750,000
• Mar 20 Receive principal
  interest 104.575,000 Pay
  principal interest - 101,825,000

43
Converting a Fixed Rate to a Floating Rate Loan

• To reduce its risks and lock in a profit, the
  bank trades as follows

Thus, the bank has locked in a profit fo
1,000,000 (4,575,000 - 3,575,000). The bank
has also effectively crated a fixed rate loan.
44
Strip and Stack Hedges

• Using the same example. Now assume that the
  construction company needs a one year loan
  instead of 6-month loan.
• The bank sets the rates to be LIBOR plus 200
  basis points. The rate will be adjusted every 3
  months to reflect any LIBOR rate changes.
• On September 15, the construction company
  observes the following rates
• Three-month LIBOR 7.00DEC Eurodollar 7.30MAR
  Eurodollar 7.60JUN Eurodollar 7.90
• The company estimates that it can finance
  100,000,000 at the following rates, for an
  average rate of 9.45.
• I Quarter 9.0II Quarte r 9.3III
  Quarter 9.6IV Quarter 9.9

45
Stack Hedges
A stack hedge occurs when futures contracts are
concentrated or stacked in a single future
expiration. The construction company enters into
a stacked hedge by transacting as shown in Table
8.14.
The hedge worked perfectly by locking in the cost
of borrowing regardless of the future course of
interest rates.
46
Stack Hedges

• Notice that in the above example all interest
  rates change by 50 basis points.
• Stack hedges may perform poorly if interest rates
  change in differing amounts. That is, the yield
  curve shifts. Figure 8.5 illustrates this
  situation.

• Insert Figure 8.5 here

47
Strip Hedge

• A strip hedge uses an equal number of contracts
  for each futures expiration over the hedging
  horizon. By doing so, the futures market hedge is
  aligned with the actual risk exposure. The
  transactions necessary to implement a strip hedge
  are demonstrated in Table 8.15.

The performance of a strip hedge is superior to
the stack hedge because the interest rates adjust
every quarter.
48
Advantages of Stacked and Striped Hedge

• Advantages of Stack Hedges
• Works better when the cash position has a single
  horizon.
• Requires trading a single contract.
• Advantage of Strip Hedges
• Can provide a more aligned hedge and better
  results with a multiple-maturity cash position.

49
Tailing The Hedge

• In a tailing hedge the trader slightly adjusts
  the hedge to compensate for the interest that can
  be earned from daily resettlement profits or paid
  on daily resettlement losses. Thus, the tail of
  the hedge is the slight reduction in the hedge
  position to offset the effect of daily
  resettlement interest.
• Tail Factor
• The tail factor is the present value of 1 at the
  hedging horizon discounted to the present (plus
  one day) at the investment rate for the
  resettlement cash flows.
• Tailed Hedge Untailed Hedge ? Tailing Factor
• .

50
Hedging with T-Bond Futures

• The effectiveness of a hedge depends on the gain
  or loss on both the spot and futures sides of the
  transaction. The change in the price of any bond
  depends on the shifts in the levels of
• Interest rates
• Changes in the shape of the yield curve
• The maturity of the bond
• Bond coupon rate
• Table 8.16 and 8.17 illustrate to effect of
  maturity and coupon rates on hedging performance.

51
Hedging with T-Bond Futures

• A manager learns on March 1 that he will receive
  5 million on June 1 to invest in AAA corporate
  bonds with a 5 coupon rate and 10 years to
  maturity. The yield curve is flat and will remain
  so. The current yield on AAA bonds as well as
  forward rates are 7.5. So the manager expects
  to acquire the bonds at 7.5. However, fearing a
  drop in rates, he decides to hedge in the futures
  market to lock-in the forward rate of 7.5.
• The manager considers hedging with T-bills or
  T-bonds. The AAA bonds have a 5 coupon rate and
  a 10-year maturity, which do not match the
  characteristics of either the T-bill or T-bond
  futures contracts. The deliverable T-bills have a
  zero coupon and a maturity of only 90 days, and
  the T-bonds have a maturity of at least 15 years
  and an assortment of semi-annual coupons. Assume
  that the cheapest-to-deliver T-bond will have a
  20-year maturity at the target date of June 1,
  and a 6 coupon.
• The manager will hedge the AAA position with
  T-bill or T-bond futures with yields of 6 and
  6.5, respectively. The manager plans to invest
  in 6,051 bonds each with a price of 826.30.

52
Hedging with T-Bond Futures
Table 8.16 illustrates the transactions and
results of hedging with T-bill futures.
Notice that this loss occurs despite the fact
that rates changed by the same amount on both
investments.
53
Hedging with T-Bond Futures
Table 8.17 illustrates the results of hedging
with T-bond futures.
Again the hedge did not produce the desired
results of isolating the portfolio.
54
Hedging with T-Bond Futures

• Simple approaches to hedging interest rate risk
  often give unsatisfactory results due to
  mismatches of coupon and maturity
  characteristics, as demonstrated in the previous
  examples.
• This section examines some of the major
  alternative strategies for hedging interest rate
  risk
• Face Value Naive (FVN) Model
• Market Value Naive (MVN) Model
• Conversion Factor (CF) Model
• Basis Point (BP) Model
• Regression (RGR) Model
• Price Sensitivity (PS) Model

55
Face Value Naive (FVN) Model

• According to FVN Model, the hedger should hedge
  1 of face value of the cash instrument with 1
  face value of the futures contract.
• Disadvantages
• Neglects potential differences in market values
  between the cash and futures positions.
• Neglects the coupon and maturity characteristics
  that affect duration for both the cash market
  good and the futures contract.

56
Market Value Naïve (MVN) Model

• The MVN Model recommends hedging 1 of market
  value in the cash good with 1 of market value in
  the futures market.
• Disadvantages
• Neglects to make adjustments for price
  sensitivity.
• Advantages
• Consider potential differences in market values
  between cash and futures positions.

57
Conversion Factor (CF) Model

• The CF Model applies only to futures contracts
  that use conversion factors to determine the
  invoice amount, such as T-bond and T-note
  futures.
• The intuition behind this model is to adjust for
  differing price sensitivities by using the
  conversion factor as an index of the sensitivity.
• The CF Model recommends hedging 1 of face value
  of a cash market security with 1 of face value
  of the futures good times the conversion factor.

58
Basic Point (BP) Model

• The BP Model focuses on the price effect of a one
  basis point change in yields on different
  financial instruments.
• To correct for the differences in sensitivity,
  the BP Model can be used to compute the following
  hedge ratio

Where BPCS dollar price change for a 1 basis
point change in the spot instrument. BPCF
dollar price change for a 1 basis point change
in the futures instrument.
59
Basic Point (BP) Model
Today, April 2, a firm plans to issue 50 million
of 180-day commercial paper in 6 weeks. For a one
basis point yield change, the price of 180-day
commercial paper will change twice as much as the
90-day T-bill futures contract, assuming equal
face value amounts. The cash basis price change
(BPCS) is twice as great as the futures basis
price change (BPCF), so the hedge ratio is
-2.0. With a -2.0 hedge ratio and a 50 million
face value commitment in the cash market, the
firm should sell 100 T-bill futures contracts.
Table 8.18 illustrates the hedging results.
60
Basic Point (BP) Model

• Sometimes rates do not change by the same
  amounts. In our previous example, suppose that
  the commercial paper rate is 25 more volatile
  than the T-bill futures rate. To consider
  differences in volatility in determining the
  hedge ratio. The hedge ratio is recomputed as

where RV volatility of cash market yield
relative to futures yield. Normally found by
regressing the yield of the cash market
instrument on the futures market yield. Assume
a RV equal to 1.25. Now the hedge ratio is
Table 8.19 shows these transactions.
61
Basic Point (BP) Model
Because more T-bill futures were sold, the
futures profit still almost exactly offsets the
commercial paper loss.
62
Regression (RGR) Model

• The hedge ratio found by regression minimizes the
  variance of the combined futures-cash position
  during the estimation period. This estimated
  ratio is applied to the hedging period.
• For the RGR Model the hedge ratio is

where COVS,F covariance between cash and
futures sF2 variance of futures
Recall from Chapter 4, the hedge ratio is the
negative of the regression coefficient found by
regressing the change in the cash position on the
change in the futures position. These changes can
be measured as dollar price changes or as
percentage price changes.
63
Price Sensitivity Model

• The PS Model assumes that the goal of hedging is
  to eliminate unexpected wealth changes at the
  hedging horizon, as defined in the following
  equation
• dPi dPF (N) 0
• where
• dPi unexpected change in the price of the cash
  market instrument
• dPF unexpected change in the price of the
  futures instrument
• N number of futures to hedge a single unit
  of the cash market asset

64
Price Sensitivity Model

• The correct number of contracts (N) is calculated
  using the Modified Duration MD

where FPF the futures contract price. Pi
the price of asset I expected to prevail at the
hedging horizon. MDi the modified duration of
asset I expected to prevail at the hedging
horizon. DF the modified duration of the asset
underlying futures contract F expected to
prevail at the hedging horizon. RYC for a
given change in the risk-free rate, the
change in the cash market yield relative to
the change in the futures yield, often
assumed to be 1.0 in practice.
65
Price Sensitivity Model
Suppose that you have accumulated the data in
Table 8.20. Assume that the cash and futures
markets have the same volatility.
For the T-bill hedge, the number of contracts to
trade is given by
For the T-bond hedge the number of contracts to
be traded is
66
Price Sensitivity Model
The performance of the T-bond and T-bill hedges
are presented in Table 8.21.
The T-bond hedge is slightly more effective as it
produces a lower hedging error.
67
Summary of Alternative Hedging Strategies
68
Immunization

• In bond investing, maturity mismatches result in
  exposure to interest rate risk.
• Consider the case of a bank.
• when the asset duration is higher than the
  liability duration, a sudden rise in interest
  rates will cause the value of the portfolio to
  decline.
• When the asset duration is less than the
  liability duration a sudden rise in interest
  rates will cause the value of the portfolio to
  rise.
• By matching the duration of asset and
  liabilities, it is possible for the bank to
  immunize itself from changes in interest rates.
• We consider two examples of immunization
• Planning Period Case
• Bank Immunization case

69
Immunization with Interest Rate FuturesPlanning
Period Case
A portfolio manager has collected the following
information

• The portfolio manager has a 100 million bond
  portfolio of bond C with a duration of 9.2853
  years and is considering two alternatives. The
  manager has a 6-year planning period.
• The manager wants to shorten the portfolio
  duration to six years to match the planning
  period, and is considering two alternatives to do
  so.

70
Immunization with Interest Rate FuturesPlanning
Period Case

• Alternative 1
• The shortening could be accomplished by selling
  Bond C and buying Bond A until the following
  conditions are met

Subject to
Where WI percent of portfolio funds committed
to asset I.
71
Immunization with Interest Rate FuturesPlanning
Period Case

• Alternative 2
• The manager could also adjust the portfolio's
  duration to match the six-year planning period by
  trading interest rate futures and keeping bond C.
• If bond C and a T-bill futures comprise the
  portfolio, the T-bill futures position must
  satisfy the condition
• PP PC NC FPT-bill NT-bill
• where
• Pp value of the portfolioPc price of
  bond CFPT-bill t-bill futures priceNc
  number of C bondsNT-bill number of T-bills
• Expressing the change in the price of a bond as a
  function of duration and the yield on the asset
• dP -Dd(1 r)/(1 r)P

72
Immunization with Interest Rate FuturesPlanning
Period Case

• Applying the equation to the portfolio value,
  bond C, and the T-bill futures, the following
  immunization condition is obtained

This can be simplified to DP PP DC PC NC
DT-bill FPT-bill NT-bill
73
Immunization with Interest Rate FuturesPlanning
Period Case

• Because immunization requires mimicking
  alternative 1, which has a total value of
  100,000,000 and a duration of six years, it must
  be that
• Pp 100,000,000Dp 6DC 9.2853PC
  449.41NC 222,514DT-bill 0.25FPT-bill
  970.00
• Solving by
• DP PP DC PC NC DT-bill FPT-bill NT-bill
• Or alternatively for a T-bond
• DP PP DC PC NC DT-bond FPT-bond NT-bond
• Table 8.24 shows the relevant data for each of
  the three scenarios.

74
Immunization with Interest Rate FuturesPlanning
Period Case
75
Immunization with Interest Rate FuturesPlanning
Period Case

• To see how the immunized portfolio performs,
  assume that rates drop from 12 to 11 percent for
  all maturities. Assume also that all coupon
  receipts during the six-year planning period can
  be reinvested at 11 percent, compounded
  semi-annually, until the end of the planning
  period. With the shift in interest rates the new
  prices are
• PA 904.98
• PC 491.32
• FPT-bill 972.50
• FPT-bond 598.85
• Table 8.25 shows the effect of the interest rate
  shift on portfolio values, terminal wealth at the
  horizon (year 6), and on the total wealth
  position of the portfolio holder.

76
Immunization with Interest Rate FuturesPlanning
Period Case
Notice that each portfolio responds similarly.
77
Transaction Costs for Planning Period Case
While each of the portfolios are equally
effective in immunizing, the cost of obtaining
the immunization varies as demonstrated in Table
8.28.
Notice that the cost of becoming immunized varies
from 949,315 to 13,240 depending upon the
strategy selected.
78
Immunization with Interest Rate FuturesBank
Immunization Case

• Assume that a bank holds a 100,000,000 liability
  portfolio in Bond B, the composition of which is
  fixed. The bank wishes to hold an asset portfolio
  of Bonds A and C that will protect the wealth
  position of the bank from any change as a result
  of a change in yields. Five different portfolio
  combinations illustrate different means to
  achieve the desired result
• Portfolio 1 Hold Bond A and Bond C (the
  traditional approach)
• Portfolio 2 Hold Bond C Sell T-bill futures
• Portfolio 3 Hold Bond A Buy T-bond futures
• Portfolio 4 Hold Bond A Buy T-bill futures
• Portfolio 5 Hold Bond C Sell T-bond futures
• The portfolios are presented in Table 8.26. For
  each portfolio, the full 100,0000,000 is put
  into a bond portfolio and is balanced out by
  cash.

79
Immunization with Interest Rate FuturesBank
Immunization Case
80
Immunization with Interest Rate FuturesBank
Immunization Case
Now consider a drop in rates from 12 to 11 for
all maturities. The effect on the portfolio is
presented in Table 8.27.
Notice that all 5 methods perform similarly.