FINANCING IN
INTERNATIONAL MARKETS
3. BOND RISK MANAGEMENT
Forward Price of a Coupon Bond
Consider the following transactions at time T=0:
i.
Borrow to buy a bond for T2 days at an interest rate r2.
ii.
Buy a coupon bond at P + A1. Receive a coupon C at T1.
iii.
Sell the bond at T2 at the forward price F, receiving also A2.
P+A1
C
F+A2
T=0
T1
T2
(Recall:
Cash price = P + A1.)
Cash flows at time T2:
i. Pay interest on (P+A1) for T2 days.
ii. Receive interest on C for T2-T1 days at the interest rate f. (f?)
iii. Receive payment (F+A2) for the bond
Forward Price of a Coupon Bond
Q: How do we determine f ?
f represents the interest rate that makes a lender indifferent between
making a deposit for T2 days at the interest rate r2 or making a deposit for
T1 days at the interest rate r1 and rolling over the deposit at time T1 for
T2-T1 days at the (unknown) rate f.
=> We call this interest rate at time T1 an implied forward rate, f:
1 + f (T2 - T1)/360 = (1 + r2 T2/360)/(1 + r1 T1/360).
No arbitrage implies:
(P+A1)(1+r2 T2/360)  C (1+r2 T2/360)/(1+r1 T1/360) + (F+A2).
Assuming equality and solving for F:
F = (P+A1)(1+r2 T2/360) - C (1+r2 T2/360)/(1+r1 T1/360) - A2.
Example: Calculation of F for a 11% T-bond to August 23.
Price = P = 84'17 --84(17/32)-Coupon = C = 11% (payable on July 15 and January 15).
As of May 23, r1 = 8% (two months or less)
r2 = 8¼ % (three months).
T2 = 92 days (May 23 to August 23),
T1 = 53 (May 23 to July 15).
Basis for T-bonds = actual/actual.
Forward price, calculated to August 23 = F = ?
Example (continuation): Recall the previous formula:
F= (P+A1)(1+r2 T2/360) - C (1+r2 T2/360)/(1+r1 T1/360) - A2.
a.- A1 (Days the seller held the coupon= January 15-May 23= 128 days)
Interest (actual/actual) = (.11/2)(128/181) = .038895.
A1 = .038895x100 = 3.8895.
b.- A2 (T2 = August 23 - July 15 = 39 days)
Interest (actual/actual) = (.11/2)(39/184) = .0116756.
A2 = 1.16756.
c.- F (Convert the bond price 84'17 to 84.53125%)
F= (84.53125+3.8895)(1+.0825x92/360)-5.5(1 +.0825(92/360))-1.16576
(1 + .08(53/360))
= 83.5686. ¶
Bond Futures
• Bond risks: price (interest rate); default; reinvestment; call; & inflation.
• Bond futures are used to manage price risk on bond portfolios.
• Design (supply) of a bond futures contract:
What is the underlying security?
Demand (Need for the contract?)
Easy to buy and sell (liquid)
• Three types: Notional Bond Futures
Cheapest to Deliver Bond (CDB) Futures
Index-based Bond Futures
(1) Notional Bond Futures
Hypothetical bond of fixed principal, coupon, and maturity.
Example: A futures contract could be based on a (nonexistent)
GBP 50,000 7% government bond and maturing in 10 years. ¶
Note: We know how to price this hypothetical bond; which s is the
notional bond behind the U.K. Long Gilt futures.
Notional government bond futures trade in many exchanges:
the CBOT, the Tokyo Stock Exchange,
the Deutsche Boerse in Frankfurst, LIFFE in London,
the Euronext in Paris, the MEFF in Barcelona
The U.S. Treasury Bond Contract
• Second most popular interest rate derivative instruments: CBOT U.S.
T-bond and T-notes contracts.
• Prices are based on a (fictional) 20-year 6% U.S. Treasury bond.
• Different bonds can be delivered to cover a short position.
 Any Treasury bond with at least 15 years to maturity or to first
call date, whichever comes first, qualifies for delivery.
Short side delivers FV=USD 100,000 of any one of the qualifying bonds.
• Delivery dates (maturity): Mar, June, Sep, Dec, and two nearby months.
• Short position decides when, in the delivery month, delivery actually
will take place (timing option).
• On the delivery day the buyer (long side) pays the seller for the bonds,
and in return receives Fed Reserve book-entry T-bonds.
• Quotes: in terms of par being 100 and in 32nds of 1%.
 Tick size and value: 1/32 of 1% -- USD 31.25.
Example: The price moves from 61'07 to 61'08. Long side makes:
(1/32) x (.01) x (USD 100,000) = USD 31.25.
If the price moves from 61'07 to 60'31, the long side loses:
(8/32) x (.01) x (USD 100,000) = USD 250. ¶
• Characteristics of three Notional Bond Futures
(2) Cheapest-to-Deliver Bond (CDB) Futures
• Several traded bond futures contracts allow for actual delivery of one of
a number of bonds against the futures contract.
Short side (the seller) makes the choice of which bond to deliver.
• Some of the deliverable bonds will cost less than others.
 the seller chooses the "cheapest to deliver.“
Equilibrium price for a deliverable T-bond future
• During the delivery month, traders will compare the value of a cash
bond (deliverable) to its equivalent in the futures market.
Long side pays an invoice price (I), representing the cash price as a
percentage of face value, plus accrued interest on the cash bond.
The invoice price is the futures price (Z), times an exchange conversion
factor (cf):
I = Z x cf.
The same bond in the cash market may be purchased at P+A1. Then:
P = Z x cf.
• Calculating cf:
- Setting the YTM on the CDB = 6% (s.a.)
- Divide the resulting price by 100.
- Rounded down maturity to the nearest quarter.
(If there is an extra quarter, subtract accrued interest.)
Note: For the UK Long gilt set the YTM on the CDB = 7% (s.a.).
Example: Calculation of cf.
(A) T-bond futures & no Accrued interest: U.S. Treasury 10% May 2030
In March 2010, we observe:
C = 10%
Maturity = 20 yrs. and two months. (Rounding down, 20 yrs).
Z = 90.
First payments to be made after 6 months.
Setting the YTM = 6%,
i=1 to 40 5/(1+.03)i + 100/(1+.03)40 = 146.2295.
Dividing by 100,
cf = 1.462295.
Calculation of I:
I = 90 x 1.462295 = 131.60659.
Actual cash payments = USD 131,606.59.
=> A buyer of the T-bond futures pays USD 131,606.59 and
receives a Federal Reserve book-entry transferring the property
of this T-bond.
(C) T-bond futures & Accrued interest.
C = 10%
Maturity = 18 yrs and 4 months (round down= 18 yrs and 3 months).
Z = 92.125
YTM = 6 %.
Discounting all the payments back 3 months from today:
5 + i=1 to 36 5/(1+.03)i + 100/(1+.03)36 = 148.665.
The interest rate for a 3-month period is ((1+.03).5 - 1) = 1.489%.
Discounting back to today: 148.665/1.0149 = 146.48398..
Subtracting the accrued interest of 2.5 gives a price of 143.98398.
Then, cf = 1.4398398.
Calculation of I:
I = 92.125 x 1.4398398 = 132.64524.
Actual cash payment = USD 132,645.24.
=> For USD 132,645.24, a T-bond futures buyer receives the
10% U.S. Treasury with 18 years and 4 months to maturity. ¶
Explanatory Factors of CDB
• YTM set at 6%.
• Option features associated with the delivery process:
(1) Delivery option: short side has a choice of cash bonds for delivery.
(2) Wild card option: futures market closes at 2PM (Chicago time) and
sets futures settlement price (I is fixed). Short side has until 8PM to
declare delivery.
 the wild card option is a six-hour put option, with X=I.
If P < I by 8PM, the short side exercises this put.
• Both the delivery option and the wild card option have positive value.
In general, P > Z x cf.
Example: Value of embedded options on Dec 31, 2000.
Today: October 2, 2000.
Underlying Instrument: Sep 27 2013 8% U.K. Long Gilt .
Hedging Instrument: U.K. Long Gilt Dec futures.
Z = 113.27.
P = 123.05
cf = 1.08356.
r2 = 5.50%
T2 = 90 (10/2/00 to 12/31/00).
Value of deliverable options (on Dec 31, 2000): ?
Example (continuation):
Steps:
(1), Calculate the carry component, which is equal to the interest income
received minus the financing cost:
Carry = A2 – (P + A1) x r2 T2/360
= 1.98895 – (123.05 + 0.110497) x x.055 x 90/360 = 0.29545
(2) Basis = P – I = Carry + delivery options value.
Basis = P – Z x cf = 123.05 - 113.27 x 1.08356 = 0.31516.
Option value = 0.31516 – 0.29545 = 0.019665.
That is, the option value represents 6.24% of the basis. ¶
• Traders cannot buy a cash bond and make immediate delivery during
the period prior to the delivery month. Thus, they compare F (forward
price of the bond) to its futures equivalent:
Basis after carry (BAC):
BAC = F - Z x cf.
• The BAC indicates the cost, in terms of forward dollars, of buying a
cash bond and delivering it against the futures contract.
• A trader wants to minimize the BAC
(cost: F; revenue: Z x cf)
 CDB is the bond with the smallest BAC.
Example: Determine CDB against the June 1990 T-bond futures
contract. Today is April 16. Assuming an 8% short rate.
Conversion Value of Selected Deliverable T-Bonds, April 16, 1990
Maturity
Coupon cf
June Future Invoice Price (I)
(or 1st call)
(%)
(Z)
(Z x cf)
Nov. 15, 2006
14
1.5400
92'03
141.82438
Aug. 15, 2015
10(5/8) 1.2820
92'03
118.06419
Nov. 15, 2016
7½
0.9453
92'03
87.05622
May. 15, 2016
7¼
0.9185
92'03
84.58811
Example (continuation):
Forward Value of Deliverable T-Bond, Calculated from April 16 to June
1, with a Short Rate of 8%
Maturity
(or 1st call)
Price
(P)
Accrued
Interest
Nov. 15, 2006
Aug. 15, 2015
Nov. 15, 2016
May. 15, 2016
143'15
118'13
87'10
84'27
5.87845
1.76105
3.14917
3.04420
Accrued Forward
Coupon Interest Price
(4/16)
(6/1)
7.0000 0.64674 143.20085
0
3.11119 118.28449
3.7500 0.34647 87.27584
3.6250 0.33492 84.81283
BAC
1.37647
0.22030
0.21962
0.22472
 The November 15, 2016 has the smallest BAC and hence is the CDB.
Example (continuation):
• Calculations for first (Nov. 15, 2006) bond:
(1)
Calculation of I = Z x cf
As of June 1, 1990, there are 16 yrs and 5 months to call.
Round down to the nearest quarter. Then,
Price = 154.00,
 cf = 1.5400.
The June future is at Z = 92 (3/32) (92.09375),
I = 92.09375 x 1.5400 = 141.82438.
(2) Calculation of F.
Data: P = 143.46875 & C = 14.
As of April 16, accrued interest from November 15 is calculated as
A1 = (14/2) x (152/181) = 5.87845.
On June 1, there will be seventeen days of accrued interest from May 15:
A2 = (14/2) x 17/184 = 0.64674
Assuming an 8% short rate, the coupon paid on May 15 may be
reinvested at the forward interest rate f given by
1 + f x (17/360) =
1 + .08 x (46/360).
1 + .08 x (29/360)
Calculation of F:
F = (143.46875+5.87845)(1+.08x46/360)-7.0 x 1+.08 x 46/360 -.64674=
1+.08x29/360
= 143.20085.
Finally, BAC = 143.20085 - 141.82438 = 1.37647. ¶
Hedging With Bond Futures
• Key concept: basis point value (bpv).
bpv = change in the bond's price for a 1 bp movement in yield.
• We want to determine the optimal hedge ratio.
Assume that the spread between the bond yields is constant.
We have two bonds: A with a with a bpvA=2 and B with a bpvB= 1.
For 1 bp rise in yields
 the price of the A will fall by 2.
 the price of the B will fall by 1.
Long position: USD 1 in Bond A,
Hedging position: We need to be short USD 2 of the bond B.
hedge ratio of bond B: number of units of bond B needed to create a
hedged position against another bond, A:
hedge ratio of bond B = -(bpvA/bpvB) = -2.
Similarly, if we use bond futures to hedge we have
hedge ratio future = -(bpvbond/bpvfuture).
Note: For CDB futures:
bpvfuture = bpvCDB / cf.
• Comparing the above equations, we see that the hedge ratio for a CDB
hedge ratio CDB future = h = -(bpvbond/bpvCDB) x cf.
Example: The bpvs of the bonds in Tables XIII.B and XIII.C are:
Maturity
Coupon
bpv
(or 1st call)
(%)
Nov. 15, 2006
14
.1133
Aug. 15, 2015
10 5/8
.1162
Nov. 15, 2016
7½
.0919 (<= CDB, with cf=0.9453)
May. 15, 2016
7¼
.0894
• Interpretation: If yields drop 10 bps (.1%), the price of the 14% Nov.
15, 2006, will rise by 1133 bps (1.133%), from 143.469 to 144.602.
• Consider a Eurobond with a bpv = .145.
hedge ratio = h =- (.145/.0919) x 0.9453 = -1.4915.
=> To hedge USD 10M of this bond, short 149 (-1.4915x100) T-futures.
If the August 15, 2015, bond subsequently became cheapest to deliver:
h =- (.145/.1162) x 1.2820= -1.5997 (sell 11 more contracts). ¶
Duration-Based Hedging Strategies
• Recall the definition of modified duration:
1 dP = -1 Σt t Ct = -D = -D*
P dr
P (1+r)t+1
1+r
• Typical situation: position in an interest rate dependent asset (a
Eurobond portfolio) is hedged using an interest rate futures contract.
• Assume that the change in interest rates, dr, is the same for all
maturities (we only allow for parallel shifts in the yield curve).
• Define:
F:
Contract price for the interest rate futures contract
DF:
Duration of asset underlying futures contract
S:
Value of the asset (eurobond portfolio) being hedged
Ds:
Duration of asset (eurobond portfolio) being hedged
• Assume that the change in interest rates, dr, is the same for all
maturities (we only allow for parallel shifts in the yield curve).
Then, the number of contracts required to hedge against an uncertain
change in interest rates, dr, is:
N = S Ds (1+rF)/[F DF(1+rs)] = (S Ds*)/(F DF*).
 This is the duration-based hedge ratio (price sensitivity hedge ratio).
Example:
Today: It is January 19.
Underlying position: EUR 20 million in Eurobonds.
Duration of EUR portfolio: 7.80 years.
YTM of EUR portfolio: 7.92%
Expectation: interest rates volatile over the next three months.
The bank manager decides to hedge: use June bond futures contracts.
Futures price = 91 (08/64)
 Futures contract price = EUR 91,250.
CDB at Euronext = 10-year 3.5% government EUR 100,000 bond
Duration of CDB = 7.20 years
YTM of CDB = 6.80%
Example (continuation):
Summary:
F: Contract price for the CDB futures contract = EUR 91,250.
DF: Duration of CDB underlying futures contract = 7.20 years
rF: YTM of the CDB = .0680
S: Value of the eurobond portfolio being hedged = EUR 20,000,000.
Ds: Duration of eurobond portfolio being hedged = 7.80 years
rs: YTM of the eurobond portfolio being hedged = .0792
The number of bond futures contracts is given by:
N = S Ds (1+rF)/[F DF(1+rs)] = (S Ds*)/(F DF*)
N = (EUR 20Mx7.80x(1+.0680))/[EUR 91,250x7.20x(1+.0792)] =
= 234.9787
The portfolio manager should short 235 futures contracts. ¶
Application: Asset Allocation
Situation: A fund manager decides to invest (long-term):
25 percent in bonds
60 percent in stocks
15 percent in real estate.
Usually, significant changes to the allocation are avoided because of
transaction costs.
Solution: Use futures to change the asset allocation indirectly.
Example: A portfolio is composed:
X dollars in stocks
Y dollars in bonds.
Manager wants to decrease her short-term bond allocation from Y to Y1
(the reduction in bonds is matched by increasing stocks).
Bond portfolio has a modified duration of DY*.
Solutions: (1) Sell bonds and buy stocks (might be too expensive)
(2) Change allocation by selling T-bond futures contracts.
• Solution (2): Manager wants to have income DY*Y1 from bond position
if interest rates change by 1 percent.
She plans to generate that amount with income from the current bond
portfolio, DY*Y, and income from a T-bond futures position, N DF*F:
DB*Y1 = DB*Y + N DF*F.
Solving for N, we get
N = DB* (Y1 - Y)/(DF*F) <0 (sell T-bonds futures contracts are sold).
Example: Ms. O'Neil has the following portfolio:
USD 90,000,000 in an index stock portfolio that tracks the S&P 500.
USD 60,000,000 in a bond portfolio with a D* of 7.8 years.
Long-term, Ms. O'Neil likes her portfolio allocation.
She forecasts that stocks are going to do well over the next six months.
Situation: Ms. O'Neil wants to take advantage of this forecast.
Transaction costs are very high.
She uses futures contract to change her portfolio allocation.
Data:
CDB has a D* of 6.5 yrs
Price of a 6-mo T-bond contract = 90.
Price of 6-mo S&P 500 futures = 350.
Example (continuation):
Steps:
(1) Eliminate the interest rate exposure: Use T-bond futures.
Number of T-bond futures to sell is
N = 7.8 (0 - 60,000,000)/(6.5 x .90 x 100,000) = -800 contracts.
 selling 800 T-bonds futures completely wipes out interest
rate risk (completely liquidating the bond investment position).
(2) Increase long stock position: Use S&P 500 futures.
Number of contracts to buy is
N = 60,000,000/(350x500) = 342.857, or 343 contracts. ¶