VII. ARBITRAGE AND HEDGING WITH FIXED INCOME

INSTRUMENTS AND CURRENCIES

A. Arbitrage with Riskless Bonds

• Riskless bonds can be replicated with portfolios of other riskless bonds if their payments are and made on the same dates.

• Consider Bond D, a 3-year, 20% coupon bond selling for $1360.

BOND CURRENT

PRICE FACE

VALUE COUPON

RATE YEARS TO MATURITY

A 1000 1000 .04 2

B 1055.5 1000 .06 3

C 889 1000 0 3

200 = 40bA + 60bB + 0bC 200 = 1040bA + 60bB + 0bC 1200 = 1060bB + 1000bC

333333.2

333333.3

0

=

C

B

A

b

b

b

=

001.00070667.018373.

00006667.0173333.

0001.001.

1200

200

200

B. Fixed Income Hedging

• Fixed income instruments provide for fixed interest payments at fixed intervals and principal repayments.

• In the absence of default and liquidity risk (and hybrid or adjustable features), uncertainties in interest rate shifts are the primary source of pricing risk for many fixed income instruments.

Bond Yields and Sources of Risk

• A bond maturing in n periods with a face value of F pays interest annually at a rate of c with yield y.

• In general, bond risk might be categorized as follows:– Default or credit risk: the bond issuer may not fulfill all of its obligations– Liquidity risk: there may not exist an efficient market for investors to

resell their bonds– Interest rate risk: market interest rate fluctuations affect values of

existing bonds.

n

tnt y

F

y

cFPV

1 )1()1(

2210 )12.1(

1000

)12.1(

100

)12.1(

10020.966

P

Fixed Income Portfolio Dedication

• Assume that a fund needs to make payments of $12,000,000 in one year, $14,000,000 in two years, and $15,000,000 in three years.

12,000,000 = 40bA + 60bB + 0bC 14,000,000 = 1040bA + 60bB + 0bC 15,000,000 = 1060bB + 1100bC

67.195586

67.198666

2000

=

C

B

A

b

b

b

=

001.00070667.018373.

00006667.0173333.

0001.001.

000,000,15

000,000,14

000,000,12

C. Fixed Income Portfolio Immunization

• Bonds, particularly those with longer terms to maturity are subject to market value fluctuations after they are issued, primarily due to changes in interest rates offered on new issues.

• Generally, interest rate increases on new bond issues decrease values of bonds that are already outstanding; interest rate decreases on new bond issues increase values of bonds that are already outstanding.

• Immunization models such as the duration model are intended to describe the proportional change in the value of a bond induced by a change in interest rates or yields of new issues.

Bond Duration• Bond duration measures the proportional sensitivity of a bond to changes

in the market rate of interest.

n

tnt

t

y

F

y

cFPV

1 )1()1(

PV

y

yd

dPV

y

yd

PV

dPVDur

y

y

PV

PV )1(

)1()1(

)1(

)1(

)1(

n

t

nt ynFytcFyd

dPV

1

11 )1()1()1(

)1(

)1()1(

)1(1

y

ynFytcF

yd

dPV

n

t

nt

0

1

0

)1()1()1(

)1( Py

nF

y

tcF

P

y

yd

dPVDur

n

tnt

907.120.966

)12.1(

10002

)12.1(

10001.2

)12.1(

10001.122

Dur

Portfolio Immunization

• Immunization strategies are concerned with matching present values of asset portfolios with present values of cash flows associated with future liabilities.

• The simple duration immunization strategy assumes:– Changes in (1 + y) are infinitesimal.– The yield curve is flat (yields do not vary over terms to

maturity).– Yield curve shifts are parallel; that is, short- and long-

term interest rates change by the same amount.– Only interest rate risk is significant.

Immunization Illustration• Assume a flat yield curve, such that all yields to maturity equal 4%. • The fund manager has anticipated cash payouts of $12,000,000,

$14,000,000 and $15,000,000.• The flat yield curve of 4% implies a value for the liability stream is

$37,816,120.• We calculate bond and liability durations:

DurA = 401+0.04+2× 1040ሺ1+0.04ሻ2−1000 = -1.96

DurB = 601+0.04+2× 60ሺ1+0.04ሻ2+3× 1060

ሺ1+0.04ሻ3−1055.5 = -2.84

DurC = 3× 1000ሺ1+0.04ሻ3−889 = -3

DurL = 12,000,0001+0.04 +2×14,000,000ሺ1+0.04ሻ2 +3×15,000,000

ሺ1+0.04ሻ3−37,817.193.9 = -1.97

Duration and Immunization• Portfolio immunization is accomplished when the duration of the portfolio of bonds equals

the duration (-1.97):

DurA ∙ wA + DurB ∙ wB + DurC ∙ wC = DurL

wA + wB + wC = 1

 

-1.96 ∙ wA – 2.84 ∙ wB – 2.84 ∙ wC = -1.97

wA + wB + wC = 1

• There are an infinity of solutions to this two-equation, three variable system. • Next, suppose that the manager already has invested $3,781,612 (10% of the total liability

value) into Bond A constant at wA = .1. We solve for investment weights as follows:

w = Dur-1 s

742164.

157836.

1.

=

C

B

A

w

w

w

=

82208.486037.17132.

17792.513963.171321.

100

1.0

1

97486.1

Convexity

• The first two derivatives can be used in a second order Taylor series expansion to approximate new bond prices induced by changes in interest rates as follows: 2

0001 )]1([)1(''!2

1)]1([)1( yyfyyfPP

2010

2

12

0

2

20

2

11

01

001

][2

1][

][)1(

)(

)1(

)(

2

1

][)1()1(

0

0 yconvexityPyDurP

yy

Fnn

y

cFtt

yy

nF

y

tcFPP

yP

n

tnt

n

tnt

0

2

2

12

2

)1(

)(

)1(

)(

P

y

Fnn

y

cFttn

n

tt

Convexity

Convexity Illustration

ConvA = 2× 40ሺ1+0.04ሻ3+6× 1040

ሺ1+0.04ሻ41000 = 5.41

ConvB = 2× 60ሺ1+0.04ሻ3+6× 60

ሺ1+0.04ሻ4+12× 1060ሺ1+0.04ሻ5−1055.5 = 10.30

ConvC = 12× 1000ሺ1+0.04ሻ5−889 = 11.09 ConvL = 2×12,000,000

ሺ1+0.04ሻ3 +6×14,000,000ሺ1+0.04ሻ4 +12×15,000,000

ሺ1+0.04ሻ5−37,816.120 = 6.38

Duration, Convexity and Immunization

• Portfolio immunization is accomplished when the duration and the convexity of the portfolio of bonds equals the duration and convexity (6.38) of the liability stream:

 

DurA ∙ wA + DurB ∙ wB + DurC ∙ wC = Duro

ConvA ∙ wA + ConvB ∙ wB + ConvC ∙ wC = Convo

wA + wB + wC = 1

 

-1.962 ∙ wA – 2.837 ∙ wB – 3 ∙ wC = -1.975

5.41 ∙ wA + 10.30 ∙ wB + 11.09 ∙ wC = 6.38

wA + wB + wC = 1

 

• The single solution to this 3 X 3 system of equations is wA = -0.481, wB = 9.358 and wC = -7.877. This system provides an improved immunization strategy when interest rate changes are finite.

D. Term Structure, Interest Rate Contracts and Hedging

• The Pure Expectations Theory:

• The Yield Curve can be bootstrapped

n

ttt

nn yy

1,1,0 )1()1(

1)1(

1,1,0

n

n

tttn yy

0 2 4 6 8 10 12 14 16 18

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

Years

Sp

ot

Rat

e

Simultaneous Estimation of Discount Functions

• Three coupon bonds are trading at known prices. Bond yields or spot rates must be determined simultaneously to avoid associating contradictory rates for the annual coupons on each of the three bills.

BOND CURRENT PRICE

FACE VALUE

COUPON RATE

YEARS TO MATURITY

E 947.376 1000 .05 2

F 904.438 1000 .06 3

G 981 1000 .09 3

002.003.0

00177.00181667.001.

0371.03815.001.

981

438.904

376.947

=

3

2

1

D

D

D

=

751316.

85734.

943377.

CF-1 P0 = d

Spot and Forward Rates• Spot rates are as follows:

• Forward rates are as follows:

1,0

1

06.11

yD

2,0

2

1

2

08.11

y

D

3,0

3

1

3

10.11

y

D

10.1

)06.1(

)08.1( 2

2,1

y

14.1

)10.1)(06.1(

)10.1( 3

3,2

y

12.1

)06.1(

)10.1( 3

3,1 y

E. Arbitrage with Currencies• Triangular arbitrage exploits the relative price difference between one currency and two other

currencies. • Suppose the buying and selling prices of EUR 1 is USD 1.2816. However, the South African

Rand (SFR) has a price (buying or selling) equal to USD 0.2000 or EUR 0.1600.• Since USD 0.20 = EUR 0.16, dividing both figures by 0.16 implies that USD1.25 = EUR1. • But, this is inconsistent with the currency price information given above, which states that

USD1.2816 = EUR1.0. • In terms of the SFR, it appears that the USD is too strong relative to the EUR, so we will start

by selling USD0.20 for SFR1 as per the price given above. We will cover the short position in USD by selling EUR0.16, which actually nets us .16 ∙ USD 1.2816 = USD0.2051. We will cover our short position in EUR by selling SFR at the price listed above.

USD SFR EUR

Sell USD0.20 for SFR1 -0.2000 +1.0000

Sell EUR0.16 for USD0.2051 +0.2051 - 0.1600

Sell SFR1.0 for EUR0.16 -1.0000 +0.1600

Totals +0.0051 0 0

Parity and Arbitrage in FX Markets

1. Purchase Power Parity (PPP)

2. Interest Rate Parity (IRP)

3. Forward rates equal expected spot rates

4. The Fisher Effect

5. The International Fisher effect.

Collectively, these conditions are often referred to as the International Equilibrium Model.