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.
BOND
A
CURRENT
PRICE
1000
FACE
VALUE
1000
COUPON
RATE
.04
YEARS TO
MATURITY
2
B
1055.5
1000
.06
3
C
889
1000
0
3
• Consider Bond D, a 3-year, 20% coupon bond selling for $1360.
200 = 40bA + 60bB + 0bC
200 = 1040bA + 60bB + 0bC
1200 =
1060bB + 1000bC
.001
0 
0

 bA    .001
 3.333333  = b  =  .0173333  .0006667 0 


  B 
 2.333333 bC   .018373 .00070667 .001
 200 
 200 


1200
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.
n
PV  
t 1
P0  966.20 
cF
(1  y)
t

F
(1  y) n
100
100
1000


1
2
(1  .12) (1  .12) (1  .12) 2
• 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.
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
.001
0 
 2000  bA    .001
 198666.67  = b  =  .0173333  .0006667

0
B

   

 195586.67 bC   .018373 .00070667 .001
12,000,000
14,000,000


15,000,000
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
PV  
t 1
cFt
F

t
(1  y ) (1  y ) n
PV (1  y)
dPV d (1  y)
dPV (1  y)

 Dur 



PV
(1  y)
PV (1  y) d (1  y) PV
n
dPV
   tcF (1  y) t 1  nF (1  y ) n1
d (1  y ) t 1
n
dPV

d (1  y )
  tcF (1  y)
t
 nF (1  y ) n
t 1
(1  y )
 tcF
 nF

t
dPV
(1  y ) 
(1  y ) n
t 1 (1  y )
Dur 


d (1  y )
P0
P0
n
 1 .11000  2  .11000  2 1000


(1  .12)
(1  .12) 2
(1  .12) 2
Dur 
 1.907
966.20
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 =
DurB =
40
1040
+2×
1+0.04
1+0.04 2
−1000
60
60
1060
+2×
+3×
2
1+0.04
1+0.04
1+0.04 3
−1055.5
DurC =
DurL =
= -1.96
3×
1000
1+0.04 3
−889
= -2.84
= -3
12,000 ,000
14,000 ,000
15,000 ,000
+2×
+3×
1+0.04
1+0.04 2
1+0.04 3
−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:
0
1 
 .1   wA   0
.157836 =  w  =  .171321 .513963  .17792


  B 
.742164  wC   .17132 .486037  .82208
w =
Dur-1
 1.97486


1


 0.1 
s
Convexity
Convexity 
 n ( t 2  t ) cF   ( n 2  n ) F 


t 2 
n2 
t

1
(1 y )




 
 (1 y )

P0
• 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:
1
P1  P0  f (1  y0 )  [(1  y )]   f ' ' (1  y0 )  [(1  y )]2
2!
 n

 tcF
nF
P1  P0  

[ y ]
t 1
n 1 
(1  y0 )
 t 1 (1  y0 )

1  n (t 2  t )  cF
( n 2  n)  F 
2
 


[

y
]

2  t 1 (1  y0 ) t  2
(1  y0 ) n  2 
1
 P0  Dur  1P0y0  [ y ]   P0  convexity  [ y ]2
2
Convexity Illustration
ConvA =
ConvB =
2×
2×
1000
= 5.41
60
60
1060
+6×
+12×
4
3
1+0.04
1+0.04
1+0.04 5
−1055 .5
ConvC =
ConvL =
40
1040
+6×
3
1+0.04 4
1+0.04
2×
12×
1000
1+0.04 5
−889
= 10.30
= 11.09
12,000 ,000
14,000 ,000
15,000 ,000
+6×
+12×
4
3
1+0.04
1+0.04
1+0.04 5
−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
5.41 ∙ wA
wA
– 2.837 ∙ wB – 3 ∙ wC = -1.975
+ 10.30 ∙ wB + 11.09 ∙ wC = 6.38
+
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:
n
(1  y0,n )   (1  yt 1,t )
y 0, n 
n
n
n
 (1  y
t 1,t
) 1
t 1
t 1
• The Yield Curve can be bootstrapped
6.00%
Spot Rate
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
0
2
4
6
8
Years
10
12
14
16
18
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
.0371  947.376  D1  .943377
 .001  .03815
 .001 .00181667  .00177 904.438 =  D  =  .85734 


  2 

 0
.003
 .002   981   D3  .751316
CF-1
P0
= d
Spot and Forward Rates
• Spot rates are as follows:
1
 1  .06  y0,1
D1
1
1
2
2
 1  .08  y0, 2
D
1
1
3
3
 1  .10  y0,3
D
• Forward rates are as follows:
(1  .08) 2
y1, 2 
 1  .10
(1  .06)
(1  .10)3
y2 , 3 
 1  .14
(1  .06)(1  .10)
y1,3 
(1  .10)3
 1  .12
(1  .06)
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
Sell USD0.20 for SFR1
-0.2000
Sell EUR0.16 for USD0.2051 +0.2051
Sell SFR1.0 for EUR0.16
Totals
+0.0051
SFR
+1.0000
-1.0000
0
EUR
- 0.1600
+0.1600
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.