Forwards  and  Swaps:  Interest  Rates              

2

Learning  Objec-ves    

¨  Understand  and  manage  interest  rate  risk  via  forward  and  swap  agreements    

¨  Understand  the  rela-onship  between  discount  rates,  swap  rates,  zero  coupon  rates,  forward  rates,  and  bond  yields    

Interest  Rate  Risk    3  

t0=0 tS=0.5 tL=1.0 yrs yrs yrs

q

f

• A  firm  requires  a  $10,000,000  loan  over  a  period  from  6  to  12  months  from  present    • Over  the  period  0.5  ≤  t  ≤  1.0  

• The  firm’s  treasurer  believes  that  the  interest  rate  offered  will  rise  over  the  next  6  months    i.e.,  interest  expense  will  be  greater  in  the  near  future    • Assume  that  the  company  can  borrow  and  deposit  funds  at  LIBOR.      • Current  6  month  LIBOR  is  4.28363%,  q  (simple  annual  rate)    • Current  12  month  LIBOR  is  4.51863%,  r  (simple  annual  rate)    

r • The  firm  might  borrow  for  12  months,  but  loan  the  funds  for  the  first  6  months  leaving  an  effec-ve  ‘forward’  rate,  f  

)tr1())tt(f1()tq1( LSLS ⋅+=−⋅+⋅⋅+

⎥⎦

⎤⎢⎣

⎡−

⋅+⋅+

−= 1

)tq(1)tr(1

)t(t1f

S

L

SL4.65395%      

10.5).0428363(11.0).0451863(1

0.5)(1.01f

=

⎥⎦

⎤⎢⎣

⎡−

⋅+

⋅+

−=

Forward  Rate  Agreement  4  

t0 tS tL qB-qO

fB-fO

rB-rO

FRA term loan term

Similar to foreign exchange risk and ‘money market’ hedges, banks have a product called a ‘FRA’ forward rate agreement which packages the interest rate hedge

The actual loan interest rate will be set at tS while the actual interest will be paid at tL The FRA will be executed at t0 and settled at tS The effective loan or forward rate is set at t0, but the relative benefit of the FRA and cost of the loan are not known The FRA includes a ‘notational principal’, and is cash settled

Forward  Rate  Agreement  5  

FRA buyer • is the loan borrower and takes the long position in the FRA • believes that interest rates may rise so seeks to hedge its interest rate risk exposure • becomes a fixed rate payer instead of a floating rate payer as it is initially • Equivalently makes the following transactions

• Borrows at the long offer rate rO over term t0 to tL • Lends at short bid rate qB over term t0 to tS • Locks in the forward offer rate fO over term tS to tL

FRA seller • is often a financial intermediary such as bank and takes the short position, but most likely will ‘lay off’ its risk • takes the short position in the FRA and becomes a floating rate payer • Equivalently makes the following transactions

• Lends at the long bid rate rB over term t0 to tL • Borrows at short offer rate qO over term t0 to tS • Locks in the forward bid rate fB over the term tS to tL

⎥⎦

⎤⎢⎣

⎡−

⋅+⋅+

−= 1

)tq1()tr1(

)tt(1f

SO

LB

SLB⎥

⎦

⎤⎢⎣

⎡−

⋅+⋅+

−= 1

)tq1()tr1(

)tt(1f

SB

LO

SLO

6  

t0 = 0 tS = .5 tL = 1.0

4.4092%        

1.5).0428363(11.0).0439363(1

0.51fB

=

⎥⎦

⎤⎢⎣

⎡−

⋅+

⋅+=

%4.7793        

10.5).0415863(11.0).0451863(1

0.51fO

=

⎥⎦

⎤⎢⎣

⎡−

⋅+

⋅+=

• If the treasurer buys a FRA with notational principal of $10M and forward offer (borrowing) rate of 4.7793% • Treasurer effectively locks in the forward offer rate for a six month loan (tS < t ≤ tL) with principal $10M commencing in 6 mo. at tS. • The FRA is actually settled in cash at FRA expiry which we assume here is also the time of loan commencement. • Note that the FRA and loan are two completely separate agreements and transactions and that a party can buy or sell a FRA for speculation and not only to hedge a natural interest rate risk.

qB = 4.15863% rB =4.39363%

qO = 4.28363% rO =4.51863%

7  

d1

z1·∆t f2·∆t f3·∆t f4·∆t f5·∆t f6·∆t

k 0 1 2 3 4 5 6

tk 0.0 0.5 1.0 1.5 2.0 2.5 3.0 d2 d3

d4 d5 d6

2t2

22

)z1(

tz

+

=⋅

6t6 )z1( +5t

5)z1( +4t4)z1( +3t

3)z1( +2t2)z1( +

Now consider a sequence of future lending requirements – semi-annual for 3 years Zero coupon rates, zk 1 ≤ k ≤ 6 Forward rates, fk Δt = .5 Discount rates, dk

8  

Δt)z(1CC1

10 ⋅+=

ktk

k0 )z(1

CC+

=

⎟⎠⎞

⎜⎝⎛ +

=Δ⋅+

=

mz1

1t)z(1

1d11

1

ktk

k )z(11

d+

=

1)-­‐(d1m

1)-­‐(d1

t1z

111 =

Δ=

1d1

z kt

kk −=

9  

Δt)z(11d1

1 ⋅+=

Δt)f(1d

d2

12 ⋅+=

Δt)f(1d

dk

1kk ⋅+= −

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−= −− 1

ddm1

dd

Δt1f

k

1k

k

1kk

YearZero  

Coupon  Rate

Discount  Factor

Forward  Rate

1 4.5000% 0.95694   4.5000%2 5.0126% 0.90681   5.5276%3 5.2723% 0.85715   5.7936%4 5.4027% 0.81020   5.7948%5 5.4671% 0.76633   5.7255%

0%

1%

2%

3%

4%

5%

6%

7%

0 1 2 3 4 5

Rat

est years

10  

Discount  factors  

dk  

Zero  coupon  rates  

zk  

Forward  rates  

fk  

Yields  for  coupon  bonds  

yj

‘Boot-­‐strapping’

1d1

z kt

kk −=

ktk

k )z(11

d+

=

Δt)f(1d

dk

1kk ⋅+= −

⎟⎟⎠

⎞⎜⎜⎝

⎛−= − 1

dd

Δt1f

k

1kk

Interest  Rate  Swaps    11  

Firm

Swap Dealer

Bank LIBOR

+2%

SWAP Rate

LIBOR

Net interest rate = LIBOR – (LIBOR +2%) – swap rate

= - (swap rate +2%)

12  

YearZero  

Coupon  Rate

Discount  Factor

Forward  Rate

Swap  Rate  

Floating  Cash  Flow

Fixed  Cash  Flow

Net  Flow  to  Swap  Buyer

1 4.5000% 0.95694   4.5000% 4.5000% 450,000$         543,750$         (93,750)$        2 5.0126% 0.90681   5.5276% 5.0000% 552,764$         543,750$         9,014$              3 5.2723% 0.85715   5.7936% 5.2500% 579,359$         543,750$         35,609$          4 5.4027% 0.81020   5.7948% 5.3750% 579,479$         543,750$         35,729$          5 5.4671% 0.76633   5.7255% 5.4375% 572,549$         543,750$         28,799$          

Present  Value 2,336,729$   2,336,729$  

13  

kkk

2k

1k dFd

mcF...d

mcFd

mcFP ⋅+⋅⋅++⋅⋅+⋅⋅=

kkk

2k

1k dFd

msF...d

msFd

msFF ⋅+⋅⋅++⋅⋅+⋅⋅=

kkk

2k

1k dd

ms...d

msd

ms1 +⋅++⋅+⋅=

∑=

=k

1j

jkk m

dsd-­‐1

∑=

= k

1j

j

kk

mdd-­‐1

s

⎟⎠⎞

⎜⎝⎛ +⋅++⋅+⋅= 1msd...d

msd

ms1 k

k2k

1k

14  

⎟⎠⎞

⎜⎝⎛ +⋅+= ∑

−

=

1msd

md

s1 kk

1k

1j

jk

⎟⎠⎞

⎜⎝⎛ +

−

=∑−

=

1ms

md

s1d

k

1k

1j

jk

k

YearZero  

Coupon  Rate

Discount  Factor

Forward  Rate

Swap  Rate  

1 4.5000% 0.95694   4.5000% 4.5000%2 5.0126% 0.90681   5.5276% 5.0000%3 5.2723% 0.85715   5.7936% 5.2500%4 5.4027% 0.81020   5.7948% 5.3750%5 5.4671% 0.76633   5.7255% 5.4375%

15  

Discount  factors  

dk  

Zero  coupon  rates  

zk  

Forward  rates  

fk  

Yields  for  coupon  bonds  

yj

‘Boot-­‐strapping’

1d1

z kt

kk −=

ktk

k )z(11

d+

=

Δt)f(1d

dk

1kk ⋅+= −

⎟⎟⎠

⎞⎜⎜⎝

⎛−= − 1

dd

Δt1f

k

1kk

Swap  rates  

sk

∑=

= k

1j

j

kk

mdd-­‐1

s

⎟⎠⎞

⎜⎝⎛ +

−

=∑−

=

1ms

md

s1d

k

1k

1j

jk

k

16  

CME begins clearing interest rate swaps CHICAGO/NEW YORK Mon Oct 18, 2010 (Reuters) - CME Group Inc said on Monday that it had begun providing clearing to the $400 trillion interest-rate swaps market, the largest of the opaque markets that lawmakers are forcing onto more transparent venues.

CME Information

Essen-al  Concepts      17