Lecture 9
Ct
Ct

t
rt
(1  r ) e
Warning:
Answers in book will be slightly different than calculator.
Bond Value =
C1
(1+r)
+
C2
(1+r)2
+
C3
(1+r)3
Example
$1,000 bond pays 8% per year for 3 years. What is the price at a
YTM of 6%
1053.46
=
80
+
(1+.06)
80
+
(1+.06)2
1080
(1+.06)3
Bond Value =
C1
er
+
C2
er2
+
C3
er3
Example
$1,000 bond pays 8% per year for 3 years. What is the price at a
YTM of 6%
1048.39
=
80
e.06
+
80
e.06x2
+
1080
e.06x3
YTM
Example
zero coupon 3 year bond with YTM = 6% and
par value = 1,000
Price = 1000 / (1 +.06)3 = 839.62
YTM
Example
zero coupon 3 year bond with YTM = 6% and
par value = 1,000
1000
Price  .063
e
 835.27
8.04
6.00
4.84
2
3
10
Maturity (years)
1
5
10
15
20
30
YTM
3.0%
3.5%
3.8%
4.1%
4.3%
4.5%
The “Pure Term Structure” or “Pure Yield Curve” are
comprised of zero-coupon bonds
These are often only found in the form of “US Treasury
Strips.”
http://online.wsj.com/mdc/public/page/2_3
020-tstrips.html?mod=topnav_2_3000
Rates
f3-1
year
0
1
2
Rn = spot rates
fn = forward rates
3
R2
R3
0
1
2
3
year
f2
f3
f3-2
example
1000
(1+R3)3
=
1000
(1+f1)(1+f2)(1+f3)
Forward Rate Computations
(1+ Rn)n = (1+R1)(1+f2)(1+f3)....(1+fn)
Example
What is the 3rd year forward rate?
2 year zero treasury YTM = 8.995%
3 year zero treasury YTM = 9.660%
Example
What is the 3rd year forward rate?
2 year zero treasury YTM = 8.995%
3 year zero treasury YTM = 9.660%
Answer
FV of principal @ YTM
2 yr 1000 x (1.08995)2 = 1187.99
3 yr 1000 x (1.09660)3 = 1318.70
IRR of ( FV= 1318.70 & PV= -1187.99) = 11%
example (using previous example )
f3 = 11%
Q: What is the 2 year forward price on a 1 yr bond?
A: 1 / (1+.11) = .9009
Example
Two years from now, you intend to begin a project that will last
for 5 years. What discount rate should be used when
evaluating the project?
2 year spot rate = 5%
7 year spot rate = 7.05%
Example (previous example)
2 yr spot = 5%
7 yr spot = 7.05%
5 yr forward rate at year 2 = 7.88%
Q: What is the price on a 2 year forward contract if the
underlying asset is a 5year zero bond?
A: 1 / (1 + 7.88)5 = .6843
coupons paying bonds to derive rates
Bond Value =
C1 +
(1+r)
C2
(1+r)2
Bond Value =
C1 +
(1+R1)
C2
(1+f1)(1+f2)
d1 =
1
(1+R1)
d2 =
1
(1+f1)(1+f2)
Example – How to create zero strips
8% 2 yr bond YTM = 9.43%
10% 2 yr bond YTM = 9.43%
What is the forward rate?
Step 1
value bonds 8% = 975
10%= 1010
Step 2
975 = 80d1 + 1080 d2 -------> solve for d1
1010 =100d1 + 1100d2 -------> insert d1 & solve for d2
example continued
Step 3 solve algebraic equations
d1 = [975-(1080)d2] / 80
insert d1 & solve = d2 = .8350
insert d2 and solve for d1 = d1 = .9150
Step 4
Insert d1 & d2 and Solve for f1 & f2.
.9150 = 1/(1+f1)
.8350 = 1 / (1.0929)(1+f2)
f1 = 9.29%
f2 = 9.58%
PROOF
Example
What is the 3rd year forward rate?
2 year zero treasury YTM = 8.995%
3 year zero treasury YTM = 9.660%
Example
What is the 3rd year forward rate?
2 year zero treasury YTM = 8.995%
3 year zero treasury YTM = 9.660%
Answer
FV of principal @ YTM
2 yr  1000  e.089952  1197.10
3 yr  1000  e.096603  1336.16
IRR of ( FV= 1336.16 & PV= -1197.10) = 11.62%
example (using previous example )
f3 = 11.62%
Q: What is the 2 year forward price on a 1 yr bond?
A:
Price 
1
e.11621
 .8903
Example
Two years from now, you intend to begin a project that will last
for 5 years. What discount rate should be used when
evaluating the project?
2 year spot rate = 5%
7 year spot rate = 7.05%
Example (previous example)
2 yr spot = 5%
7 yr spot = 7.05%
5 yr forward rate at year 2 = 8.19%
Q: What is the price on a 2 year forward contract if the
underlying asset is a 5year zero bond?
A:
Price 
1
e.08195
 .6640
coupons paying bonds to derive rates
C1 C2
Bond Value  r  r 2
e
e
C1
C2
Bond Value  f1  f1 f 2
e
e e
1
d1  f1
e
1
d 2  f1 f 2
e e
Example – How to create zero strips
8% 2 yr bond YTM = 9.43%
10% 2 yr bond YTM = 9.43%
What is the forward rate?
Step 1
value bonds 8% = 975
10%= 1010
Step 2
975 = 80d1 + 1080 d2 -------> solve for d1
1010 =100d1 + 1100d2 -------> insert d1 & solve for d2
example continued
Step 3 solve algebraic equations
d1 = [975-(1080)d2] / 80
insert d1 & solve = d2 = .8350
insert d2 and solve for d1 = d1 = .9150
Step 4
Insert d1 & d2 and Solve for f1 & f2.
1
.9150  f1
e
.8350 
f1 = 8.89%
1
e.0889e f 2
f2 = 9.15%
PROOF
Purchase of shares
April: Purchase 500 shares for $120
-$60,000
May: Receive dividend
+500
July: Sell 500 shares for $100 per share
+50,000
Net profit = -$9,500
Short Sale of shares
April: Borrow 500 shares and sell for $120
May: Pay dividend
July: Buy 500 shares for $100 per share
Replace borrowed shares to close short position
+60,000
-$500
-$50,000
.
Net profit = + 9,500
S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for maturity
T
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007


The price of a non interest bearing asset futures
contract.
The price is merely the future value of the spot
price of the asset.
F0  S0e
rT
Example
 IBM stock is selling for $68 per share. The zero
coupon interest rate is 4.5%. What is the likely price
of the 6 month futures contract?
F0  S0e
rT
.045.50
F0  68e
F0  $69.55
Example - continued
If the actual price of the IBM futures contract is selling for
$70, what is the arbitrage transactions?
NOW
 Borrow $68 at 4.5% for 6 months
 Buy one share of stock
 Short a futures contract at $70
Month 6
Sell stock for $70
Repay loan at $69.55
Profit
+70.00
-69.55
$0.45
Example - continued
If the actual price of the IBM futures contract is selling for
$65, what is the arbitrage transactions?
NOW
 Short 1 share at $68
 Invest $68 for 6 months at 4.5%
 Long a futures contract at $65
Month 6
Buy stock for $65
Receive 68 x e.5x.045
Profit
-65.00
69.55
$4.55



The price of a non interest bearing asset futures
contract.
The price is merely the future value of the spot
price of the asset, less dividends paid.
I = present value of dividends
F0  ( S0  I )e
rT
Example
 IBM stock is selling for $68 per share. The zero
coupon interest rate is 4.5%. It pays $.75 in
dividends in 3 and 6 months. What is the likely price
of the 6 month futures contract?
I
.75
e.045.25
I  $1.47

.75
e.045.50
F0  ( S 0  I )e
rT
F0  (68  1.47)e.045.50
F0  $68.04


If an asset provides a known % yield, instead of a
specific cash yield, the formula can be modified to
remove the yield.
q = the known continuous compounded yield
F0  S0e
( r  q )T
Example
 A stock index is selling for $500. The zero coupon
interest rate is 4.5% and the index is known to
produce a continuously compounded dividend yield
of 2.0%. What is the likely price of the 6 month
futures contract?
F0  S0e( r q )T
F0  500e(.045.02).50
F0  $506.29

The profit (or value) from a properly priced futures
contract can be calculated from the current spot
price and the original price as follows, where K is
the delivery price in the contract (this should have
been the original futures price.
Long Contract Value
( F 0 K )
Value 
rT
e
Short Contract Value
( K  F 0)
Value 
rT
e
Example
 IBM stock is selling for $71 per share. The zero
coupon interest rate is 4.5%. What is the likely value
of the 6 month futures contract, if it only has 3
months remaining? Recall the original futures price
was 69.55.
F0  S0e rT
F0  71e
.045.25
F0  $71.80
(71.80  69.55)
Value 
.045.25
e
 $2.22



Commodities require storage
Storage costs money. Storage can be charged as either a constant yield or a
set amount.
The futures price of a commodity can be modified to incorporate both, as in
a dividend yield.
Futures price given
constant yield storage
cost
F 0 S 0U e
U
t Storage Cost
e rT
rT
Futures price given set
price storage cost
F 0 S 0 e
( r  u )T
u =continuously compounded
cost of storage, listed as a
percentage of the asset price
Example
 The spot price of copper is $3.60 per pound. The 6 month cost to store
copper is $0.10 per pound. What is the price of a 6 month futures contract
on copper given a risk free interest rate of 3.5%?
U
.10
e.035.50
 .098
F 0 S 0U e
rT
 (3.60  .098)e
 $3.76
.035.50
Example
 The spot price of copper is $3.60 per pound. The annual cost to store copper
is quoted as a continuously compounded yield of 0.5%. What is the price of a
6 month futures contract on copper given a risk free interest rate of 3.5%?
F 0 S 0e
( r  u )T
(.035.005).50
 3.60e
 $3.67



Shortages in an asset may cause a lower than
expected futures price.
This lower price is the result of a reduction in
the interest rate in the futures equation.
The reduction is called the “convenience
yield” or y.
F 0 S 0e
( r u  y )T






The cost of carry, c, is the storage cost plus the
interest costs less the income earned
For an investment asset F0 = S0ecT
For a consumption asset F0  S0ecT
The convenience yield on the consumption asset, y,
is defined so that
F0 = S0 e(c–y )T
c can be thought of as the difference between the
borrowing rate and the income earned on the asset.
C=r-q
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
5.45