Interest Rate Futures
Banikanta Mishra
Xavier Institute of Management
“Risk Free” Rates
Treasury Bill, Treasury Note, Treasury Bond
LIBOR: Rate at which large banks willing to lend
(keep deposits with other banks)
LIBID: Rate at which large banks willing to borrow
(accept deposits from other banks)
Professor Banikanta Mishra, XIMB
2
Repurchase Offer (Repo) Rate
t=0
t=T
Lend TB to Mr. X
Buy back TBs from Mr. X
Borrow money (B) from Mr. X
[B < PTB = Value of TB]
Pay Mr. X back B (1 + RR)
RR (Repo Rate) is the implied interest-rate
RR is quite close to risk-free rate,
as the borrowing is virtually risk-free
Professor Banikanta Mishra, XIMB
3
Short Selling
t=0
t=T
Borrow the asset from Ms. X
Return Asset to Ms. X
Sell asset in open market @ S0
Buy asset in open @ ST
Invest S0 in another asset
Get S0 (1+R)
=> Net Gain = S0 (1+R) – ST
So, short iff S0 (1+R) – ST > 0
Professor Banikanta Mishra, XIMB
4
Compounding
If APR = R and compounding is m times per year,
then EAR = (1 + R/m)m
If m tends to infinity (contious compounding),
then EAR = eR
If Rc with continuous compounding gives as much as
Rm with m compoundings per year, then
Rc = m ln (1 + Rm/m) OR
Rm = m (eRc/m – 1)
Professor Banikanta Mishra, XIMB
5
Discrete & Continuous Compounding
Suppose RN = 10% and N = 4
Then, if one invests $1 now,
she ends up at year-end with
1 X (1 + 10%/4)4 = 1.1038
To have 1 x eR = 1.1038, need
R = ln (1.1038) = 9.876%
Verify that,1 x e9.876% = 1.1038
Similarly, if R is given as 9.876%,
then to find out what RN with N=4
would give us same amount at end,
we get RN = 4 x (e9.876%/4 – 1) = 10%
Professor Banikanta Mishra, XIMB
6
Bond and Par Yield
Bond Yield: The discount-rate that
makes bond’s PV = P
Example: 6% coupon 2-yr bond with
semiannual interest and P = 98.39
=> YTM = 6.88% and Effective Yield = 6.99%
Par Yield: Coupon Rate that makes
PV = Par
Example: $100 par 2-yr bond
C/2 e-0.5x15.01% + C/2 e-14% + C/2 e-1.5x13.34%
+ (C/2 +100) e-2x13% = 100
=> C = 13.52
Professor Banikanta Mishra, XIMB
7
Zero Coupon Yield Curve
Annualized
Maturity
Coupon
Price
Rate
3m
0
96.08
6m
0
92.77
1y
0
86.94
18m
10%
94.95
2y
12%
97.43
16.00%
15.01%
14.00%
13.34%
13.00%
Professor Banikanta Mishra, XIMB
8
Forward Rates
Cash Inflow at t =
Invest
Mat (Y)
Zero Rate
1
2
3
4
5
14.00%
13.00%
11.75%
10.25%
8.50%
t=0
100
100
100
100
100
One-Year Forward Rate -->
Two-Year Forward Rate -->
Three-Year Forward Rate -->
If Borrow $100 for 2-yr and Lend for 4-yr
=> Implied Rate (of lending for 2y at t=2)
1
115.03
2
3
4
5
129.69
142.26
150.68
152.96
12.00%
9.25%
5.75%
10.6250%
7.50%
3.63%
9.0000%
5.5000%
-129.69
7.50%
Professor Banikanta Mishra, XIMB
1.50%
150.68
9
FRA and Arbitrage
Cash Inflow at t =
Invest
Mat (Y)
Zero Rate
1
2
3
4
5
14.00%
13.00%
11.75%
10.25%
8.50%
t=0
100
100
100
100
100
If actual 2-y Forward Rate at t=2 =
1
115.03
2
3
4
5
129.69
142.26
150.68
152.96
7.2500%
then arbitrage-strategy involves
Borrow
Lend
Borrow
-100
Total CFs
0
100
-129.69
150.68
129.69
-149.93
0
0.75
Professor Banikanta Mishra, XIMB
10
If Borrowing & Lending Rates Differ
Zero Rate
Mat (Y)
Deposit
Lending
1
2
3
4
5
14.00%
13.50%
12.75%
12.50%
12.00%
14.10%
Cash Inflow at t =
1
115.03
2
12.25%
Lending
1
2
3
4
5
14.00%
14.10%
13.65%
12.90%
12.70%
12.25%
12.50%
12.00%
One-Year Forward Rate(D) -->
Forward Rate(L) -->
Two-Year Forward Rate(D) -->
Forward Rate(L) -->
Three-Year Forward Rate(D) -->
Forward Rate(L) -->
t=0
182.21
100
100
100
100
100
164.87
12.70%
Deposit
5
146.59
12.90%
Mat (Y)
12.75%
4
131.00
13.65%
Zero Rate
13.50%
3
Investment
Cash Inflow at t =
1
115.14
2
3
Investment
4
5
t=0
184.50
100
100
100
100
100
131.39
147.26
166.20
12.90%
10.95%
11.30%
9.20%
13.30%
11.70%
12.55%
11.25%
12.0750%
11.35%
10.65%
12.3500%
11.90%
11.50%
11.9667%
10.9000%
12.2667%
11.4167%
Professor Banikanta Mishra, XIMB
11
Arbitrage when RB < RL*
One-Year Forward Rate(D) -->
Forward Rate(L) -->
Two-Year Forward Rate(D) -->
Forward Rate(L) -->
Three-Year Forward Rate(D) -->
Forward Rate(L) -->
12.90%
10.95%
11.30%
9.20%
13.30%
11.70%
12.55%
11.25%
12.0750%
11.35%
10.65%
12.3500%
11.90%
11.50%
11.9667%
10.9000%
12.2667%
11.4167%
If Borrow $100 for 2-yr and Lend for 4-yr
=> Implied Rate (of Depositing for 2y at t=2)
If actual 2-y Forward Borrowing Rate at t=2 =
-131.39
164.87
11.35%
11.30%
then arbitrage-strategy involves
Borrow
Deposit
Borrow
-100.00
Total CFs
0
100.00
-131.39
164.87
131.39
-164.71
0
0.16
Professor Banikanta Mishra, XIMB
12
Pricing of Forward (& Futures?)
Professor Banikanta Mishra, XIMB
13
Day Count Conventions
8% Interest Payment dates: 1 Mar & 1 Sep
How much interest earned (AI) by 3rd July?
Actual / Actual
Actual number of days elapsed is used for T-Bonds in US
124 days between 1 Mar & 3 Jul  AI = 4 x (124/184)
184 is the number of days from 1 Mar to 1 Sep
30 / 360
Each month taken as 30 days, one year 360 days
Used for corporate & municipal bonds in US
Total no of days from 1 Mar to 1 Sep = 30 x 6 = 180
Total no of days from 1 Mar to 3 Jul = (30 x 4) + 2 = 122
=> AI = 4 x (122 / 180)
Actual / 360
AI = 4 x (124/180) [Can exceed 4, since Actual can = 366]
Used for money-market instruments in US
Professor Banikanta Mishra, XIMB
14
Discount-Rate Price Quotation
360
P
(100  Y )
n
where Y is the cash price, but P the quoted “price”,
which is the interest-rate on face-value, not price
Example: Consider 91-day MM instrument quoted at 10
360
 10 
(100  Y ) 
91
 91 
Y  100  10 x 
  97.47222
 360 
=> Rs.2.52778 interest earned on Rs.97.4222 over 91 days
APR = 10.26% or 10.40% and EAR = 10.66% or 10.81%
depending on whether a year is taken as 360 or 365 days
Professor Banikanta Mishra, XIMB
15
Treasury-Bond Price Quotation
Cash Price = Quoted Price + Accrued Interest
Example: Consider a 10% s.a. coupon bond
Pays interest every 5th June and 5th December
What is the accrued interest (AI) on 20th October?
No of days from 5th June to 20th October = 137
AI = 100 x (10%/2) x (137/183) = 3.74317
5
If the Quoted Price = 92-05  92 = 92.15625
32
Then, the Cash Price = 92.15625 + 3.74317 = 95.90
=> If face value is 100/1000, then it would sell for 95.90/959.00
Professor Banikanta Mishra, XIMB
16
Treasury-Bond Futures
One contract involves delivery of
$100,000 face-value of T-Bond
Cash Received for $100 face-value =
(Most Recent Settlement Price x Conversion Factor) + AI
CF (conversion factor) is equal to the
value of the bond per $1 principal
on the first day of the delivery month
assuming interest rate is flat at 6.00%
Professor Banikanta Mishra, XIMB
17
Conversion Factor
Round bond maturity to nearest 3 month
If divisible by 6 also, next coupon after 6 months
Compute PV of bond @3% semi-annual rate
Divide by Face Value to get Conversion Factor
If not divisible by 6, next coupon after 3 months
Compute PV of bond @3% semi-annual rate
(the last discounting is for three months, not six)
 Subtract Accrued Interest for 3 months elapsed
On next slide, we determine Conversion Factor for
a 10% s.a. coupon bond with 20y 4m to maturity
and its cash price (most recent settlement is 98.50)
Professor Banikanta Mishra, XIMB
18
Conversion Factor Example
Years to Mat
20
1/3
Bond Coupon Rate %
10.00 paid s.a.
Months to Mat
Ex-Interest Bond Value
at end of 3m (100 Par)
244
146.23
No of 3 mths
Cum-Interest Bond Value
at the end of 3m
81.33
151.23
Rounddown
Bond Value Now at t=0
81
149.01
No of Months Taken
Bond Value - Accr Int
243
146.51
No of 6 Months in it
Conversion Factor
40
1.4651
Residual 3 Mths, if any
Cash Recd per 100 Par
1
146.81
Professor Banikanta Mishra, XIMB
19
Conversion Factor: Another Example
Years to Mat
18
Bond Coupon Rate %
1/6
8.00 paid s.a.
Months to Mat
Bond Value now at t=0
(for 100 Par)
218
121.83
No of 3 mths
Bond Value now at t=0
72.67
121.83
Rounddown
Bond Value Now at t=0
72
121.83
No of Months Taken
Bond Value - Accr Int
216
121.83
No of 6 Months in it
Conversion Factor
36
1.2183
Residual 3 Mths, if any
Cash Recd per 100 Par
0
120.00
Professor Banikanta Mishra, XIMB
20
Cheapest-to-Deliver Bond
Pay  Quoted Price + Accrued Interest
Receive  (Most Recent Settlement x CF) + AI
Gain = (MR Settlement x CF) - Quoted Price
Highest Gain (or Lowest Loss) => Cheapest
Most Recent Settlement Price = 98-16
Which Equals 98.50
Quoted Price
Conversion
Factor
Gain
146.11
121.15
132.01
1.4651
1.2183
1.3232
-1.7977
-1.1475
-1.6748
Chpeast to Deliver bond is the middle one
Professor Banikanta Mishra, XIMB
21
Cheapest to Deliver “Rules”
When bond-yields > benchmark rate,
CF favors low-coupon, long-maturity bonds
but, when bond yields < benchmark rate,
CF favors high-coupon, short-maturity bonds
When yield-curve is upward-sloping,
long-maturity bonds are CTD
but, when yield-curve is downward-sloping,
short-maturity bonds are CTD
Professor Banikanta Mishra, XIMB
22
Futures Price Determination: Input
Consider a Treasury-Bond futures contract
Delivery would take place in 270 days
CTD bond has 10% Coupon & 1.4651 CF
Coupons paid semiannually, as is typical
Last paid 60 days back, next due after 122;
the coupon after that after 183 days
Interest rate flat at 12% for all maturities
Quoted Price of the bond is 146.11
Professor Banikanta Mishra, XIMB
23
Determining Futures Price
Quoted Price
Coupon Rate
146.11
10.00%
Last Coupon
Paid
Next Coupon
Due
60 days back 122 days after
Accrued Interest
1.6484
RRR
PV of Next
Coupon
Days to
Maturity of
Futures
Cash Futures
Price (using F for
Known Income)
147.7584
12.00%
4.8034
270
156.2249
Days Left At
Next Coupon
AI at Fut
Maturity
Implied Fut Pr
at Maturity
(i.e. Net of AI)
Conversion
Factor
Quoted Fut Price
148
4.0437
152.1812
1.4651
103.8708
Current Cash
Price
Professor Banikanta Mishra, XIMB
24
Euro-Dollar Futures
Futures on 3-month Eurodollar interest-rate
on notional amount of $1 million
Contract Price =
10,000 x [100 – 0.25 x (100 – Q)]
where Q is the quote for settlement-price
1 bp increase in Q  $25 gain/loss for
a long/short position in one contract
Also implies that annualized interest-rate is
locked in at (100 – Q) through the futures
Professor Banikanta Mishra, XIMB
25
Eurodollar Futures Example
Q= Quote
Contract Value
Locked-in Interest Rate %
94.79
986,975
5.21
Contract Size
Days to Futures Mat
Maturity LIBOR
5,000,000
160
4.00%
Maturity Settlement
Maturity Contract Val
96.00
990,000
Futures Gain/Loss at
Maturity
Interest Earned (at t=3m) Total Cash Flow at t=3m
15,125
Actual Int Rate Earned
50,000
65,276
5.22%
Number of Contracts
D in Each Contr Val
Total Change
5
3,025
15,125
D interest (bp)
Gain/Loss Per Contract
Total Implied Gain/Loss
121
3025
15,125
Professor Banikanta Mishra, XIMB
26
Convexity Adjustment
1 2
F  f   T1 T2
2
where T1 is time to maturity of futures and
T2 is TTM of asset underlying the futures
 is standard-deviation in short-term interest-rate
Example
Let  = 1.3%
If 10-year Eurodollar futures is quoted at 90.00,
(=> continuously compounded rate 10.0142%)
Conv-Adj = ½ x 0.0132 x 10 x 10.25 = 0.8661%
=> Forward-Rate = 10.0142% - 0.8661% = 9.1481%
Professor Banikanta Mishra, XIMB
27
Euro$ Futures Extends LIBOR-Zero Curve
Let ith Eurodollar futures matures at Ti
If Ri is the Zero Rate for maturity Ti and
Fi the forward-rate implied by ith futures
(deemed to be the rate from Ti to Ti+1),
then Ri+1 is given by following equation
Fi (Ti  1  Ti )  R i Ti
R i 1 
Ti  1
Euro$ Futures Rate
Beginning Day
Days Out
LIBOR Zero
Days
5.30%
400
90
5.50%
491
90
5.60%
589
90
4.80%
400
4.8927%
4.9937%
Professor Banikanta Mishra, XIMB
28
Duration and Convexity
LIABILITY
Loan of 20337 at a coupon-rate of 8.8%
(equivalent to 8.44% APR, Continuously Compounded)
-> 100.0% of Total Liabilities
ASSETS
a. Lease payments to be received: PV=19239.43
-> 96.1% (19239.43/20022.37) of Total Assets
b. Cash and Securities: PV = 83.26
-> 0.4% (83.26 / 20022.37) of Total Assets
c. CF from leasing of new machine (cost = 699.68)
@PV of lease-payments = 699.68 -> 3.5% Total Assets
Professor Banikanta Mishra, XIMB
29
Duration & Convexity of Liability
t->
1
2
3
4
5
Sum
CF
1790
1790
22127
0
0
-
PV=C
CF x e-rt
1636
1495
16891
20022.37
PV/PV
Sum
0.08
0.07
0.84
1.00
xt
0.08
0.15
2.53
2.76
x t2
0.08
0.30
7.59
7.97
Professor Banikanta Mishra, XIMB
30
Duration & Convexity of Asset
t->
1
2
3
4
5
Sum
CF
5000
5000
5000
5000
5000
-
PV=
CF x e-rt
4570
4176
3817
3488
3188
19239.43
PV/PV
Sum
0.24
0.22
0.20
0.18
0.17
1.00
xt
0.24
0.43
0.60
0.73
0.83
2.82
x t2
0.24
0.87
1.79
2.90
4.14
9.94
Professor Banikanta Mishra, XIMB
31
One and Three Year Leases
t->
1
2
3
4
Cf
765.57
PV=
CF x e-rt
699.68
699.68
PV/PV
Sum
1.00
1.00
xt
1.00
1.00
x t2
1.00
1.00
t->
1
2
3
CF
278.47
278.47
278.47
-
PV=
CF x e-rt
255
233
213
699.68
PV/PV
Sum
0.36
0.33
0.30
1.00
xt
0.36
0.66
0.91
1.94
x t2
0.36
1.33
2.73
4.43
4
Professor Banikanta Mishra, XIMB
5
5
Sum
Sum
32
Interest-Rate Sensitivity of A&L
Duration = 2.76
Convexity = 7.97
Duration = 2.75
weighted
averages
Duration = 2.76
weighted
averages
Duration = 2.78
weighted
averages
Convexity = 9.58 Convexity = 9.63 Convexity = 9.70
Value of Assets
Value of Assets
Value of Assets
With 1-year lease With 2-year lease With 3-year lease
(% change)
(% change)
(% change)
Interest Rate
Value of Liability
(% change)
8.0%
20583.43
(2.80)
20581.76
(2.79)
20585.15
(2.81)
20588.45
(2.83)
8.9%
20077.75
(0.276)
20077.43
(0.275)
20077.77
(0.277)
20078.09
(0.278)
9.0%
20022.37 20022.37 20022.37 20022.37
9.1%
19967.15
(-0.276)
19967.49
(-0.274)
19967.16
(-0.276)
19966.84
(-0.277)
10.0%
19477.27
(-2.72)
19482.16
(-2.70)
19478.87
(-2.71)
19475.70
(-2.73)
Professor Banikanta Mishra, XIMB
33