Chapter 5: “Interest Rate Futures: Introduction”
Interest Rate Contracts:
T-Bills Futures – contract reqs delivery of $1mil face value T-bill w 90
days to maturity (DTM). Contracts trade for MAR, JUN, SEP, DEC –
w delivery on 3 business days after last day of trading. P quotes use
IMM index (IMMI).
(5.1) IMMI = 100.0 – DY; DY = discount yield. If DY = 8.32%  IMMI
= 91.68%. P flucs are no smaller than 1 tick = 1 basis point = 1 BP 
 P = $25.00 (5.2) gives T bill P
(5.2) P = $1mil – (DY($1mil)(DTM))/360
DY = 8.32  P = $1mil – (.0832($1mil)(90))/360 = $979,200
- no limit exist on daily P flucs. Fig 5.1 shows how futs Ps are
quoted, 91+92 day deliveries are also poss. At delivery the SP
delivers T-bills while LP pays invoice amt (IA)
(5.3) IA = $1mil – (T-bill yield($1mil)(DTM))/360
Eurodollar Futures - $deposits in comm banks outside US. Rates are
LIBOR + delivery is via cash settlement (CS) Euro$ were 1st to use
CS + they are not transferable (quote on p.116 tells how LIBOR is
detd via polling of CBs)
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- yields are quoted on an add-on basis; add-on-yield (AOY)
(5.4) AOY = (discount/P)(360/DTM)
if DY = 8.32  discount = ($1mil-P) = ($1mil – 979,200) = 20,800
 AOY = (20,800/979,200)(360/90) = .0850
 AOY>DY but  BP = 1   value of Euro$ + T-bills contracts is
same = $25 (for 3 mo maturities)
 Quotes on Euro$ + T-bills both use IMMI but Euro$ yields are
AOY. Fig 5.3 shows settlement yields for Euro$. The rel bet  yield
on the two is strong as shown on p. 117. (discuss this regression eq)
Fig 5.3 shows rel (p.118) + as chps 20-22 show “...the Euro$ futs cont
plays an imp role ... to hedge int rate risk” (p.117).
Treasury bond futures – has diff delivery procedure than T-bills. They
begin in Aug 1977 + have been most successful contract. Its
vol>Euro$ vol but its open int is less  T bonds used more for spec.
while Euro$ for Hing! Fig 5.1 shows quotes. Min P fluc = 1/32 of 1%
point of value = $31.25 (1/32 x .01 x 100,000) Max P fluc = 3 points =
96/32 = $3000 (s in pers of high volatility)
- Delivery is on any day of delivery mo (MAR, JUN, SEP, DEC) SP
chooses day: fist position day – 1st permissible day for SP to
declare delivery date (2 days later); any other day is called position
day. This is followed by notice of intention day then delivery day –
SP delivers fin inst + receives PMT from LP.
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- Table 5-2 (p.121) shows large # of deliverable bonds ag any
specific futs con. The SP wants to deliver the cheapest so to
facilitate choice (to ensure more than 1 type is delivered)
conversion factors (CF) are used to simulate all bonds as having a
20 yr maturity @ 8% 
IA = DSP($100,000)(CF) + AI
(5.6)
DSP = decimal SP (eg. 96-16  0.965)
AI = accrued interest
 w flat term structure + yields @ 8% no bond has an advan for
delivery  in real world there is a bond which is cheapest-to-deliver
(chp. 6 explains this). Having such a large supply of avail bonds for
delivery helps offset chance for any one trader to corner the mkt. But
SP has options w regard to which bond + when to deliver – which
then lowers futs mkts Ps.
Treasury Note Futures – have 10,5 2 yr contracts w a range of
maturites deliverable ag ea contract. 5 + 10 yr contracts have
$100,000 denom but 2 yr has $200,000 denom CF are same as for T
bonds + delivery systems are also same. Fig 5.5 shows close rel bet
T-note and T-bond Ps. (also see regression eq)
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Municipal Bond Futures – based on Municipal Bond Index (MBI) of 40
tax exempt municipalities in US. The index uses CF to det a
coefficient which corrects for changes in bond membership
Pj
j 1 CFj
40
(5.8) MBI = coeff 1/40

Pj = P of bond j
CFj = CF for j
- futs P = 90-16  90.5% of par  value of futs contract = 1000 x
90.5 = $90,500. Tick size is 1/32 or $31.25 per contract. Daily
trading limit is $3000. CS closes all contacts.
Pricing Interest Rate Futures Contracts – the features which promote
full carry – from chp 3, ease of short selling, large supply of
underlying, nonseasonal prod + consp., ease of storage – are easily
met for int rate futures. Although, Kolb doesn’t recog that the trad
abundance of S of T-bills may no longer be true. Acc to Kolb for Tbills, bonds, + notes. “These instruments are avail in huge supply and
trade in a highly liq mkt” (p. 125) But now with the US govt running a
budget surplus   issue of Treasury securities  S  P  i 
S  liq  can ret on T-secs still be RF?
- to extent that mkt approx full carry  egs 3.3, 3.6 + 3.7 will apply
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COC in Perfect Markets – assumps: perfect mkts; COC is only fin
cost; ignore seller’s options; ignore diff bet futs + FWD Ps. Maturity
req of deliverable T-bills applies on the delivery date  when T-bill is
purchased prior to delivery purchaser must be sure that it has 90-92
days of maturity remaining on delivery date. The perf mkt assup 
borrowing + lending at RF T-bill yields!
- Tab 5.4 (p.127) shows a COCARB opp since the cash bill will have
90 DTM on Mar 22 when delivery is reqd on futs contract. Now the
perf mkt assump  borrow or lend at RF repd by T-bill DY. This is
then also the financing cost to acquire the 77 day bill (ea DY for ea
maturity reps RF for that maturity)
- Tab 5.5 shows trans nec to execute COCARB: to fin holding of 77
day bill trader must borrow at RF = 6% (‘issuing’ a 77 day bill at
6%). The P of this bill (=amt borrowed) = $953,611. But its face
value (PM) in 77 days is (using 5.2):
$953,611 = PM[1 – (DY x DTM)/360] = PM[1 – (.06x77)/360]  PM
= $966,008
- the proceeds of the 77 day bill are used to purchase the 167 day
bill deliverable ag the futures. As Tab 5.5 shows the ARB  =
$2742
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ARB  >0 since “rel to the ST rate the futs yield + the LT T-bill yield
were too high…” (p.127). Trader acquires funds @ 6% + re-Is at
10%
If 77 day bill had DY = 8%  ST rate too high rel to LT + futs rates 
RCOCARB in Tab 5.6 (p.128). “With this new set of rates the ARB is
more complicated…”
Trader will want to buy the MAR futs for $968,750 (payable in 77
days)  needs to borrow amount P0 which will accrue to 968,750
when ‘lending’ via purchasing a 77 day T-bill yielding 8%
 P0 = 968,750[1-((.08)(77))/360] = $952,174
use proceeds (968,750) of 77 day bill to buy futs  on JUN 20 =
$1mil. Repay loan of 952,174  must repay maturity value (PM) of
167 day bill 
P0 = 952,174 = PM[1-((.10)(167))/360] = PM(.9536111)
 PM = 998,493   = $1mil – PM = $1507. Here borrowing @ 10%
while lending @ 8%  must take entire set of rates into consid when
det ARB conds. Here the SS is the borrowing or debt issue (or T-bill
may be sold from inventory)
eq. 5.2  DY = [(PM-P0)/PM] x [360/DTM]
(5.2a)
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for the 167 day bill (5.2a)  DY = [($1mil – 953,611)/$1mil] x
[360/167] = 10%, now trader could hold this bill or hold the 77 day bill
following by holding the 90 day bill delivered on the futs. Either one of
these alts must yield the same the avoid ARB  it is as if the amount
$953,611 (=P0) were to grow to 968,750 (=PM) on day 77  what
then would be DY or 77 day bill to prevent ARB 
DY = [(PM – P0)/PM] x [360/DTM] = [(968,750 – 953,611)/968,750] x
[360/77]  DY = 7.3063%
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Cost of Financing and the Implied Repo Rate (IREPOR)
Assuming that fin cost is the only COC
 (1+C) = PF/P167 = 968,750/953,611 = 1.015875
 C = IREPOR = 1.5875%, which is COC for 77 days
now fin charge (CF) is given by (PM/P0)77 = 966,008/953,611 = 1+CF
 CF = 1.3%
In Table 5.4 CF = 1.3% and this led to COCARB w >0 
If C=IREPOR>fin cost = CF COCARB
If C<fin cost (CF)  RCOCARB, e.g. in above eg. of Table 5.6
(PM/P0)167 = 1+CF = 998,493/952,174 = 1.048645  CF = 4.865%
 CF = 4.865% >C=1.5875%  RCOCARB
 given the yields on futs and T-bills and their prices, the ARB (or
no-ARB) opps can be identified via comparison of C and CF
Futures Yield and FWD Rate of Interest (FWDr)
In this same example consider the following time line:
t=0
77
t=167
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- assuming no-ARB, we have seen that the int rate from t=0 t=77
= 7.3063% and that from t=0t=167 = 10.00%. Now, int rate from
t=77t=167 is the FWDr which should = futs yield
No-ARB  (PM/P0)167 = (PM/P0)77(PM/P0)90
 ret on holding 167 day bill must equal ret on holding 77 day bill
followed (times) ret on holding 90 day bill. Here, (PM/P0)90 = FWDr
 ($1mi/953,611)167=(968,750/953,611) x FWDr
 1.048646 = (1.015875)FWDr
 FWDr = 1.032259 = (PM/P0)90 = $1mil/968,750  3.2259% 
FWDr = futs yield
 DY on futs contract = [($1mil-968,750)/$1mil](360/90) = 12.50%
 futs rate must = 12.50% on DY basis to avoid ARB (assuming (*);
perfect mkts, fin cost is only COC; ignore sellers options + diff bet
FWD and futs Ps)
COC for T Bond Futures – adj must be made for AI on T bonds, e.g.
7.3063% fin rate (to buy 100,000 bond @ 8%) for 77 days  AI =
(77/182)(.04)(100,000) = $1692
 invoice amt = $101,692  this = PM via issuing 77 day T bill 
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P0 = 101,692[1- (.073063)(77)/360]= $100,103
Table 5.7 (p.131) shows results of COC trans  =0
unless P0<$100,103 (This is not clear in book since here it seems
CF=1.5% + C=1.69% at cost = 100 + treatment differs from problems
11+17.
COC in Imperfect Markets – relaxing perf mkt assump but
maintaining rest of * (also, CB>CL)
- what effect will  CB have on ARB trans? Since COCARB  buy
spot + sell futs  S + F +  fut yield. CB  lower S + higher F
 lower futs yield
Tab. 5.8 (p. 132) shows sit when CB = 7.5563%, w the same 10% 167
day bill 
P0 = PM[1-((.075563)(77)/360]
PM = $969,277  loan repayment is more  this lowers opp for
COCARB  higher F 969,275>968,750 and lower futs yield nec to
produce no-ARB conds
- for RCOCARB, selling spot + buying futs the higher CB  lower F
due to  D to buy futs + higher futs yield. Table 5.9 shows in this
sit how no-ARB produces this result
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- the conseq is that a no-ARB band arises for the futs P + yield: in
this case the yield is the range 12.29-12.97%. While the P is
967,525-969,275; tc also  no-ARB band. Rest on SS are
unimportant in int rate futs if
1)
supply of deliverable insts is large (?)
2)
existence of large inv of these insts  ease of SS
- Since bond Ps + int rates are neg reld  “… we would expect the
futs P to be less then the FWD P… However, most studies
indicate that this is not a serious problem in general…” (p.133).
Sellers rights are particularly imp in T bond futs where later
delivery  more value to seller due to  AI (sellers options can
account for 15% of F, see chp 6)
- If a trader accepts delivery on a JUN T-bond contract + carries it
FWD to SEP  pay invoice P, fin at JUN T-bill rate, rec AI + sell
SEP futs 
Eq. 5.9 should hold: F0,d + AI – F0,n(1+C) = 0
(5.9)
Figure 5.6 shows that this eq  0 (as it should in a perf mkt)
Speculating with Interest Rate Futures
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- “… spec  is a very slippery notion… A specr may earn accounting
profits that constitute a justifiable ret to K…” vs econ profit (…)
“…which would be a  in excess of ret for the use of K + bearing of
risk. Acc profits are consistent w mkt efficiency but econ profits are
not.” (p.134)
Outright Position – if you think int rates will () in fut take LP (SP).
Table 5.10 (p.135) shows a specr who thought int rates would  
sell futs + after r  offset   =$450
Spreads – an intra comm spread is a specl on the term structure of
int rates (TSr); an intercomm spread is a specl on TSr or on diff risk
bet insts (eg. T bills vs Euro)
- Tab 5.11 (p. 136) shows a pos TSr (upward sloping YC) whereby
“… the futs yield are consistent w. the term structure given by the
spot rates, in the sense that the fut yields = the FWD rate from the
term structure…” (p.135)
- if YC is steep  specr may think it will flatten  spread bet
successive contracts will narrow  buy distant FC + sell nearby
FC  if YC flattens  >0 indep of level of r ( or ). Tab 5.12 
=12BP x $25 = $300 (here r   T-bill Ps). This arises since
spread prior to trans = 13,50%-12.50% = 100BP but now =
11.86%-10.98% = 88BP  BP = 12 = 
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- Now same sit except assume r  SEP = 13%  T bill P = 87.00
+ DEC=13.8%  P = 86.20  Spread = 13.8%-13% = 80BP  a
 spread should  >0. Lets see assuming same Mar 20 futs
trans. For the DEC FC the  = 86.20-86.50 = -.30 While for the
SEP FC  = 87.50-87.00 = .50  net  = .50-.30 = .20  =20BP
 =20x$25 = $500  >0 whether r  or .  as long as spread
narrows  >0
T Bill/T Bond Spread – Table 5.13 shows a flat TSr but perhaps specr
thinks YC will slope upward  for intracomm spread on T bills  sell
distant + buy nearby. But greatest diff in yield would arise bet T-bills +
longer maturity insts like T bond. In this sit you would sell T bond futs
(+ simul buy T bill futs) and then, if you are right about d slope of YC
  T bond Ps  buy lower P futs + sell acc to higher P FC. Table
5.14 shows trans.
- Table 5.14  for intercomm spread sell longer maturity T-bonds +
buy T bills. In Oct r  yield on T bills  .20% while on bonds  =
.78%  loss on T Bill but greater gain on T Bond. Now ea 32nd of
a point on T bonds  $31.25
-  on T bonds = 4-05  4 5/32 = 133/32  133 x $31.25 =
$4,156.25
-  on T bills = -20BP  -20x25 = -500  net  = 4,156.25 – 500 =
$3,656.25. Here since  yields on bonds = 4x that on bills  insts
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have diff P sensitivities which also should be considered. This eg.
also shows that a spec pos foc using on  YC need not employ diff
futs maturities.
T Bill/Eurodollar Spreads – Kolb uses eg. of TW debt probs as  
default risk  banks exposed to such debt would have to  yield on
time deposit   Euro-$ yields (say relative to T-bills) “…if the full
riskiness of the banks pos has yet to be understood… whether int
rates were rising or falling …” (p.137)  sell Euro$ futs + buy T-bill
futs (a ‘TED’ spread)
- Table 5.15 (p.138) shows conseq of such a successful spec  in
Oct when Euro$ spread d  =27BP x $25 = $675. As in all specul
the trader thinks they know course of fut event better than mkt does
 to make >0 trader must expect spread to  more than mkt and
have correct exps  trader must ‘outguess’ mkt.
‘Notes Over Bonds’ (NOB) – pos for trading spreads bet T-notes + Tbonds: based on fact that T-bonds have a longer duration - %  P
guess a %  yield – than T-notes. If yield  by same on both insts 
gain on LP in bonds  loss on LP in notes. NOB also used for YC
specl:  of YC  sell bonds + buy notes
 of YC  buy bonds + sell notes
Hedging with Interest Rate Futures
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- Hr seeks to  risk to assure a future CF  gain (loss) in futs mkt
offsets loss (gain) in cash mkt.
Long Hedge – on Dec 15 a fin mgr learns that he will have $970,000
to I in T-bills on June 15 + wants to secure $1mil face value of bills @
12% (YC if flat but mgr thinks rates may fall). Using DY=12% 
P0=970,000=PM[1-((.12)(90)/360]  PM=$1mil
- as Table 5.16 (p. 139) shows the fin mgr buys the JUN 15 futs.
When June 15 comes DY = 10% 
P0=$1mil[1-((.10)(90)/360]  P0=$975,000
 to obtain $1mil worth of bills must pay $975,000  loss = $5000
(wo hedge). But since mgr hedged “…just before the FC matures, the
mgr sells one June T-bill contract…”  futs =$5000. Since
S=975,000  to obtain bill mgr must pay $5000 more than expd. –
but loss is offset by futs =$5000.
Short Hedge – in 4 mos a securities dealer wants to buy $1mil. FV of
T-bills (+ has agreed to buy from another dealer @ DY=14.37%). Acc
to YC spot yield = 13% + 4 mos DY = 14.37%  P.. in 4 mos =
964,075. Del is worried that if r s  “… the value of the bills will fall
below the expd P… + [he] will have to pay more…than they will be
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worth …” (p.140). Since he must honor his commitment to the other
dlr  sell T-bill futs to ‘lock-in’ DY=14.37%
- Table 5.17 (p.141) shows trans when rates do go up but in
accordance w mkt exps   on both pos = 0 since rates moved
exactly as mkt had expd. Fig 5.8 shows how basis moved but
since it was anticpd  hedge was effective  in using int rate futs
Hrs must only be concerned w UAd s in the basis!
- Effective H  FWDr is estimd from TSr @ t=0 FWDr “… is the rate
pertaining to the instrument being hedged…” eff H reqs diff bet
FWDr + futs yield (FY) to be constant (p.140-1)
Cross Hedge – fin VP, fearing r wants to issue $1bil worth of comm
paper (CP) which yields 2% above 90 day T-bill rate of 15%. The
quest is what to H the CP with?  if Hd inst + Hing inst differ in
1) risk
2) coupon
3) maturity
4) time span
then it must be a cross-hedge. Here VP uses T-bills (which are
correld w CP) to H issue of CP in 90 days  sell T-bill futs  Table
5.18 (p. 142) (CP priced on DY basis)
- FY = 16%  mkt expects 1%  in T-bill DY – but mgr has exp of
17% for CP  exp of DY = 15%  mgrs exps < mkt exps 
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apparent loss in cash mkt (‘opp loss’)  mgr thought he was
‘locking in’ curr spot rate but was actually getting the mkt expd
yield of 18% on the CP. Since mkt exps were RZd and the t=0 exp
of mgr should have been $955mil  there is no ‘opp loss’.
Now, even if mgr correctly understand how mkt works (since before
mgr had “misunderstood the nature of the futs mkt”)(p.144) bet time
of initiation of FC (t=0) + issue of CP, mkt risk premium (or premia)
may  as well as underlying econ conds (most likely sit). Table 5.19
shows sit when spread bet CP/T-bills s + inf exps .
- t=0  DY on FC = 16% but at t=1 RZd rate = 16-25%   inf exps
= 0.25%  (this was UAd by mgr). In cash mkt t=1 
CP=18.5%>18%  UAd  in RP (now = 2.25%)  net result is that
pos is not ‘perfectly Hd’ since gain on FC < less in cash mkt. There
could be losses of FC too if DY on T-bills < 16% at t=1!
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