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Implicit Cost of Carry in Inter-Index Arbitrage
It is well known that since the BSE and NSE operate different settlement
cycles it is possible to do a form of carry forward (or badla) trading
by continuously shifting positions from one exchange to the other to avoid
delivery. A person who has bought on BSE can square his position on that
exchange on or before Friday and simultaneously buy on NSE. Since he has
squared up on BSE, he does not have to take delivery there. On or before
Tuesday, he can square up on NSE and buy on BSE avoiding delivery at NSE.
He can keep repeating this cycle as long as he likes. Since this is very
similar to carry forward trading (or rolling a futures contract), it is
clear that this person would implicitly pay a carry forward charge (contango
or backwardation) in the form of a price difference between the two exchanges.
To model this, this study assumes that a trade in the BSE could be regarded
as a futures contract for Friday expiry while a trade on the NSE could
be regarded as a futures contract for Tuesday expiry. The cost of carry
model of futures prices tells us that the futures price equals the cash
price plus the cost of carry till the expiry date. Two futures contract
with different expiry dates will be priced to yield a price difference
equal to the cost of carry for the difference between the two expiry dates.
The table below summarises the impact of the differing settlement cycles.
(Throughout this study, day means trading day and yesterday means last
trading day).
| Day of week |
Yesterday
Days to expiry
|
Today
Days to expiry
|
Change in differential
|
|
BSE
|
NSE
|
Difference
|
BSE
|
NSE
|
Difference
|
days to expiry
|
| Monday |
0
|
2
|
-2
|
4
|
1
|
3
|
5
|
| Tuesday |
4
|
1
|
3
|
3
|
0
|
3
|
0
|
| Wednesday |
3
|
0
|
3
|
2
|
4
|
-2
|
-5
|
| Thursday |
2
|
4
|
-2
|
1
|
3
|
-2
|
0
|
| Friday |
1
|
3
|
-2
|
0
|
2
|
-2
|
0
|
The last column of this table is crucial. It tells us that the relation
between BSE and NSE undergoes a change on Monday and Wednesday.
-
From Friday close to Monday close the BSE contract changes from an expiry
2 days ahead of NSE to an expiry 3 days after NSE - a net positive change
of 5 trading days or one week. From being priced two days’ carry below
NSE, the BSE contract will now be priced three days’ carry above the NSE
price causing a net change of 5 trading days’ or one week’s cost of carry
in the difference between the two prices. Therefore Monday's return on
BSE should exceed that in NSE by one week’s cost of carry.
-
Similarly from Tuesday close to Wednesday close the BSE contract changes
from an expiry 3 days after NSE to an expiry 2 days ahead of NSE - a net
negative change of 5 trading days or one week. This is the reverse of the
above situation and therefore Wednesday's return on BSE should be lower
than that in NSE by one week’s cost of carry.
To estimate the cost of carry, the Nifty index was used. The Nifty Index
based on Last Traded Prices (LTP) at the NSE was obtained from the NSE
and the returns on this index were computed. The returns on the Nifty Index
was computed separately using BSE prices for the period from January 1,
1998 to June 30, 1998.
It turns out that on average on Mondays, the return in BSE exceeds that
in NSE by 0.61% while on Wednesdays, it is the other way around - the return
in NSE exceeds that in BSE by 0.71%. This implies that one week’s cost
of carry is approximately 0.6-0.7% or that the annual cost of carry is
about 30-35% on a simple interest basis or 35-45% on a compound interest
basis. These rates are far above any money market rate and indicates very
strong barriers to the flow of money into financing stock market transactions.
A closer look at Table 1 suggests a way of measuring the volatility
of the cost of carry as well:
-
Both on Monday close and on Tuesday close the BSE contract is for expiry
3 days after NSE. The difference in the returns between the two exchanges
is
therefore only due to the change in the cost of carry during Tuesday. Standard
deviation of the differential return is therefore the standard deviation
of daily change in 3 days' cost of carry.
-
Similarly the standard deviation of the differential return on Thursday
and Fridays is equal to the standard deviation of daily change in 2 days'
cost of carry of carry.
The critical assumption in the above is that the differences in prices
between the BSE and NSE is due only to the difference in the two expiry
dates and that various other differences in market microstructure in the
two exchanges do not have any impact. In reality perhaps a lot of the fluctuation
in the price differences is attributable to these microstructure differences.
Nevertheless, the empirical results based on the above analysis are
instructive:
| Day of week |
Standard Deviation of Differential Return
|
Standard Deviation of Daily Change in One Day's Carry
|
Standard Deviation of Daily Change in Annual Carry
|
| Tuesday |
0.33%
|
0.11%
|
28.50%
|
| Thursday |
0.21%
|
0.11%
|
27.31%
|
| Friday |
0.20%
|
0.10%
|
25.51%
|
The results indicate an incredibly high volatility in the cost of carry
- daily standard deviation of over 25%. To put these numbers in perspective,
the estimated standard deviations of daily changes in some important money
market rates are as follows:
| Interest Rate |
Estimated Standard Deviation Of Daily Change (Percent)
Jul 1994 - Dec 1997
|
Estimated Standard Deviation Of Daily Change (Percent)
Jan 1997 - May 1998
|
| Call Rate (RBI Weekly Average) |
2.08
|
3.48
|
| T-bill - 14 days (Primary) |
|
0.29
|
| T-bill - 91 days (Primary) |
0.16
|
0.11
|
| Commercial Paper (Primary) |
0.27
|
0.65
|
| Certificate of Deposit (Primary) |
0.32
|
0.55
|
| Forward Premium - 1 Month |
|
1.66
|
| Forward Premium - 3 Month |
0.77
|
0.79
|
| Forward Premium - 6 Month |
0.69
|
0.51
|
| Forward Premium - 12 Month |
|
0.41
|
| Note: Estimated Daily Standard
Deviations are derived by rescaling weekly changes in these interest rates
by dividing by square root of 5 (one week equals five trading days). |
The volatility of the cost of carry has profound implications for margining
calendar spreads. For example, the margin on a 90 day calendar spread in
a futures market would be obtained by applying a three sigma change in
the cost of carry (the 90 day interest rate) to the notional principal
involved (say the mark to market value of the far side of the spread).
If the standard deviation of daily changes in the 90 day interest rate
is about 1%, then a three sigma event would be a change of 3% in the cost
of carry which for a 90 day spread would imply 0.75% of the notional principal
involved in the spread. In other words, the margin on 90 day calendar spreads
(as a percentage of the mark to market value of the far side of the spread)
should be 0.75 times the standard deviation of daily changes in the 90
day interest rate.
The crucial question is that of estimating the volatility of a 90 day
cost of carry. The data given above shows that the estimated volatility
of the implicit cost of carry is about 7 times that of the overnight call
rate and about 15 times that of the 1 month forward premium rate during
1997-98. (The money market volatilities in 1997-98 are themselves much
higher than in the 1994-97 period partly because of the increasing reliance
on interest rates to defend the currency). Probably a large part of the
estimated volatility of the implicit cost of carry reflects the effect
of various differences in market microstructure between the BSE and NSE
rather than a fluctuation in the cost of funds itself. Still we must assume
that the cost of carry itself would be several times more volatile than
the 90 day rate in the organised money market. A margin of 1.5% on a 90
day calendar spread implicitly assumes a standard deviation of 2% in the
90 day interest rate which is about 3 times the standard deviation of the
commercial paper rates and over 2 times the standard deviation of the 90
day forward premium (which itself is affected by many things other than
the interest rate itself).
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