Appendix 2
Implicit Cost of Carry in Inter-Index Arbitrage
    J. R. Varma
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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