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Spot asset carry cost rates and futures hedge ratios

Abstract

Since the 1970s, futures hedge ratios have traditionally been calculated ex-post via economically structure-less statistical analyses. This paper proposes an ex-ante, more efficient, computationally simpler, general “carry cost rate” hedge ratio. The proposed hedge ratio is biased, but its bias is readily mitigatable via a stationary Bias Adjustment Multiplier (BAM). The 2-part intuition for the BAM and its stationarity is as follows. First, the paper reasons that the “traditional” hedge ratio should uncover the carry cost rate and shows that it does, albeit inefficiently. Then, since both the “traditional” and “carry cost rate” hedge ratios are driven by the carry cost rate, it may be that their ratio (for implementation in the same prior periods) is stationary and useful as an ex-ante BAM for the “carry cost rate” hedge ratio; the paper tests these conjectures and finds support for both. Specifically, the paper shows that the “bias-adjusted carry cost rate” hedge ratio, defined as the average product of the ex-post BAMs from prior periods and the current ex-ante “carry cost rate” hedge ratio, has higher hedge-effectiveness than that for either the “traditional” or “naive” benchmark hedge ratios in diverse real and simulated markets.

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Data availability

The data employed in this paper is available subject to third party (i.e. Bloomberg Inc.) restrictions.

Notes

  1. For these BAMs, the distinction between the ratio of averages and the average of ratios is inconsequential.

  2. For example, suppose one wanted to implement a 2-week hedge on Aug 15 using the Sep futures and that on Aug 29, the hedge’s lift date, the Sep futures will have 3 weeks-to-maturity. To strictly control for the futures’ time-to-maturity effect on the hedge ratio, the hT approach should go back and gather data on say, for example, the prior 16 quarterly maturing futures and look at the futures price changes between their 5 and 3 week times-to-maturity. Thus, the hT approach uses data going back 4 years prior to the hedge’s implementation, whereas the hc approach data goes back only 2 weeks. Very crudely, this makes the hT approach 4 years/2 weeks ≈ 4*26 = 104 times as susceptible to regime change.

  3. The goal was to get a long (at least back to 12/7/1990) series for the short term (ideally overnight for the daily data and weekly for the weekly data) US nearly riskless interest rate. Bloomberg’s US 1-week Repo rate data begins on 07/23/98 and has several gaps. Their overnight repo rate data begins at the same time as the 1-week repo rate data but is missing for about 100 more dates. Surprisingly, the 1-week rate average was about 2.5 basis points less than the overnight rate, but still this seems a minor difference. From these 2 series, we created a merged 1-week/1-day repo series: it is the 1-week repo rate unless it is unavailable, then it is the overnight repo rate − 2.5 basis points. Bloomberg also has data on two 1-month interest rate series (repo rate and Libor) that go back farther than the above discussed (preferred, but unavailable) shorter (1-week/1-day) interest rates. The 1-month repo rate was about 20 basis points lower than the Libor on average. From these two 1-month rates, we created a 1 month merged rate series—it was the repo rate when available and Libor-20 basis points, otherwise. Finally, we created an overall series from the two merged series we just created (i.e. the 1-week/1-night series and the 1-month series). Since the merged 1 month series averaged 0.5 basis points less than the merged 1 week/1night series, it is the merged 1 week/day repo series unless it is unavailable, in which case it is the 1 month series + 0.5 basis points. This final merged series is the short- term nearly riskless US interest rate used in CCR calculations for the various assets.

  4. Gold’s financing cost rate is essentially its c since its other c components: storage cost rate (a positive component), convenience yield (a negative component), and their net value are all ≈ 0. Gold’s storage cost rate per $ is ≈ 0, since gold doesn’t spoil and its volume per $ is tiny. Gold’s convenience yield is ≈ 0, since its “production–consumption” rate is small (and ≈ constant) relative to the amount of gold outstanding.

  5. The S&P500 futures switch to the e-mini futures on 12/15/97 as the liquidity switched to the e-mini contract.

  6. This avoids the adverse impact on hT’s hedging performance of differential futures time-to-maturity distributions between its estimation and implementation.

  7. To equally-weight the assets’ results in the aggregation, the number of observations in the aggregation contributed by each asset is restricted to the number of observations for the asset with the fewest pre-restriction observations. For the assets whose number of aggregation observations are to be restricted, the most recent observations are those that are contributed. For example, gold has the fewest 1- and 2-week profit period observations at 87 and 43, respectively. Consequently, only the most recent 87 and 43 observations for each asset enter into the aggregations for the 1- and 2-week profit periods, respectively. Thus, the number of observations is 3*87 = 261 and 3*43 = 129, in the 1- and 2-week profit period aggregations, respectively. This approach for aggregating asset results is used in the Stjfjable 5, 6, 7 equal-weight aggregations as well.

  8. For example, consider the S&P500’s weekly profits. Recall that its quarterly maturing futures begin with the Mar ‘91 futures and end with the Jun’20 futures. This represents 118 futures contracts. Each approaches’ first out-of-sample HE is based on the 13 weekly unhedged and hedged profits where the Jun’91 futures is the hedging instrument; across approaches the unhedged profits are the same, only the hedged profits differ They differ since the different hedge approaches employ different hedge ratios. The process rolls forward in time, so each approaches’ second HE is based on the 13 weekly unhedged and hedge profits where the Sep’91 futures is the hedging instrument. Thus, for each hedge ratio approach, 117 = 118–1 futures HEs are calculated and paired with their futures. For the 117 futures, 117 paired HE differences are calculated and used to calculate the paired HE difference: means, standard deviations, and T-values shown in the Table.

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Acknowledgements

The authors thank Wei Zhang, Zhiyan Jiang, Xun Liu, Yueying Tang, Jingwen Yuan, Jiajian Zhu, and Xilun Zhu for research assistance and Steve Figlewski for his helpful suggestions. Any errors are the authors’ responsibilities.

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Correspondence to Dean Leistikow.

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Appendix

Appendix

The tests that were performed on the real data discussed in the body of the paper will also be performed on simulated data generated from known stochastic processes. The goal of this analysis is to determine if the real data test results also hold in robust simulated markets where the futures price is determined using the carry cost model with mean-reversion and basis risk.

Specifically, using Monte Carlo simulation, we simulate data for 1,000 runs of 100 futures contracts per run. The futures contract times-to-maturities distribution is the same for each contract. At the start (end) of a futures contract’s use as the hedging instrument it has 14 (1) weeks until its maturity. Each profit period is a week. Thus, each run is 100 futures contracts*13 weeks/futures contract = 1,300 weeks long. The carry cost rate (c) and spot price (S) at the end of one futures contract’s use as the hedging instrument (when the futures has 1 week until its maturity) are also the c and S at the start of the next maturing futures contract’s use as the hedging instrument (when it has 14 weeks to its maturity).

To avoid potential distortions when the c values are very close to 0, for each run, the time t annual c incremented by 1, rather than c, is simulated. It is denoted cit (c incremented by 1) and it is simulated as:

$$ {\text{ci}}_{{\text{t}}} = ci_{t - 1} * \exp \left( { - \Theta * \left( {{\text{ci}}_{{{\text{t}} - {1}}} - \mu {\text{i}}} \right) * \Delta t + \sigma_{{\text{c}}} *\Delta {\text{t}}^{{0.{5}}} * \epsilon_{{\text{c}}} } \right) $$
(6)

ϴ is the mean-reversion speed for c, which is the same as it is for ci,

μi is 1 + the long-run mean c (μ), i.e. the long-run mean ci,

σc is the annual volatility of c, which is the same as the volatility for ci,

∆t is 1/52, and

εc is a Wiener process realization component of c.

The time t theoretical spot price, Stheoreticalt , is given as:

$$ {\text{S}}^{{{\text{theoretical}}}}_{{\text{t}}} = {\text{S}}_{{{\text{t}} - {1}}} *{\text{exp}}\left( {\left( {{\text{c}}_{{{\text{t}} - {1}}} - \sigma_{{\text{s}}}^{{2}} /{2}} \right)*\Delta {\text{t}} + \sigma_{{\text{s}}} *\Delta {\text{t}}^{{0.{5}}} *\left( {\rho\epsilon_{{\text{c}}} + \left( {{1} - \rho^{{2}} } \right)^{{0.{5}}} * \epsilon_{{\text{s}}} } \right)} \right) $$
(7)

St-1 is the time t-1 actual spot price as defined in equation 11) below,

ct = cit -1

σs is the annual spot return volatility,

ρ is the correlation between ct and St, and

 εs is a Wiener process realization idiosyncratic spot return component.

Within a run, after every 13 weeks, the futures’ hedging instrument switches/rolls to a new futures contract that initially has 14 weeks-to-maturity. On the switch/roll date, 2 futures prices are determined, that for the old futures contract that has 1 week to its maturity and that for the new futures contract that has 14 weeks to its maturity. The timestep t theoretical futures’ price is:

$$ {\text{F}}^{{{\text{theoretical}}}}_{{{\text{t}} }} = {\text{S}}_{{\text{t}}} *\left( {{1} + {\text{c}}_{{\text{t}}} *\left( {{\text{WTM}}_{{\text{t}}} *\Delta {\text{t}}} \right)} \right) $$
(8)

where: WTMt is the time t futures hedging instrument’s Weeks-To-Maturity; it runs from 14 to 1.

$$ {\text{The carry cost from time t}} - {\text{1 to time t is S}}_{{{\text{t}} - {1}}} *{\text{c}}_{{{\text{t}} - {1}}} *\Delta {\text{t}} $$
(9)

There is basis risk; the spot and futures price processes are noisily linked by c. The spot and futures price time t errors (deviations from their above theoretical prices) are based on the prior time’s actual prices and denoted with the superscript “error”. They are:

$$ {\text{S}}^{{{\text{error}}}}_{{{\text{t}} }} = {\text{ S}}_{{{\text{t}} - {1}}} * \, \left( {{\text{exp}}\left( {\left( {\sigma_{{{\text{err}}\_{\text{s}}}} *\Delta {\text{t}}^{{0.{5}}} } \right)* \epsilon_{{{\text{err}}\_{\text{s}}}} } \right) - {1}} \right) $$
(10)
$$ {\text{F}}^{{{\text{error}}}}_{{\text{t}}} = {\text{ F}}_{{{\text{t}} - {1}}} *\left( {{\text{exp}}\left( {\left( {\sigma_{{{\text{err}}\_{\text{f}}}} *\Delta {\text{t}}^{{0.{5}}} } \right)* \epsilon_{{{\text{err}}\_{\text{f}}}} } \right) - {1}} \right) $$
(11)

 εerr_s and  εerr_f are idiosyncratic Wiener process realization components of the spot and futures price changes, and their volatilities are σerr_s and σerr_f, respectively.

The time t actual spot and futures prices are:

$$ {\text{S}}_{{\text{t}}} {\text{ = S}}^{{{\text{theoretical}}}}_{{\text{t}}} {\text{ + S}}^{{{\text{error}}}}_{{\text{t}}} $$
(12)
$$ {\text{F}}_{{\text{t}}} {\text{ = F}}^{{{\text{theoretical}}}}_{{\text{t}}} {\text{ + F}}^{{{\text{error}}}}_{{\text{t}}} $$
(13)

Two baseline parameterization simulations are run: one where c’s long-run mean (μ) is positive and another where it is negative. For each baseline parameterization simulation, we also perform simulations incorporating ceteris paribus increases and decreases in each parameter. The tests performed on the real data (that were discussed in the body of the paper) are performed on the simulated data with the goal of assessing the test results’ robustness to data generated via alternative parameter specifications.

The baseline (positive long-run c mean) simulation’s parameterization is: c0 = 0.02, S0 = 50, σc = σs = 0.2, ϴ = 0.5, μ = 0.02, ρ = 0, and σerr_s = σerr_f = 0.02. For this parameterization’s simulated data, hT’s mean is 0.9692, its mean standard deviation is 0.0133, and its mean HE is 0.9611. These values are very similar to those from the real data. For each of the 1,000 runs, the average of the variable to be tested is determined; it is their: mean, standard deviation, count, T value, and P value across these 1000 runs that is reported in Tables. However, in the stationarity test, the β’s and γ’s significance levels are based on their values for each run averaged across the 1000 runs.

The test results based on the positive long-run c mean baseline simulated data (presented in Appendix Tables

Table 8 Test results based on data generated via the simulation’s baseline (positive long-run c mean) parameters

8 and 16A) are consistent with the paper’s real data test results. First, on average, hc is higher (by ≈ 0.0288) and less volatile (by ≈ 62%) than hT. Second, the negative correlation between the paired average c and hT is recovered, albeit inefficiently; while the correlation between the paired average c and hT had a mean and standard deviation of -0.4473 and 0.1335, respectively, the corresponding values for hc were about -0.954 and essentially 0, respectively. Third, the BAM is stationary. Fourth, hc and hc-BA reduce risk better than do hT and h1. With the exception of the time coefficient estimate in the BAM stationarity test which was expected to be and was found to be statistically insignificant, all the results are statistically significant at the 1% confidence level.

These positive long-run c mean baseline results also hold for data generated by ceteris paribus changes in the c0, S0, σc, σs, ϴ, μ, ρ, and σerr_s = σerr_f simulation parameter values (presented in Appendix Tables

Table 9 Test results based on data generated via from changing the baseline c0 input (0.02) by 0.01, ceteris paribus

9,

Table 10 Test results based on data generated from changing the baseline S0 input (50) to 100 and 10, respectively, ceteris paribus

10,

Table 11 Test results based on data generated from changing the baseline σc input (0.2) by 0.05, ceteris paribus

11,

Table 12 Test results based on data generated from changing the baseline σs input (0.2) by 0.05, ceteris paribus

12,

Table 13 Test results based on data generated from changing the baseline ϴ (c mean-reversion speed) input (0.5) by 0.15, ceteris paribus

13,

Table 14 Test results based on data generated from changing the baseline ρ, correlation(ct, St), input by 0.1, ceteris paribus

14,

Table 15 Test results based on data generated from changing the baseline σerr_s and σerr_f inputs (0.02) by 0.01, ceteris paribus
Table 16 BAM Stationarity test results based on data generated for the positive long-run c mean baseline parameters and for ceteris paribus changes in the baseline parameters

15, 16). Tables 9 and 16, Panels A and B for each, show the results when c0 is increased and decreased by 0.01, respectively, from its baseline value of 0.02. Tables 3 and 9, Panels A and B for each, show the results when S0 is doubled and halved, respectively, from its baseline value of 50. Tables 4 and 9, Panels A and B for each, show the results when σc is increased and decreased by 0.05, respectively, from its baseline value of 0.2. Tables 5 and 9, Panels A and B for each, show the results when σs is increased and decreased by 0.05, respectively, from its baseline value of 0.2. Tables 6 and 9, Panels A and B for each, show the results when ϴ is increased and decreased by 0.15, respectively, from its baseline value of 0.5. Tables 7 and 9, Panels A and B for each, show the results when ρ is increased and decreased by 0.1, respectively, from its baseline value of 0. Tables 8 and 9, Panels A and B for each, show the results when σerr_s = σerr_f is increased and decreased by 0.01, respectively, from its baseline value of 0.02.

The negative long-run c mean baseline parameterization simulation and comparative static simulations are the same as the positive long-run c mean’s baseline, except for them: c0 = µ = − 0.02. For the negative long-run c mean baseline parameterization simulation, the mean hT is 0.9746; as expected, this is a little higher than it was for the above positive long-run c mean simulation (since hc is also higher than it was for the above positive long-run c mean simulation). The hT’s mean standard deviation = 0.0127 and its mean HE is 0.9625. These values nearly match their values for the data generated by the positive long-run c mean simulation.

The test results based on the negative long-run c mean baseline and comparative static simulation data (Appendix Tables

Table 17 Test results based on data generated from the simulation’s baseline negative long-run c mean parameters

17,

Table 18 A and B below show the test results based on data generated from changing the baseline c0 input (− 0.02) by 0.01, ceteris paribus

18,

Table 19 A and B below show test results based on data generated from changing the baseline S0 input (50) to 100 and 10, respectively, ceteris paribus

19,

Table 20 A and B below show test results based on data generated from changing the baseline σc input (0.2) by 0.05, ceteris paribus

20,

Table 21 A and B below show test results based on data generated from changing the baseline σs input (0.2) by 0.05, ceteris paribus

21,

Table 22 A and B below show test results based on data generated from changing the baseline ϴ input (0.5) by 0.15, ceteris paribus

22,

Table 23 A and B below show test results based on data generated from changing the baseline ρ, correlation (ct, St), input (0) by 0.1, ceteris paribus

23,

Table 24 A and B below show test results based on data generated from changing the baseline σerr_s and σerr_f inputs (0.02) by 0.01, ceteris paribus

24,

Table 25 A and B below show the BAM Stationarity test results based on data generated from the negative long-run c mean baseline parameters and for ceteris paribus changes in the parameters

25) are similar to those based on the positive long-run c mean simulations data. They are available upon request from the authors.

In summary, the real data test results also hold when applied to simulated data generated from a wide range of alternative simulation parameter specifications.

Appendix Tables 9, 10, 11, 12, 13, 14, 15, 16, show test results based on data generated via ceteris paribus changes to the positive long-run c mean baseline set of parameter inputs.

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Leistikow, D., Chen, RR. & Xu, Y. Spot asset carry cost rates and futures hedge ratios. Rev Quant Finan Acc 58, 1741–1779 (2022). https://doi.org/10.1007/s11156-022-01037-z

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Keywords

  • Carry cost rate
  • Ex-ante futures hedge ratio
  • Ex-post hedge ratio

JEL Classification

  • G13
  • G11