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.

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 ci_t (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,

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

∆t is 1/52, and

ε_c is a Wiener process realization component of c.

The time t theoretical spot price, S^theoretical_t , 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)

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

c_{t =} ci_t -1

σ_s is the annual spot return volatility,

ρ is the correlation between c_t and S_t, and

 ε_s 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: WTM_t 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)

 ε_{err_s and}  ε_{err_f} are idiosyncratic Wiener process realization components of the spot and futures price changes, and their volatilities are σ_{err_s} and σ_{err_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, σ_c = σ_s = 0.2, ϴ = 0.5, μ = 0.02, ρ = 0, and σ_{err_s} = σ_{err_f} = 0.02. For this parameterization’s simulated data, h_T’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, h_c is higher (by ≈ 0.0288) and less volatile (by ≈ 62%) than h_T. Second, the negative correlation between the paired average c and h_T is recovered, albeit inefficiently; while the correlation between the paired average c and h_T had a mean and standard deviation of -0.4473 and 0.1335, respectively, the corresponding values for h_c were about -0.954 and essentially 0, respectively. Third, the BAM is stationary. Fourth, h_c and h_c-BA reduce risk better than do h_T 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, σ_c, σ_s, ϴ, μ, ρ, and σ_{err_s} = σ_{err_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 σ_c input (0.2) by 0.05, ceteris paribus

11,

Table 12 Test results based on data generated from changing the baseline σ_s 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(c_t, S_t), input by 0.1, ceteris paribus

14,

Table 15 Test results based on data generated from changing the baseline σ_{err_s} and σ_{err_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 σ_c 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 σ_s 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 σ_{err_s} = σ_{err_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 h_T is 0.9746; as expected, this is a little higher than it was for the above positive long-run c mean simulation (since h_c is also higher than it was for the above positive long-run c mean simulation). The h_T’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 σ_c 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 σ_s 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 (c_t, S_t), 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 σ_{err_s} and σ_{err_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.