Conversion factor risk and hedging in the treasury-bond futures market

Conversion Factor Risk and Hedging in the Treasury-Bond

Futures Market Alex Kane

Alan J. Marcus

he underlying asset on a Treasury-bond futures contract in the Chicago T Board of Trade (CBT) is not a real asset, but is rather a hypothetical 15-year- maturity government bond bearing an 8% coupon. Because the contract is settled using actual government bonds, the CBT is required to establish conversion factors which relate the settlement in terms of actual bonds to the benchmark asset. These factors are established by imperfect rules of thumb, however, and create basis risk for a hedger in T-bond futures, because the amount of money required to settle the contract and the profits or losses on the contract will depend upon the bond used as the delivery vehicle. The delivering short-side trader can choose to settle the contract with the bond which maximizes his profits. When future interest rates are stochastic and many bonds are eligible as delivery vehicles, neither the bond which actually will be chosen nor the conversion factor can be determined with certainty at the time the contract is established. Kilcollin (1982) investigates the prevailing delivery rules in the CBT and the New York Futures Exchange (NYFE) and concludes that the uncertainty about the delivery instrument affects equilibrium futures prices.

In this article we examine the extent to which conversion factor risk can affect the variability of returns on a futures contract. We also suggest a minor change in the procedure by which contracts are settled which can substantially reduce the magnitude of conversion factor risk. Our suggested procedure requires no computa- tions beyond those already performed by the CBT and can significantly increase the efficacy of the T-bond futures market as a means of hedging interest rate risk.

In the next section we describe the process by which T-bond futures contracts are settled. We demonstrate the effects of conversion factor risk and suggest a way to mitigate that risk. In Section I1 we illustrate thk range of the magnitude of conversion factor risk in simulations based on hypothetical term structures. The results in- dicate that our suggestion for contract settlement can substantially decrease rate-of- return variability. Section I11 concludes.

Alex Kane and Alan J. Marcus are Assistant Professors of Finance at the Boston University School of Management.

The Journal of Futures Markets, Vol. 4, No. 1, 55-64 (1984) D 1984 by John Wiley & Sons, Inc. CCC 0270-73 14/84/0 10055- 10S04.00

I. THE T-BOND FUTURES CONTRACT

The basic trading unit in the CBT T-bond futures market is a $100,000 face value U.S. Treasury bond. Any such T-bond with maturity of at least 15 years which also is not callable for at least 15 years from the first day of the delivery month may be delivered to settle the futures contract. In order to adjust for differences in the values of deliverable grade bonds, the CBT establishes conversion factors to determine actual payments at the delivery of the contract.’ The long side of each contract pays the short side an amount equal to the futures contract settlement price times the conversion factor (CF). The CF is chosen to adjust for the relative values of the admissable delivery grade bonds (CBT, 1982).

The CF equals the fraction of par value for which the delivery bond would sell if it were priced to offer an 8% yield to maturity. Bonds with coupon rates above 8% will have a CF greater than one, and conversely for bonds below 8%. This adjustment is motivated by the 8% coupon rate on the hypothetical benchmark bond upon which the contract is written (Yardeni, 1981). If at maturity the market exhibits a flat term structure with a yield of 8%, then the benchmark bond will sell at par and the CF will equal the ratio of the value of the delivery bond to that of the benchmark bond. However, if the market yield is not 8% at maturity, or if the term structure is not flat, then the CF will not equal the ratio of bond prices, and different delivery bonds will result in different profits or losses on the contract.

Once the short-side traders choose the optimal delivery bond, the conversion factor is calculated and the long-side trader is invoiced for an amount equal to qFwhere q is the conversion factor and F is the futures settlement price? Letting B denote the actual market value of the deliverable bond, the profit of the short trader is

qF - B. (1)

Note that the size of the contract is determined in part by the CF in the sense that the invoice amount is proportional to the conversion factor, q. This fact is of great impor- tance in the choice of the delivery bond Because the CF is calculated using a discount rate of 8%, the term qF, in (l), is not affected by the prevailing interest rates at the maturity of the contract. It is determined solely by the choice of the delivery bond’s coupon and maturity combination (c,T). In contrast, the actual market value of the delivery bond, B, is affected not only by (c, T), but is also affected by the realized interest rates at maturity. When interest rates are high (and delivery bonds priced to yield more than 8%) the profit to the short position will be maximized by delivering the highest possible coupon and maturity of the deliverable bonds. The reverse is true for low (below 8%) market yields. To the extent that the short-side trader has the option of choosing the optimal delivery bond, the equilibrium futures price must adjust to reflect this option (Kilcollin, 1982). Equilibrium in the futures market ensures that the short-trader will not earn excess expected profits, nor will the long-side suffer expected losses.

However, the treatment of conversion factors still poses a problem to hedgers in futures markets because traders do not know what the scale of their contract will be

‘Actual delivery is rarely made. Instead, contracts are usually liquidated through offsetting transactions shortly before maturity. However, profits or losses can be calculated as though delivery were made.

*The actual settlement procedure is more complex because of the issue of any accrued interest on the delivery bond. Accrued interest if computed by multiplying the coupon payment which accrues on the bond per day by the number of days between the last coupon payment and the actual delivery date. The accrued interest is added to the invoice amount, qF, to obtain the total amount due.

56/ KANE AND MARCUS

at maturity. Institutions with fixed dollar income or liability streams, such as pension funds or insurance companies, could more fully hedge against changes in the value of their portfolios if the conversion factor procedure eliminated the uncertainty regarding the final scale of the contract. It is important to emphasize that both sides of the contract face quantity uncertainty, Although the short-side trader chooses the delivery bond, the optimal delivery bond is uncertain at the time the contract is established, Thus even a trader hedging the value of a nominal liability stream cannot completely eliminate risk because the ultimate value of the CF, q, and thus the ultimate scale of the contract is uncertain.

To fix ideas, consider an institution with fixed dollar liabilities3 which (for simplicity) are equivalent to a short position in the 8%, 15-year maturity benchmark bond. One would think that one long position in a T-bond futures contract should protect the institution from interest rate risk. In fact, if the delivery vehicle were required to be the benchmark bond, then this hedged portfolio would have a nonstochastic end- of-period value. At maturity, the CF would equal unity (1.0). The futures contract would thus yield a profit equal to

B - F and with the underlying (liability) position, - B, the total hedged portfolio will reduce to the nonstochastic hedging cost - F, the price of the long position in the futures contract. However, in practice, the delivery vehicle can be one of several bonds, with conversion factors that range between 0.5 and 1.5 (Yardeni, 1981). The actual value of the total hedged portfolio will equal

-B + q(B - F ) = (1 - q ) B - qF

which is stochastic prior to maturity. Even if the CF perfectly reflected the relative values of the delivery and benchmark bonds, the hedged position would not be riskless. The final profits or losses from the hedged position would in fact be precisely equal to those which would result from purchasing q futures contracts which mandated delivery of the benchmark bond. Depending upon the realized value of q, the number of “benchmark-contract-equivalents” ultimately purchased can be either excessive or insufficient to offset the change in value of the initial portfolio.

In contrast to the current settlement procedure, consider an alternative delivery arrangement which adjusts the number of bonds delivered, rather than the settlement price of the contract as in (1). Specifically, suppose that the contract calls for a settlement of Fdollars, and delivery of llq units of the delivery bond. In this case, the profit of the short trader per contract would equal

F - Blq. (2) If the conversion factor, q, were equal to the ratio of the price of the delivery bond, B, to the actual market value of the benchmark bond, then the profit would be F - B*, where B* is the value of the benchmark bond at the settlement date. Anyone holding that bond (or a short position) could achieve a perfect hedge by trading in futures contracts. The contract would be precisely equivalent to one for which the delivery vehicle is exactly one unit of the benchmark bond. The elimination of the

3Although the discussion focuses on the problem of a trader with long-term nominal liabilities, it should be clear than an analogous argument can be made for hedging portfolios of fixed-dollar income streams. The variance of the profits (losses) on the short and long sides of the contract are, of course, equal.

CONVERSION FACTOR RISK / 5 7

scale effect4 due to CF variability should be expected to improve the hedging effectiveness of the T-bond contract; our simulation results confirm this expectation.

If the initial asset or liability of the trader is equivalent to the 8% benchmark bond, then a position in the modified futures contract would provide a perfect hedge. For other initial portfolio positions, a perfect hedge is not possible using either contract, since the underlying asset of the contract differs from the asset held in that initial portfolio, and the interest sensitivity of different bonds are not all equal. However, even in these cases, we will demonstrate that the modified contract generally provides superior hedging potential than does the current contract.

In practice, an added complication arises from the fact that the CF as presently computed does not equal the actual ratio of the values of the delivery to the benchmark bond. An “ideal” CF would equal the ratio of the value of the delivery bond to the value of the benchmark bond, if that bond existed. Since the benchmark bond might not be traded, its price cannot, in general, be observed. Thus, the actual CF is only an approximation to the ideal one. The actual CF is calculated by first computing the value each bond would attain if the term structure were flat at a level of 8% and then computing the ratio of the two values.

Table I compares actual conversion factors to the true price ratios (ideal conversion factors) for various market yields. We assume in Table I that the market exhibits a flat term structure of interest rates. Notice that the actual and ideal conversion factors are equal only when the market yield is 8%. The ratio of the actual to the ideal CF is printed below the ideal CF for each entry in Table I. The table shows that the ratio of the ideal CF to the actual CF varies systematically with the market yield curve. The key variable here is the sensitivity of the value of each bond to the interest rate (which is proportional to the bond’s duration). When the market interest rate equals 8%, the actual and ideal conversion factors are equal. When the market rate is below 8%, the ideal CF exceeds the actual CF if the value of the delivery bond is more sensitive to the interest rate (i.e., has a longer duration) than is the value of the benchmark bond. This will occur for longer maturity and lower coupon rate delivery bonds. This accounts for the difference between the two panels in Table I, in which the effects of longer maturity (30 years) in panel B dominate the effect of the coupon rates.

For the more extreme values of coupon rates and the market yields, the actual CF can differ from the ideal one by more than 10%. With an uncertain end-of-period term structure, the resultant uncertainty in the ultimate CF could present a significant source of risk even if our suggested modification of the futures contract, ex- pressed in (2), were adopted. However, total risk under the modified settlement procedure would still be less than it is using the current procedure.5

One would like to use the ideal CF in (2). In that case, the T-bond futures contract always would be equivalent to one for which the delivery vehicle was the benchmark bond. The T-bond contract would be similar to other futures contracts in the sense that the deliverable commodity is known in advance of the settlement.

4The number of bonds to be delivered is inversely proportional to the value of each delivery bond. This eliminates the scale uncertainty that is present in the current system. One additional complication, however, is the fact that the modified procedure will generally involve delivery of fractional quantities of bonds. This problem can be handled using either a cash delivery system OT a mixed system, in which actual bonds are delivered in round lots to the closest possible approximation to the contract terms, and then any residual amount is covered using cash.

’Many other future contracts, especially in agricultural markets, also involve basis risk due to differences in the quality of deliverable commodities. However, the conversion factor risk in the T-bond market as currently organized is of a different nature since it involves quantity as well as quality uncertainty.

58/ KANE AND MARCUS

Table I ACTUAL AND IDEAL CONVERSION FACTORS FOR VARIOUS BONDS

AND FLAT YIELD CURVES

A. Maturity of delivery bond = 15 years

Ideal Conversion Factors and Conversion Ratios for the

Actual Following Market Yields: Coupon Conversion Rates Factors 4.00 6.00 8.00 10.00 12.00 14.00

4.00 0.65

6.00 0.83

8.00 1.00

10.00 1.17

12.00 1.35

14.00 1.52

4.00 0.55

6.00 0.77

8.00 1 .oo

10.00 1.23

12.00 1.45

14.00 1.68

Ideal CF: 0.69 0.67 0.65 IdeallActual: 1.06 1.03 1.00

Ideal CF: 0.85 0.84 0.83 I de allActua1: 1.02 1.01 1.00




Ideal CF: 1.46 1.49 1.52 IdealiActual: 0.96 0.98 1 .OO

B. Maturity of delivery bond = 30 years







0.64 0.62 0.60 0.97 0.95 0.92 0.82 0.81 0.80 0.99 0.98 0.97 1.00 1.00 1.00 1.00 1.00 1.00 1.18 1.19 1.20 1.01 1.01 1.02 1.36 1.38 1.40 1.01 1.00 1.04 1.54 1.57 1.59 1.02 1.03 1.05

0.51 0.49 0.47 0.93 0.89 0.87 0.73 0.71 0.70 0.95 0.92 0.90 0.96 0.93 0.92 0.96 0.93 0.92 1.18 1.16 1.15 0.96 0.94 0.93 1.41 1.38 1.37 0.97 0.95 0.94 1.63 1.60 1.59 0.97 0.95 0.95

Unfortunately, as mentioned earlier, the value of the benchmark bond often cannot be observed. Moreover, estimating its value from actual data would be difficult because of problems in measuring the term structure (McCulloch, 1971). Almost any method for calculating the ideal CF would require some subjective decisions which would affect traders’ profits and likely engender intense controversy. Although the current method for computing the CF is imperfect, it has the virtues of predictability, objectivity, and limited informational requirements, all of which contribute to the smooth functioning of the futures market. All these qualities are retained by the proposed modification.

CONVERSION FACTOR RISK /59

11. SIMULATION RESULTS

We have performed several simulations to determine the extent to which the effectiveness of T-bond futures contracts as hedging vehicles could be increased by substituting the modified contract for the current one. We consider traders who start with fixed dollar liabilities, the present values of which are affected by interest rate fluctuations. For ease of presentation, we represent these initial positions as short positions in one of several long-term bonds.

We assume that the term structure at the end of the year is uncertain, so that the value of the initial liability plus the profits on any position in futures contracts is also stochastic. Specifically, we assume that the term structure at year end will be of the form

~ ( t ) = u + b[l0g(t+5) - I0g(5)] (3)

where y(t) is the yield to maturity for a discount bond of maturity t . This functional form exhibits the flattening tendency of actual curves (Livingston and Jain, 1982) and allows us to choose possible values for the intercept, a, and the slope, b. We assume that there are 18 equally likely potential outcomes for the term structure cor- responding to all possible combinations of a = 7,8,9% and b = - 0.05, - 0.25,0,5, 1, 2. This distribution is admittedly arbitrary, but it serves to illustrate the qualitative attributes of the hedging capabilities provided by the current and by our proposed futures contract. Experimentation with other distributions yielded results similar to those reported below.

Both the optimal hedging position and the equilibrium futures price at the beginning of the period will depend upon the set of bonds which will be eligible as delivery bonds. The set of deliverable bonds affects the market equilibrium because the short-side futures traders can choose the delivery bond which maximizes profits. For simplicity, we set the current price equal to the value which makes the expected profits on the futures contract equal to zero, given the ability of the short trader to choose the delivery vehicle. This amounts to setting the expected rate of return on investment in T-bond futures contracts to zero. This zero-expected-profit condition is not likely to be a characteristic of market equilibrium (Cox, Ingersoll, and Ross, 1981), but it serves as a useful simplification, and should not greatly affect the estimates of the variance rates of hedge portfolios. Note that the equilibrium futures price will be different for the two types of contract settlement procedures because the “scaling option” of the short trader is minimized in the modified contract?

We assume that the investor takes a long position in that number of futures contracts which minimizes the ex ante variance of his portfolio, given the distribution of possible end-of-year yield curves. The optimal position will depend upon the contract delivery mechanism, which can be either the current method or our suggested modification.

The set of deliverable bonds available to futures traders will be a function of the

6The difference in equilibrium futures prices has important implications for studies of the efficiency of the T-bond futures market. The relationship between the futures price and realized spot price is a function in part of the set of admissible delivery bonds. Our zero expected rate of return condition is essentially a “no risk premium” restriction. While this may be somewhat unrealistic, it should not significantly affect the variance rates or hedging properties of the contracts, which is the focus of this article. We experimented with nonzero risk premia and found extremely similar results to those reported here.

60/ KANE AND MARCUS

recent history of interest rates. After a period of rising rates, for example, longer maturity bonds will tend to have higher coupon rates. We have performed simulations for several configurations of coupon rates and maturities, and report here results from two of those sets. Other results were similar to the ones reported.

Our first set of results, summarized in Table 11, examines hedging possibilities after a period of rising interest rates. The longer maturity outstanding bonds have the highest coupon rates. We assume that there will be six outstanding deliverable bonds, with maturitylcoupon combinations given in the first two columns of panel B. As noted, we assume for simplicity that the position of an institution with long-term nominal liabilities can be taken as equivalent to a short position in a long-term bond. For variety, we consider short positions in each of the outstanding government bonds as possible initial unhedged positions of the futures market trader.

For each possible initial unhedged position, there exists a variance-minimizing long position in a T-bond futures contract. The optimal position, and the variance of the optimized portfolio, depend of course on the contract settlement procedure. In no case, however, can a perfect hedge be achieved, because the delivery bond is uncertain in all cases and because the underlying asset of the T-bond futures contract is not the bond for which the trader has a current (implicit) short position.

Table I1 HEDGING POTENTIAL OF FUTURES CONTRACTS (HISTORY OF RISING

INTEREST RATES) ~~ ~

A. Equilibrium Futures Price 1. Standard Contract: 87.80 2. Modified Contract: 93.61

Initial Term Structure: y(t) = 0.09 B. Standard Deviation of Various Portfolios

Initial Bond Std. Dev. of Std. Dev. of Std. Dev. of

Maturity Coupon Bond (Current Contract) (Modified Contract) Unhedged Hedged Bond Hedged Bond

C.

15 4 16.04 ia 6 16.25 21 8 16.40 24 10 16.66 27 12 17.03 30 14 17.46

Standard Deviations of Various Portfolios Initial Term Structure: y( t ) = 0.09 + ln(t + 5) -

15 4 18.93 ia 6 22.16 21 a 19.34 24 10 19.66 27 12 20.13 30 14 20.70

3.66 3.94 4.19 4.44 4.70 4.97

ln(5) 4.29 4.62 4.91 5.21 5.53 5.87

0.64 0.36 0.30 0.42 0.60 0.78

0.72 0.35 0.27 0.44 0.67 0.90

CONVERSION FACTOR RISK /61

Columns 3 through 5 in Panel B give the standard deviations of the rates of return of three possible portfolios. The first portfolio is the unhedged implicit short position in the government bond listed in the first two columns. The second portfolio consists of the initial short position plus the variance-minimizing long position in the standard futures contract. The last portfolio consists of the initial position plus the variance-minimizing position in our modified futures contract. The variance- minimizing position in futures is easy to compute, given the distribution of potential end-of-period term structures, because the variances of the bonds and futures contracts and their common covariance are readily calculated. To calculate the rates of return for each portfolio the initial value of the implicit short position is required. We therefore must assume some term structure for the beginning of the period. In panel B, the assumed yield curve is flat at 9%.

The results of Panel B are striking. Both futures contracts provide significant hedging potential: The standard deviation of the optimized portfolio is drastically reduced using either contract. However, the modified contract is obviously the superior hedging instrument. The minimized standard deviation is smaller by factors of 6 to 14 using our delivery procedure instead of the standard one. Moreover, the hedging effectiveness of the modified contract relative to the standard one does not depend upon the initial yield curve. Panel C presents results for an upward slop- ing initial term structure which are qualitatively identical to those in Panel B. All other initial yield curves resulted in extremely similar results.

Panel A presents the equilibrium futures prices for the two types of futures contracts. The futures prices are chosen to set the expected value of the profits on each contract to zero. These prices are revealing, for they show that the equilibrium futures price using the modified contract is 6% higher than for the current contract. This difference reflects the value of the short-side trader’s option to choose the delivery bond. That option is significantly more valuable for the standard contract than for the modified one. In the modified contract, the option to choose the delivery bond is important only because of the differences between the ideal and actual conversion factors. In the standard contract, the CF also scales the size of the contract and thus conveys extra value to the option. The equilibrium futures price on the standard contract is bid down because traders take into account the value of the delivery option when negotiating the terms of the contract.

Table 111 presents results for a different history of interest rates. In contrast to the pattern in Table II, which exhibited a strong trend in historical interest rates, we consider in Table I11 an extremely volatile and erratic history of interest rates which results in no correlation between coupons and maturities of outstanding bonds. The qualitative features of Table I11 are similar to those of Table IT. The equilibrium futures price is again lower for the standard contract than for the modified one. Both contracts still offer effective hedging potential. However, the modified contract is again the more effective hedging instrument, although in this case the discrepancy in the two contracts is not as large.

The general patterns exhibited in Tables I1 and I11 were representative of those for other configurations of outstanding bonds. The history most unfavorable to the modified contract is one of falling interest rates, which results in long maturity bonds having the lowest coupon rates. In this instance, the variance of the hedged portfolio is sometimes smaller using the standard contract to hedge instead of the modified contract. However, the overwhelming majority of the results favors the use of the modified contract.

6 2 / KANE AND MARCUS

Table 111 HEDGING POTENTIAL OF FUTURES CONTRACTS (HISTORY OF

ERRATIC INTEREST RATES)

A. Equilibrium Futures Price 1. Standard Contract: 89.20 2. Modified Contract: 92.61

Initial Term Structure: y(t) = 0.09 B. Standard Deviation of Various Portfolios

Initial Bond Std. Dev. Std. Dev. of Std. Dev. of Hedged of Unhedged Hedged Bond Bond (Modified

Maturity Coupon Bond (Current Contract) Contract)

15 4 16.04 2.76 18 14 17.76 3.57 21 6 18.44 4.16 24 12 20.20 4.29 27 8 16.05 3.30 30 10 12.99 2.36

C. Standard Deviations of Various Portfolios Initial Term Structure: y(t) = 0.09 + ln(t+ 5) - ln(5)

15 4 18.93 3.28 18 14 2 1.23 4.29 21 6 22.08 5.0 1 24 12 21.96 5.15 27 8 18.82 3.89 30 10 14.83 2.72

0.98 1.24 1.72 1.85 1.21 0.95

1.13 1.46 2.04 2.20 1.40 1.06

IV. CONCLUSION

We have shown that the hedging value of standard futures contracts in Treasury bonds is critically affected by the manner in which the contract is settled. Standard contracts are settled by delivery of an admissible bond in exchange for the futures price times a conversion factor which relates the value of the delivered bond to the underlying asset of the contract. Because the CF at the contract settlement date depends upon the delivered bond and can vary substantially, the ultimate dollar magnitude of the contract is unknown at the time the contract is established. This feature of the standard contract reduces its value as a hedging instrument.

A minor modification of the standard contract, however, can eliminate the scaling uncertainty and greatly reduce the residual risk of an optimally hedged portfolio. Using the conversion factor to determine the number of bonds to be delivered requires no information beyond that already collected by the CBT. Further, the only source of risk to traders in the modified contract is basis risk attributable to im- perfections in the calculation of the CF and to differences in the underlying asset of the contract and the actual portfolio of nominal assets or liabilities which is to be hedged.

Finally, our simulations show that the delivery option of the short-side trader can

CONVERSION FACTOR RISK / 6 3

be of significant value, and will be reflected in the equilibrium futures price. This option is more valuable the more volatile are future term structures.

Bibliography Chicago Board of Trade, (1981): An Introduction to Financial Futures. Cox, J. C., Ingersoll, J. E., and Ross, S. A. (1981): “The Relation Between Forward

Kilcollin, T. E. (1982): “Difference Systems in Financial Futures Markets,” Journal of

Livingston, M., and Jain, S., (1982): “Flattening of Bond Yield Curves for Long Ma-

McCulloch, J. H. (1971): “Measuring the Term Structure of Interest Rates,” Journal

Yardeni, E. E. (1981): “Managing Interest Rate Risk with Bond Futures,” Center for

Prices and Futures Prices,” Journal of Financial Economics, 9:32 1-346.

Finance, 37:1183-1198.

turities,” Journal of Finance, 37:157-168.

ofBusiness, 44:19-31.

the Study of Futures Markets (Columbia Business School) Working Paper #CSFM-10.

64 / KANE AND MARCUS

Documents

Conversion factor risk and hedging in the treasury-bond futures market