Relative pricing of French Treasury inflation-linked and nominal bonds: an empirical approach using arbitrage strategies

Abstract

This paper investigates whether arbitrage opportunities exist between inflation-linked bonds and nominal bonds on the French Treasury market. Following arbitrage theory, we apply the risk hedging concept: we set up self-financing portfolios hedged against risks through durations of different orders. Perfectly hedged portfolios are those with a zero initial and a zero final value. The results show arbitrage gains when the first three duration orders are implemented, but they are not significantly different from zero when a fourth-order duration is added. Furthermore, a regression of arbitrage gains on the illiquidity measure of nominal and index Treasury bonds provides evidence that the illiquidity of inflation-linked bonds significantly explains arbitrage gains, whereas the illiquidity measure of nominal bonds does not.

Appendices

Appendix 1

1.1 Nominal and IL bond prices in terms of zero-coupon bond prices

Our notation is as follows.

r_n: nominal spot interest rate.

r_r: real spot interest rate.

P_n (t,τ): price at time t of a nominal zero-coupon bond (i.e., nominal discount function of a coupon bond payoff) maturing at time t + τ.

P_r(t, τ): price at time t of a real zero-coupon bond (i.e., real discount function) maturing at time t + τ.

IF_t: indexation factor of an indexed bond, which is the ratio between the value of the harmonized Consumer Price Index (CPI), excluding tobacco, at time t (I_t) and the value of the same index at the bond issuance date t₀ (I_t0).

B_nom(t, m): price on date t of a nominal bond that periodically pays C euros on each date t + τ between t and t + m, and C + VF euros on maturity date t + m; the bond’s current price is equal to the sum of the present value of its payoffs, given by

$$ {B}_{nom}\left(t,m\right)=\sum \limits_{\tau =1}^m{CP}_n\left(t,\tau \right)+ VF{P}_n\left(t,m\right) $$

(11)

B_ilb (t,m) is the price on date t of a Treasury inflation-indexed bond. The bond periodically pays C units of the CPI on each date t + τ between t and t + m, and C + VF units of the CPI on maturity date t + m. Taking into account the current value of the bond indexation factor IF_t, we find that the bond price is equal to the sum of the present value of its nominal payoffs, given by

$$ {B}_{ilb}\left(t,m\right)={IF}_t\left[{\sum}_{\tau =1}^mC\ {P}_r\left(t,\tau \right)+ VF\ {P}_r\left(t,m\right)\right] $$

(12)

The duration of B_ilb(t,m) (denoted D_ilb,IF) with respect to IF is therefore

$$ {D}_{ilb, IF}=\frac{1}{IF_t} $$

Appendix 2

1.1 Methodology of the nominal and real zero-coupon yield estimations

No data on real zero-coupon yields are public or available on the French bond market. Therefore, the real zero-coupon yield curve must be estimated from the market prices of French IL bonds. To avoid differences in estimation methods, we apply the same method to estimate the nominal zero-coupon yield curve.

JY propose a method to estimate zero-coupon bond prices that relies on a discrete time approach to modeling forward interest rates. Additionally, their method assumes that forward rates are constant within piecewise segments of the maturity spectrum. With this method, the theoretical price function of a coupon bond, both nominal and indexed, can be written as follows:

$$ B\left(t,m\right)=\sum \limits_{\tau =1}^m{C}_{\tau}\exp \left(-\left(\sum \limits_{i=1}^K{f}_i\phi \left(\tau, i\right)\right)\right) $$

(13)

where C_τ is the bond payoff on date t + τ, K is the number of piecewise maturity segments of constant forward rates, f_i is the constant forward rate to be observed within the ith maturity segment, and ϕ(τ,i) is the part of the ith maturity segment covered by the maturity of the payoff C_τ . The lower limit of the shortest maturity segment is zero and its upper limit is m(1). Similarly, the upper limits of the other maturity segments are m(i), for i = 2, …, K. In many cases, the C_τ payoff maturity covers more than one maturity segment. Hence, the part of the ith maturity segment, m(i), covered by the C_τ payoff maturity is defined as follows:

$$ {\displaystyle \begin{array}{l}\phi \left(\tau, i\right)=m(i)-m\left(i-1\right)\mathrm{if}\ \tau \ge m(i)\\ {}\phi \left(\tau, i\right)=\tau -m\left(i-1\right)\mathrm{if}\ m(i)>\tau \ge m\left(i-1\right)\\ {}\phi \left(\tau, i\right)=0\ \mathrm{if}\ \tau <m\left(i-1\right)\end{array}} $$

(14)

This paper proposes an alternative to the JY method that consists of setting the limits between two piecewise segments at the maturity dates of the coupon bonds. Hence, the upper limit of the first maturity segment is the maturity of the coupon bond with the shortest maturity, represented by m(b(1)). Similarly, the upper limits of the other maturity segments m(b(i)), for i = 2, …, K, where K is the number of bonds in the sample, are adjusted to the maturity of the K coupon bonds. Under this approach, the portion of the ith maturity segment covered by the maturity of C_τ, ϕ(τ, i), is defined as follows:

$$ {\displaystyle \begin{array}{l}\phi \left(\tau, i\right)=m\left(b(i)\right)-m\left(b\left(i-1\right)\right)\mathrm{if}\ \tau \ge m\left(b(i)\right)\\ {}\phi \left(\tau, i\right)=\tau -m\left(b(i)\right)\mathrm{if}\ m\left(b\left(i-1\right)\right)>\tau \ge m\left(b\left(i0-1\right)\right)\\ {}\phi \left(\tau, i\right)=0\ \mathrm{if}\ \tau <m\left(b\left(i-1\right)\right)\end{array}} $$

(15)

While the piecewise maturity limits in the JY model, m(i), are chosen arbitrarily, in the innovative procedure proposed in this article, the corresponding m(b(i)) limits are adjusted to the maturities of the coupon bonds in the sample. This procedure has the advantage of providing estimated prices that perfectly match the market prices.

1.2 Comparison of the zero-coupon yield estimations for the two models

Fig. 1

Nominal term structures, 2013

Fig. 2

Real term structures, 2013

Fig. 3

Nominal term structures, 2014

Fig. 4

Real term structures, 2014

Fig. 5

Nominal term structures, 2015

Fig. 6

Real term structures, 2015

Appendix 3

1.1 Data presentation and correlation coefficients between indexation factor changes

We constructed a database that comprises daily prices covering the period from January 1, 2013, through December 31, 2015, for a total of 783 daily market prices for nominal and IL bonds issued by the French Treasury. Market prices and inflation factors were extracted from the Thomson Reuters Datastream database. The inflation factor is the ratio between the value of the harmonized CPI, excluding tobacco, on date t and its value on the issuance date t₀ of the IL bond. This inflation factor, IF, is applied to IL bonds following eq. (2) and allows us to protect investor cash flows against inflation. The same database is used to both extract zero-coupon yields and set up hedging strategies.

The period begins in 2013 because too few French IL bonds were traded before that year. Two types of IL bonds are available on the French market. Some are indexed to the domestic CPI and are denoted OATi. Others were issued more recently (since July 2001) and are indexed to the euro area Harmonised Index of Consumer Prices and denoted OAT€i. As Pericoli (2014), we include both types of IL bonds in our sample. Before including them, however, we took the precaution to calculating the correlation coefficients between the variations of the indexation factors (IF). As shown in Table 7, these coefficients are very high, not only within each group (OATis/OAT€is), but also between the two groups.

Table 7 Correlation coefficients between indexation factor changes