How fair-value accounting can influence firm hedging

Abstract

The potential influence of accounting regulations on hedging strategies and the use of financial derivatives is a research topic that has attracted little attention in both the finance and the accounting literature. However, recent surveys suggest that company hedging can be substantially influenced by the accounting for financial instruments. In this study, we illustrate not only why but also how the accounting regulations may affect hedging behavior. We find that under mark-to-market accounting, most firms concerned with earnings smoothness adopt myopic hedging strategies relative to the benchmark, cash flow hedging. The specific influence of the accounting regulations depends on market and firm-specific characteristics, but, in general, the firms dramatically reduce the extent of hedging addressing price risk in future accounting periods. We illustrate that the change in hedging behavior significantly dampens the increase in earnings volatility stemming from fair value accounting of derivatives. However, the adjusted hedging strategies may substantially increase the firms’ cash flow volatility.

Appendices

Appendix A: Proof of proposition 1

Let $\tilde{S}_t$ and $\tilde{Q}_t$ denote the random spot price and the random quantity produced in period $t$, respectively, $c$ the constant marginal cost, $C$ fixed costs, $F_{0,t}$ the forward price at $t = 0$ with maturity at time $t$, and $a_t$ the number of long positions in forward contracts with delivery at time $t$ entered into at $t = 0$. A firm faces random cash flows $\widetilde{CF_1 }$ equal to

$$\begin{aligned} \widetilde{CF_1 }=\tilde{S}_1 \tilde{Q}_1 -c\tilde{Q}_1 -C+a_1 \left[ {\tilde{S}_1 -F_{0,1} } \right] \end{aligned}$$

(18)

The variance of the cash flow is equal to

$$\begin{aligned} \mathrm{var}\left( {C\tilde{F}_1} \right)&= \mathrm{var}\left( {\tilde{S}_1 \tilde{Q}_1 } \right)+c^{2}\mathrm{var}\left( {\tilde{Q}_1 } \right)+a_1^2 \mathrm{var}\left( {\tilde{S}_1 } \right) \nonumber \\&\quad -2c \,\mathrm{cov}\left( {\tilde{S}_1 \tilde{Q}_1 ,\tilde{Q}_1 } \right)+2a_1 \,\mathrm{cov}\left( {\tilde{S}_1 \tilde{Q}_1 ,\tilde{S}_1 } \right) -2a_1 c \,\mathrm{cov}\left( {\tilde{S}_1 ,\tilde{Q}_1 } \right)\nonumber \\ \end{aligned}$$

(19)

A firm minimizing the volatility of the cash flow solves the following condition (interior solution):

$$\begin{aligned} \frac{d\,\mathrm{var}\left( {\widetilde{CF}_1 } \right)}{da_1 }=0 \end{aligned}$$

(20)

Under bivariate normality, it follows from Lemma 2 in Sévi (2006) that the optimal number of forward contracts for period 1 is equal to

$$\begin{aligned} a_1^{CF} =-\mu _Q -\left( {\mu _S -c} \right)\rho _{\tilde{Q}_1 ,S_1 } \frac{\sigma _{\tilde{Q}_1 } }{\sigma _{\tilde{S}_1 } } \end{aligned}$$

(21)

Because forward contracts covering future periods do not affect cash flows in the previous periods, this expression for the optimal number of forward contracts is general and holds for any random year t.

We now assume that we have a two-period setting. Under mark-to-market accounting, the earnings in period (year) 1 is equal to

$$\begin{aligned} \widetilde{EARN}_1 =\tilde{S}_1 \tilde{Q}_1 -c\tilde{Q}_1 -C+a_1 \left[ {\tilde{S}_1 -F_{0,1} } \right]+a_2 \left[ {\tilde{F}_{1,2} -F_{0,2} } \right] \end{aligned}$$

(22)

The variance of earnings in period 1 equals

$$\begin{aligned} \mathrm{var}\left( {\widetilde{EARN}_1 } \right)&= \mathrm{var}\left( {\tilde{S}_1 \tilde{Q}_1 } \right)+c^{2}\mathrm{var}\left( {\tilde{Q}_1 } \right)+a_1^2 \mathrm{var}\left( {\tilde{S}_1 } \right)+a_2^2 \mathrm{var}\left( {\tilde{F}_{1,2} } \right) \nonumber \\&\quad -2c\,\mathrm{cov}\left( {\tilde{S}_1 \tilde{Q}_1 ,\tilde{Q}_1 } \right)+2a_1 \,\mathrm{cov}\left( {\tilde{S}_1 \tilde{Q}_1 ,\tilde{S}_1 } \right) \nonumber \\&\quad +2a_2 \,\mathrm{cov}\left( {\tilde{S}_1 \tilde{Q}_1 ,\tilde{F}_{1,2} } \right) -2a_1 c \,\mathrm{cov}\left( {\tilde{Q}_1 ,\tilde{S}_1 } \right) \nonumber \\&\quad -2a_2c\,\mathrm{cov}\left( {\tilde{Q}_1 ,\tilde{F}_{1,2} } \right)+2a_1 a_2 \,\mathrm{cov}\left( {\tilde{S}_1 ,\tilde{F}_{1,2} } \right) \end{aligned}$$

(23)

A firm minimizing the volatility of earnings under mark-to-market accounting at $t = 0$ solves the following conditions (interior solution):

$$\begin{aligned} \frac{\partial \mathrm{var}\left( {\widetilde{EARN}_1 } \right)}{\partial a_1 }&= 0 \nonumber \\ \frac{\partial \mathrm{var}\left( {\widetilde{EARN}_1 } \right)}{\partial a_2 }&= 0 \end{aligned}$$

(24)

Following Theorem 17.10 in Sydsaeter and Hammond (1995) and the fact that

$$\begin{aligned} \frac{\partial ^{2}\mathrm{var}\left( {\widetilde{EARN}_1 } \right)}{\partial a_1^2 }&= 2\sigma _{\tilde{S}_1 }^2 >0 \nonumber \\ \frac{\partial ^{2}\mathrm{var}\left( {\widetilde{EARN}_1 } \right)}{\partial a_2^2 }&= 2\sigma _{\tilde{F}_1 }^2 >0\nonumber \\ \frac{\partial ^{2}\mathrm{var}\left( {\widetilde{EARN}_1 } \right)}{\partial a_1 \partial a_2}&= \frac{\partial ^{2}\mathrm{var}\left( {\widetilde{EARN}_1 } \right)}{\partial a_2 \partial a_1}=0 \end{aligned}$$

(25)

the variance function is strictly convex. Therefore, the solutions of the two first-order conditions define the unique global minimum value (Theorem 17.11). Given Eq. (22), these two solutions are defined by the following two simultaneous equations:

$$\begin{aligned} a_1&= -\frac{\mathrm{cov}(\tilde{S}_1 \tilde{Q}_1 ,\tilde{S}_1 )+a_2 \,\mathrm{cov}\left( {\tilde{S}_1 ,\tilde{F}_{1,2} } \right)-c\,\mathrm{cov}\left( {\tilde{Q}_1 ,\tilde{S}_1 } \right)}{\mathrm{var}\left( {\tilde{S}_1 } \right)} \nonumber \\ a_2&= -\frac{\mathrm{cov}(\tilde{S}_1 \tilde{Q}_1 ,\tilde{F}_{1,2} )+a_1 \,\mathrm{cov}\left( {\tilde{S}_1 ,\tilde{F}_{1,2} } \right)-c\,\mathrm{cov}\left( {\tilde{Q}_1 ,\tilde{F}_{1,2} } \right)}{\mathrm{var}\left( {\tilde{F}_{1,2} } \right)} \end{aligned}$$

(26)

$$\begin{aligned} \text{ Setting} \;a_1 =-\frac{A+a_2 C-cF}{B}=-\frac{A-cF}{B}-a_2 \frac{C}{B} \text{ and} \;a_2&= -\frac{D+a_1 C-cG}{E}\\&= -\frac{D-cG}{E}-a_1 \frac{C}{E}, \end{aligned}$$

where $A\!=\!\mathrm{cov}(\tilde{S}_1 \tilde{Q}_1 ,\tilde{S}_1 ), B\!=\!\mathrm{var}\left( {\tilde{S}_1 } \right), C\!=\!\mathrm{cov}\left( {\tilde{S}_1 ,\tilde{F}_{1,2} } \right), D\!=\!\mathrm{cov}(\tilde{S}_1 \tilde{Q}_1 ,\tilde{F}_{1,2} ), E=\mathrm{var}\left( {\tilde{F}_{1,2} } \right)$, $F=\mathrm{cov}\left( {\tilde{Q}_1 ,\tilde{S}_1 } \right)$, and $G=\mathrm{cov}\left( {\tilde{Q}_1 ,\tilde{F}_{1,2} } \right)$ yields the solutions

$$\begin{aligned} a_1 =\frac{\frac{DC}{EB}-\frac{A}{B}+c\left( {\frac{F}{B}-\frac{GC}{EB}} \right)}{1-\frac{C^{2}}{EB}} \end{aligned}$$

(27)

$$\begin{aligned} a_2 =\frac{\frac{A}{B}\frac{C}{E}-\frac{D}{E}+c\left( {\frac{G}{E}-\frac{FC}{BE}} \right)}{\left( {1-\frac{C^{2}}{BE}} \right)} \end{aligned}$$

(28)

Reinserting the definitions of the constants yields the optimal number of forward contracts entered into at $t = 0$ under general distributional assumptions:

$$\begin{aligned} a_1^{M2M*}&= -\frac{\mathrm{cov}(\tilde{S}_1 \tilde{Q}_1 ,\tilde{S}_1 )}{\sigma _{\tilde{S}_1 }^2 \left( {1-\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} }^2 } \right)}+\frac{\mathrm{cov}(\tilde{S}_1 \tilde{Q}_1 ,\tilde{F}_{1,2} )}{\sigma _{\tilde{F}_{1,2} } \sigma _{\tilde{S}_1 } }\frac{\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} } }{1-\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} }^2 }\nonumber \\&\quad +c\left( {\frac{\rho _{\tilde{Q}_1 ,\tilde{S}_1 } -\rho _{\tilde{Q}_1 ,\tilde{F}_{1,2} } \rho _{\tilde{S}_1 ,\tilde{F}_{1,2} } }{1-\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} }^2 }} \right)\frac{\sigma _{\tilde{Q}_1 } }{\sigma _{\tilde{S}_1 } } \end{aligned}$$

(29)

$$\begin{aligned} a_2^{M2M*}&= -\frac{\mathrm{cov}(\tilde{S}_1 \tilde{Q}_1 ,\tilde{F}_{1,2} )}{\sigma _{\tilde{F}_{1,2} }^2 \left( {1-\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} }^2 } \right)}+\frac{\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} } }{1-\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} }^2 }\frac{\mathrm{cov}(\tilde{S}_1 \tilde{Q}_1 ,\tilde{S}_1 )}{\sigma _{\tilde{S}_1 } \sigma _{\tilde{F}_{1,2} } }\nonumber \\&\quad +c\frac{\left( {\rho _{\tilde{Q}_1 ,\tilde{F}_{1,2} } -\rho _{\tilde{Q}_1 ,\tilde{S}_1 } \rho _{\tilde{S}_1 ,\tilde{F}_{1,2} } } \right)\frac{\sigma _{\tilde{Q}_1 } }{\sigma _{\tilde{F}_{1,2} } }}{\left( {1-\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} }^2 } \right)} \end{aligned}$$

(30)

These optimal numbers of forward contracts collapse into the following equations under multivariate normality by using Lemma 2 in Sévi (2006)

$$\begin{aligned} a_1^{M2M}&= -\mu _{\tilde{Q}_1 } -\left( {\mu _{\tilde{S}_1 } -c} \right)\frac{\sigma _{\tilde{Q}_1 } }{\sigma _{\tilde{S}_1 } }\left[ {\frac{\rho _{\tilde{Q}_1 ,\tilde{S}_1 } -\rho _{\tilde{Q}_1 ,\tilde{F}_{1,2} } \rho _{\tilde{S}_1 ,\tilde{F}_{1,2} } }{1-\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} }^2 }} \right] \end{aligned}$$

(31)

$$\begin{aligned} a_2^{M2M}&= -\left( {\mu _{\tilde{S}_1 } -c} \right)\frac{\sigma _{\tilde{Q}_1 } }{\sigma _{\tilde{F}_{1,2} } }\left[ {\frac{\rho _{\tilde{Q}_1 ,\tilde{F}_{1,2} } -\rho _{\tilde{Q}_1 ,\tilde{S}_1 } \rho _{\tilde{S}_1 ,\tilde{F}_{1,2} } }{1-\rho _{\tilde{S}_1 ,\tilde{F}_{1,2} }^2 }} \right] \end{aligned}$$

(32)

Appendix B: Detailed specification of the simulation assumptions of Section 4

 

• 1) Earnings targets (predictions at time t for the earnings of time t +1 and t+2) Before:

  $$\begin{aligned} \widehat{CF}_{t,t+1}&= \left( {E_t \left[ {\tilde{S}_{t+1} } \right]-c^{*}T_{t+1} \left( \Phi \right)} \right)\mu _Q +\sigma _\varepsilon \sigma _{\tilde{Q}} \rho _{\tilde{Q},\tilde{S}} -C^{*}T_{t+1} \left( \Phi \right)\nonumber \\&\quad +a_{1,t}^{CF} \left( {E_t \left[ {\tilde{S}_{t+1} } \right]-F_{t,t+1} } \right)+a_{2,t-1}^{CF} \left( {F_{t,t+1} -F_{t-1,t+1} } \right) \end{aligned}$$

  (33)
  $$\begin{aligned} \widehat{CF}_{t,t+2}&= \left( {E_t \left[ {\tilde{S}_{t+2} } \right]-c^{*}T_{t+2} \left( \Phi \right)} \right)\mu _Q +\sigma _\varepsilon \sigma _{\tilde{Q}} \rho _{\tilde{Q},\tilde{S}} -C^{*}T_{t+2} \left( \Phi \right) \nonumber \\&\quad +a_{2,t}^{CF} \left( {E_t \left[ {\tilde{S}_{t+2} } \right]-F_{t,t+2} } \right) \end{aligned}$$

  (34)

  After, same strategy:

  $$\begin{aligned} \widehat{Earn}_{t,t+1}^{CF}&= \left( {E_t \left[ {\tilde{S}_{t+1} } \right]-c^{*}T_{t+1} \left( \Phi \right)} \right)\mu _Q +\sigma _\varepsilon \sigma _{\tilde{Q}} \rho _{\tilde{Q},\tilde{S}} -C^{*}T_{t+1} \left( \Phi \right) \nonumber \\&+ a_{1,t}^{CF} \left( {E_t \left[ {\tilde{S}_{t+1} } \right]\!-\!F_{t,t+1} } \right) \!+\!a_{2,t}^{CF} \left( {E_t \left[ {\tilde{S}_{t+2} } \right]\!-\!F_{t,t+2} } \right) \end{aligned}$$

  (35)
  $$\begin{aligned} \widehat{Earn}_{t,t+2}^{CF}&= \left( {E_t \left[ {\tilde{S}_{t+2} } \right]\!-\!c^{*}T_{t+2} \left( \Phi \right)} \right)\mu _Q \!+\!\sigma _\varepsilon \sigma _{\tilde{Q}} \rho _{\tilde{Q},\tilde{S}} \!-\!C^{*}T_{t+2} \left( \Phi \right) \end{aligned}$$

  (36)

  After, new strategy:

  $$\begin{aligned} \widehat{Earn}_{t,t+1}&= \left( {E_t \left[ {\tilde{S}_{t+1} } \right]-c^{*}T_{t+1} \left( \Phi \right)} \right)\mu _Q +\sigma _\varepsilon \sigma _{\tilde{Q}} \rho _{\tilde{Q},\tilde{S}} -C^{*}T_{t+1} \left( \Phi \right) \nonumber \\&+a_{1,t}^{M2M} \left( {E_t \left[ {\tilde{S}_{t+1} } \right]\!-\!F_{t,t+1} } \right)\!+\!a_{2,t}^{M2M} \left( {E_t \left[ {\tilde{S}_{t+2} } \right]\!-\!F_{t,t+2} } \right)\quad \end{aligned}$$

  (37)
  $$\begin{aligned} \widehat{Earn}_{t,t+2}&= \left( {E_t \left[ {\tilde{S}_{t+2} } \right]-c^{*}T_{t+2} \left( \Phi \right)} \right)\mu _Q +\sigma _\varepsilon \sigma _{\tilde{Q}} \rho _{\tilde{Q},\tilde{S}} -C^{*}T_{t+2} \left( \Phi \right) \end{aligned}$$

  (38)

  Other assumptions:

  $$\begin{aligned} E_t \left[ {\tilde{S}_{t+1} } \right]&= T_{t+1} \left( \Phi \right)+\varphi S_t^*\end{aligned}$$

  (39)
  $$\begin{aligned} E_t \left[ {\tilde{S}_{t+2} } \right]&= T_{t+2} \left( \Phi \right)+\varphi ^{2}S_t^*\end{aligned}$$

  (40)
  $$\begin{aligned} c_t&= c^{*}T_t \left( \Phi \right) \end{aligned}$$

  (41)
  $$\begin{aligned} C_t&= C^{*}T_t \left( \Phi \right) \end{aligned}$$

  (42)

  Consistent with Brown and Toft’s (2002, p. 1291) base case assumptions, we set $c^{*} = 0.25$ and $C^{*} = 0.4$. Note that both $\widehat{Earn}^{CF}$ and $\widehat{Earn}$ above represent earnings under fair-value accounting, the former using cash flow hedging and the latter using a hedging strategy designed to manage earnings risk under fair-value accounting.

• 2) Net profit decomposition under deferral accounting (FCP = forward contract payoff):

  $$\begin{aligned} \widetilde{FCP}_{1,t}^{DA}&= a_{1,t-1}^{CF} \left( {\tilde{S}_t -F_{t-1,t} } \right) \end{aligned}$$

  (43)
  $$\begin{aligned} FCP_{2,t}^{DA}&= a_{2,t-2}^{CF} \left( {F_{t-1,t} -F_{t-2,t} } \right) \end{aligned}$$

  (44)
  $$\begin{aligned} \widetilde{NP}_t^{NoHedge}&= \tilde{S}_t \tilde{Q}_t -c\tilde{Q}_t -C \end{aligned}$$

  (45)
  $$\begin{aligned} \widetilde{CF}_t&= \widetilde{NP}_t^{NoHedge} +\widetilde{FCP}_{1,t}^{DA} +\widetilde{FCP}_{2,t}^{DA} \end{aligned}$$

  (46)

• 3) Net profit decomposition under fair-value accounting and cash flow hedging: Let F be the forward lag operator. In this case,

  $$\begin{aligned} \widetilde{Earn}_t^{CF} =\widetilde{NP}_t^{NoHedge} +\widetilde{FCP}_{t,1}^{DA} +F\left( {\widetilde{FCP}_{t,2}^{DA} } \right) \end{aligned}$$

  (47)

• 4) Net profit decomposition under fair-value accounting and a hedging strategy designed to manage earnings risk under fair-value accounting:

  $$\begin{aligned} \widetilde{FCP}_{1,t}^{M2M}&= a_{1,t-1}^{M2M} \left( {\tilde{S}_t -F_{t-1,t} } \right) \end{aligned}$$

  (48)
  $$\begin{aligned} \widetilde{FCP}_{2,t}^{M2M}&= a_{2,t-1}^{M2M} \left( {F_{t,t+1} -F_{t-1,t+1} } \right) \end{aligned}$$

  (49)
  $$\begin{aligned} \widetilde{Earn}_t&= \widetilde{NP}_t^{NoHedge} +\widetilde{FCP}_{1,t}^{M2M} +\widetilde{FCP}_{2,t}^{M2M} \end{aligned}$$

  (50)

• 5) Average root mean squared prediction errors in the sample of M  = 10,000 replications over the N  = 10 years 2012–2021: Define $\Theta = \{ \text{ t}: \text{ t}\in \{2012,\ldots ,2021\} \}$. In this case, root mean squared errors are defined as follows for one-year and two-year forecasts, respectively:

  $$\begin{aligned} RMSE_{1YR}^{\widehat{CF_{t\in \Theta } }}&= \frac{1}{M}\sum _{i=1}^M {\sqrt{\frac{1}{N}\sum \limits _{t\in \Theta } {\left( {\widetilde{CF}_{t,i} -\widehat{CF}_{t-1,t,i} } \right)^{2}} }} \end{aligned}$$

  (51)
  $$\begin{aligned} RMSE_{2YR}^{\widehat{CF}_{t\in \Theta } }&= \frac{1}{M}\sum _{i=1}^M \sqrt{\frac{1}{N}\sum \limits _{t\in \Theta } {\left( {\widetilde{CF}_{t,i} -\widehat{CF}_{t-2,t,i} } \right)^{2}} } \end{aligned}$$

  (52)
  $$\begin{aligned} RMSE_{1YR}^{\widehat{Earn}_{t\in \Theta }^{CF} }&= \frac{1}{M}\sum \limits _{i=1}^M {\sqrt{\frac{1}{N}\sum _{t\in \Theta } {\left( {\widetilde{Earn}_{t,i}^{CF} -\widehat{Earn}_{t-1,t,i}^{CF} } \right)^{2}} }} \end{aligned}$$

  (53)
  $$\begin{aligned} RMSE_{2YR}^{\widehat{Earn}_{t\in \Theta }^{CF} }&= \frac{1}{M}\sum _{i=1}^M {\sqrt{\frac{1}{N}\sum \limits _{t\in \Theta } {\left( {\widetilde{Earn}_{t,i}^{CF} -\widehat{Earn}_{t-2,t,i}^{CF} } \right)^{2}} }} \end{aligned}$$

  (54)
  $$\begin{aligned} RMSE_{1YR}^{\widehat{Earn}_{t\in \Theta }}&= \frac{1}{M}\sum \limits _{i=1}^M {\sqrt{\frac{1}{N}\sum _{t\in \Theta } {\left( {\widetilde{Earn}_{t,i}-\widehat{Earn}_{t-1,t,i}} \right)^{2}} }} \end{aligned}$$

  (55)
  $$\begin{aligned} RMSE_{2YR}^{\widehat{Earn}_{t\in \Theta } }&= \frac{1}{M}\sum _{i=1}^M {\sqrt{\frac{1}{N}\sum \limits _{t\in \Theta } {\left( {\widetilde{Earn}_{t,i} -\widehat{Earn}_{t-2,t,i} } \right)^{2}} }} \end{aligned}$$

  (56)

  Percentage errors (relative to the targets) are obtained as in Eq. (17).

• 6) Procedure for generating the unhedgeable quantity innovations ((Hull 1997, p. 363)).

The unhedgeable quantity innovations are presumed to be i.i.d. and correlated (corr) with the innovation terms of the ARMAX price dynamics; that is, the quantity innovations are correlated with the $\varepsilon$-terms of the price process.

• Step 1: Scale each of the price innovations by multiplying $\varepsilon$ with the ratio $\frac{\sigma _{\tilde{Q}}}{\hat{\sigma }_\varepsilon } (\hat{\sigma }_{\varepsilon } =125.1)$. Denote this rescaled series of price innovations $e_{1}$.

• Step 2: Generate an independent set of normally distributed RVs with zero (expected) mean and standard deviation $\sigma _{\tilde{Q}}$. Denote this series $e_{2}$.

• Step 3: Generate a correlated set of bivariately normal innovations price and quantity innovations with zero mean and standard deviations equal to $\hat{\sigma }_{\varepsilon } =125.1$ and $\sigma _{\tilde{Q}}$, respectively, by calculating the new series of $\tilde{Q}$-innovations as follows: corr * $e_{1}$ + sqrt($1-\text{ corr}^{2}$) * $e_{2}$. This is the set of randomly generated quantity innovations.