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Some characteristics of an equity security next-year impairment

Abstract

In this paper, we propose some characteristics of next-year impairments in a generic Black and Scholes framework, with one equity security, and under International Financial Reporting Standards (IFRS) rules. We derive expression for the probability of impairment event for an equity-security recognized in the available-for-sale category. Our decomposition of this event is also useful to retrieve barrier options valuation methods. From there, we obtain an explicit formula for the first moment of impairment value and its cumulative distribution function, as well as sensitivities. Numerical studies are carried out on concrete securities. We also study a mean-preserving one-criterion proxy used by some insurance practitioners for the next-year impairment losses and discuss its relevance. More generally, our study paves the way for applications of financial mathematics techniques to accounting issues related to impairments in the IFRS framework.

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Notes

  1. 1.

    International Financial Reporting Standards Interpretations Committee.

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Acknowledgements

This work has been mainly supported by the BNP Paribas Cardif Chair Management de la modélisation. The views expressed in this document are the authors owns and do not necessarily reflect those endorsed by BNP Paribas Cardif. The second author acknowledges support from the Milliman France Chair Actuariat Durable. The authors acknowledge financial support from the Europlace Institute of Finance (EIF) for research on impairment of financial assets (DéCAF project: http://isfa.univ-lyon1.fr/decaf).

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Correspondence to Pierre-E. Thérond.

Appendices

Appendices

In the sequel, \(\Upphi\) denotes the c.d.f. of a standard normal distribution, and \(\Uppsi_\rho\) is the bivariate normal distribution function: for all \({x,y, \Uppsi_\rho (x,y) = \mathbb{P}_t \left[ {X\leq x, Y\leq y} \right]}\) where (XY) is a Gaussian vector with standard marginals and correlation ρ.

Appendix 1: Proof of results about the probability that an impairment occurs next year

We would like to evaluate, for \(S_{t_a}\)m t and K t , this following quantity:

$$\begin{aligned} P&={\mathbb{P}}_t \left[ { S_{t+1} \leq m_t} \right] + {\mathbb{P}}_t \left[ { \max_{t+1-s\leq u\leq t+1} S_u \leq S_{t_a} , S_{t+1} \leq K_t } \right]\\ &\quad- {\mathbb{P}}_t \left[ { \max_{t+1-s\leq u\leq t+1} S_u \leq S_{t_a} ,S_{t+1} \leq m_t } \right]. \end{aligned}$$

It is possible to quickly retrieve an expression using the drifted Brownian motion \((B_t)_t=\left( \ln \frac{S_t}{S_0} \right)_{t}\). Indeed, we apply the following property with the previous decomposition and obtain Theorem 1

Proposition 2

For all real z ≤ a, for all times 0 < s < t:

$$\begin{aligned} &{\mathbb{P}}_t \left[ { \max_{t+1-s \leq u \leq t+1} B_u \geq a ,\ B_{t+1} \leq z } \right]\\ &\quad={\exp}\left( \frac{2(\mu-\frac{\sigma^2}{2}) a}{\sigma^2} \right)\left[ \Upphi\left( \frac{ -2a +z -\mu}{\sigma } + \frac{\sigma}{2} \right) \right.\\ &\quad\left. - \Uppsi_\rho \left( \frac{-a - (1-s)\mu }{ \sigma \sqrt{1-s} } - \frac{\sigma\sqrt{1-s}}{2} , \frac{- 2a +z -\mu}{ \sigma } +\frac{\sigma}{2}) \right) \right]\\ &\quad+ \left[ 1-\Upphi\left( \frac{a -(1-s)\mu}{\sigma\sqrt{1-s}} +\frac{\sigma\sqrt{1-s}}{2} \right) \right]\\ &\quad- \Uppsi_\rho \left( \frac{-a +(1-s)\mu}{\sigma \sqrt{1-s}} -\frac{\sigma\sqrt{1-s}}{2} , \frac{-z +\mu}{\sigma} -\frac{\sigma}{2} \right), \end{aligned}$$
(13)

with \(\rho=\sqrt{1-s}\).

In the following two subsections, we shall prove this property.

Proposition 2

Let X t  = v t + σ W t , t ≥ 0 be a drifted Brownian motion. For time t, for some 0 < s < 1, for all a and z, we would like to express the following quantity:

$${\mathbb{P}}_t \left[ { \max_{t+1-s \leq u \leq t+1} X_u \geq a ,\ X_{t+1} \leq z } \right]$$
(14)

The joint law of a drifted Brownian motion and its running maximum is well known. For example it can be find in Hull (2011), Shreve 2004), Harrison (1985).

We use the following classical result about joint law of a drifted Brownian motion and its running maximum in order to obtain the result:

Lemma 1

Let X t  = v t + σ W t , t ≥ 0 be a drifted Brownian motion. For all a > 0 = X 0 we have:

$${\mathbb{P}} \left[ { \max_{0 \leq u \leq t} X_u \geq a , X_t \leq z } \right] = \left\{\begin{array}{ll} e^{ \frac{2\mu a}{\sigma^2} } \Upphi \left( \frac{z-2a-v t}{\sigma \sqrt{t} } \right) &,\quad z\leq a\\ \Upphi\left( \frac{z-v t}{\sigma \sqrt{t}} \right) -\Upphi\left( \frac{a-v t}{\sigma \sqrt{t}} \right) + e^{ \frac{2\mu a}{\sigma^2} } \Upphi \left( \frac{-a-v t}{\sigma \sqrt{t}} \right) &,\quad z>a. \end{array}\right.$$
(15)

The case a ≤ 0 is simple because X 0 = 0, so, \({\forall z \in \mathbb{R}}\):

$${\mathbb{P}} \left[ { \max_{0 \leq u \leq t} X_u \geq 0 , X_t \leq z } \right] = {\mathbb{P}} \left[ { X_t \leq z} \right] = \Upphi\left( \frac{z-v t}{\sigma \sqrt{t}} \right).$$
(16)

Application to our problem

In our problem, we focus on the maximum over a period ]t + 1 − s ,t + 1], for some given s and t. To retrieve the above problem, one can take probabilities conditioned to X t+1−s , it is similar to shift time and space axis in order to make X t+1−s the new origin. Then the barrier value 0 that appears in Equation (16) above is from now the value of X t+1−s .

Remark 7

(Notations) Here above, we assume that we know the value of X t+1−s . For clarity, we denote:

$${\mathbb{P}}_s [A] = {\mathbb{P}}_t \left[ A / X_{t+1-s} \right],\quad A \subset \Upomega$$

and we shall use

$${\mathbb{P}}_s [A]_{\vert X_{t+1-s} = x} = {\mathbb{P}}_t \left[ A / X_{t+1-s} =x \right],\quad A \subset \Upomega.$$

One can then easily get

  • for a > X t+1−s ,

    $${\mathbb{P}}_s \left[ \max_{t+1-s \leq u \leq t+1} X_u \geq a ,X_{t+1} \leq z \right] =\left\{\begin{array}{ll} e^{ \frac{2\mu (a-X_{t+1-s})}{\sigma^2} } \Upphi \left( \frac{z-2a+X_{t+1-s}-v s}{\sigma \sqrt{s} } \right) &, z\leq a\\ \Upphi\left( \frac{z-X_{t+1-s}-v s}{\sigma \sqrt{s}} \right)&\\ -\Upphi\left( \frac{a-X_{t+1-s}-v s}{\sigma \sqrt{s}} \right) &\\ + e^{ \frac{2\mu (a-X_{t+1-s})}{\sigma^2} } \Upphi \left( \frac{-a+X_{t+1-s}-v s}{\sigma \sqrt{s}} \right) &, z>a. \end{array}\right.$$
    (17)
  • For \({a{\leq} X_{t+1-s}, \forall z \in \mathbb{R}}\), we have

    $$\begin{aligned} {\mathbb{P}}_s \left[ \max_{t+1-s \leq u \leq t+1} X_u \geq a , X_{t+1} \leq z \right] &= {\mathbb{P}}_s \left[ \max_{t+1-s \leq u \leq t+1} X_u \geq X_{t+1-s}, X_{t+1} \leq z \right]\\ &= \Upphi\left( \frac{z-X_{t+1-s} -v s}{\sigma \sqrt{s}} \right). \end{aligned}$$
    (18)

But we would like to get rid of this conditioning. So the next step is to take the integral among all possible values of X t+1−s .

Conditional law and integration

Explicitly, we have to evaluate, for all \({a\in \mathbb{R}}\) and z ≤ a:

$$\begin{aligned} {\mathbb{P}}_t &\left[ \max_{t+1-s \leq u \leq t+1} X_u \geq a ,\ X_{t+1} \leq z \right]\\ &= \int\limits_{x=-\infty}^{+\infty} {\mathbb{P}}_s \left[ \max_{t+1-s \leq u \leq t+1} X_u \geq a , \quad X_{t+1} \leq z \right]_{\vert X_{t+1-s} = x} d{\mathbb{P}}_t \left[X_{t+1-s} \leq x \right]\\ &= \int\limits_{x=-\infty}^{a} {\mathbb{P}}_s \left[ \max_{t+1-s \leq u \leq t+1} X_u \geq a ,X_{t+1} \leq z \right]_{\vert X_{t+1-s} = x} d{\mathbb{P}}_t \left[X_{t+1-s} \leq x \right]\\ &\quad+ \int\limits_{x=a}^{+\infty} {\mathbb{P}}_s \left[ \max_{t+1-s \leq u \leq t+1} X_u \geq a ,X_{t+1} \leq z \right]_{\vert X_{t+1-s} = x} d{\mathbb{P}}_t \left[X_{t+1-s} \leq x \right]\\ &= \Upxi_1 (a,z) + \Upxi_2 (a,z), \end{aligned}$$
(19)

where the expression of \({d\mathbb{P} \left[X_{t+1-s} \leq x \right], \quad x\in \mathbb{R}}\) is as follows:

$$d{\mathbb{P}}_t \left[X_{t+1-s} \leq x \right] =\frac{1}{\sigma\sqrt{1-s}} \varphi \left( \frac{x-v (1-s)}{\sigma\sqrt{1-s}} \right) dx.$$

Then it is possible to use Proposition 2.1 in (Chuang 1996) (p.83) to evaluate both integrals. Define \(\rho = \sqrt{1-s}\). We first introduce the following intermediate variables:

  • c =  − 2μ /σ2,

  • δ1 = v (1 − s),

  • \(\eta_1= \sigma \sqrt{1-s}\),

  • δ2 = 2a + v s − z,

  • and \(\eta_2= \sigma \sqrt{s}\).

We can now recognize the result of (Chuang 1996):

$$\begin{aligned} \Upxi_1 (a,z) &= {\exp}\left( \frac{2v a}{\sigma^2} \right)\left[ \Upphi\left( \frac{-v -2a +z}{\sigma } \right) \right.\\ &\left. - \Uppsi_\rho \left( \frac{-a- v (1-s)}{ \sigma \sqrt{1-s} } , \frac{-v - 2a +z }{ \sigma} \right) \right]. \end{aligned}$$

We do the same with the following intermediate variables:

  • c = 0,

  • δ1 = v (1 − s),

  • \(\eta_1= \sigma \sqrt{1-s}\),

  • δ2 = z − v s,

  • and \(\eta_2= \sigma \sqrt{s}\).

We obtain:

$$\Upxi_2 (a,z) = \left[ 1-\Upphi\left( \frac{a- v (1-s)}{\sigma \sqrt{1-s}} \right) \right] - \Uppsi_\rho \left( \frac{ -a + v (1-s) }{\sigma \sqrt{1-s}} , \frac{v -z}{\sigma} \right).$$

Appendix 2: Proof of results about expected value of next year impairment

Thanks to the decomposition introduced in the Property 1, we are able to use results about some exotic option, the Rear-End up-and-out Put Option. This option is studied in Carr and Chou (1997), Cox and Rubinstein (1985), Carr (1995), Hui (1997) for example. Here after, we present some of its characteristics.

Rear-End up-and-out Put Option

The barrier of a rear-end put option exists from an intermediate time t between the option time start 0 and the option maturity T. The value of the option at this intermediate time is the value of a up-and-out put option with maturity the time that left T − t (it will be s in our study).

  • Its price is given by

    $$\begin{aligned} P_{2uo} &= K e^{-rT} \Uppsi_\rho \left( -B(1-s), - A(T) \right) - S_0 \Uppsi_\rho \left( -B^{\prime}(1-s), - A^{\prime}(T) \right)\\ &-\left( \frac{b}{S_0} \right)^{k_1 -1} K e^{-rT} \Uppsi_{-\rho} \left( C(1-s), - D(T) \right)\\ &+ \left( \frac{b}{S_0} \right)^{k_1 -1} S_0 \Uppsi_{-\rho} \left( C^{\prime}(1-s), - D^{\prime}(T) \right), \end{aligned}$$

    where

    • \(A(t)=\frac{\ln(S_0/K) + r t}{\sigma \sqrt{t}} - \frac{\sigma\sqrt{t}}{2}, A^{\prime}(t)=A+\sigma\sqrt{t}\),

    • \(B(t)=\frac{\ln(S_0/b) + r t}{\sigma \sqrt{t}} - \frac{\sigma\sqrt{t}}{2}, B^{\prime}(t)=B+\sigma\sqrt{t}\),

    • \(C(t)=\frac{\ln(b/S_0) + r t}{\sigma \sqrt{t}} - \frac{\sigma\sqrt{t}}{2}, C^{\prime}(t)=C+\sigma\sqrt{t}\),

    • \(D(t)=\frac{\ln(b^2/S_0 K) + r t}{\sigma \sqrt{t}} - \frac{\sigma\sqrt{t}}{2}, D^{\prime}(t)=D+\sigma\sqrt{t}\),

    • \(k_1 = \frac{2\mu}{\sigma^2}\),

    and \(\rho=\sqrt{t/T}\).

  • Expectation of its payoff under P measure is

    $$\begin{aligned} P_{2uo} &= K \Uppsi_\rho \left( -B(1-s), - A(T) \right) - S_0 e^{\mu T} \Uppsi_\rho \left( -B^{\prime}(1-s), - A^{\prime}(T) \right)\\ &\quad-\left( \frac{b}{S_0} \right)^{k_1 -1} K \Uppsi_{-\rho} \left( C(1-s), - D(T) \right)\\ &\quad+\left( \frac{b}{S_0} \right)^{k_1 -1} S_0 e^{\mu T} \Uppsi_{-\rho} \left( C^{\prime}(1-s), - D^{\prime}(T) \right), \end{aligned}$$

    where

    • \(A(t)=\frac{\ln(S_0/K) + \mu t}{\sigma \sqrt{t}} - \frac{\sigma\sqrt{t}}{2}, A^{\prime}(t)=A+\sigma\sqrt{t}\),

    • \(B(t)=\frac{\ln(S_0/b) + \mu t}{\sigma \sqrt{t}} - \frac{\sigma\sqrt{t}}{2}, B^{\prime}(t)=B+\sigma\sqrt{t}\),

    • \(C(t)=\frac{\ln(b/S_0) + \mu t}{\sigma \sqrt{t}} - \frac{\sigma\sqrt{t}}{2}, C^{\prime}(t)=C+\sigma\sqrt{t}\),

    • \(D(t)=\frac{\ln(b^2/S_0 K) + \mu t}{\sigma \sqrt{t}} - \frac{\sigma\sqrt{t}}{2}, D^{\prime}(t)=D+\sigma\sqrt{t}\),

    • \(k_1 = \frac{2\mu}{\sigma^2}\),

    and \(\rho=\sqrt{t/T}\).

Application

We can now directly derive an expression of

$${\mathbb{E}}_t \left[X_{t+1}\right] ={\mathbb{E}}_t \left[(K_t-S_{t+1})^+ {\bf 1} \left\{ \max_{t+1-s\leq u\leq t+1} S_u \leq S_{t_a} \right\} \right].$$
(20)

Proposition 3

We have

$$\begin{aligned} {\mathbb{E}}_t \left[ X_{t+1} \right] &= K_t \Uppsi_\rho \left( -B, - A(K_t) \right) - S_t e^{\mu} \Uppsi_\rho \left( -B^{\prime}, - A^{\prime}(K_t) \right)\\ &\quad-\left( \frac{S_{t_a}}{S_t} \right)^{k_1 -1} K_t \Uppsi_{-\rho} \left( C, - D(K_t) \right) + \left( \frac{S_{t_a}}{S_t} \right)^{k_1 -1} S_t e^{\mu} \Uppsi_{-\rho} \left( C^{\prime}, - D^{\prime}(K_t) \right), \end{aligned}$$

where

  • \(A(K_t)=\frac{\ln(S_t /K_t) + \mu}{\sigma} - \frac{\sigma}{2}, A^{\prime}(K_t)=A(K_t)+\sigma\),

  • \(B=\frac{\ln(S_t /S_{t_a}) + \mu (1-s)}{\sigma \sqrt{(1-s)}} - \frac{\sigma\sqrt{(1-s)}}{2}, B^{\prime}=B+\sigma\sqrt{(1-s)}\),

  • \(C=\frac{\ln(S_{t_a}/S_t) + \mu (1-s)}{\sigma \sqrt{(1-s)}} - \frac{\sigma\sqrt{(1-s)}}{2}, C^{\prime}=C+\sigma\sqrt{(1-s)}\),

  • \(D(K_t)=\frac{\ln(S_{t_a}^2/S_t K_t) + \mu}{\sigma} - \frac{\sigma}{2}, D^{\prime}(K_t)=D(K_t)+\sigma\),

  • \(k_1 = \frac{2\mu}{\sigma^2}\)

and \(\rho=\sqrt{(1-s)}\).

Terms Y t+1 and Z t+1 are both easy to compute. In fact we only have to get rid of the indicator on {S t  ≤ (1 − α)\(S_{t_a}\)} to make known expressions appear. The previously used variable m t  = min(K t , (1 − α)\(S_{t_a}\)) is involved in the next computations.

For Y t+1, ∀ K t , α, we have

$$Y_{t+1} = (K_t - S_{t+1} )^+ {\bf 1} \left\{ S_{t+1} \leq (1-\alpha )S_{t_a} \right\} = (m_t - S_{t+1} )^+ + (K_t-m_t) {\bf 1}\left\{ S_{t+1} \leq m_t \right\}$$

and we obtain the following proposition.

Proposition 4

We have

$$\begin{aligned} {\mathbb{E}}_t \left[ Y_{t+1} \right] &= P (S_t , t+1 , m_t) + {\mathbb{E}}_t \left[ (K_t-m_t){\bf 1}\left\{ S_{t+1} \leq m_t \right\} \right]\\ &= -S_t e^\mu \Upphi \left( -A^{\prime}(m_t) \right) + K_t \Upphi\left( -A(m_t) \right). \end{aligned}$$
(21)

For the last term, we have

$$\begin{aligned} Z_{t+1} &= (K - S_{t+1} )^+ {\bf 1} \left\{ \max_{t+1-s\leq u\leq t+1} S_u \leq S_{t_a} \right\} {\bf 1} \left\{ S_{t+1} \leq (1-\alpha )S_{t_a} \right\}\\ &= (m - S_{t+1} )^+ {\bf 1} \left\{ \max_{t+1-s\leq u\leq t+1} S_u \leq S_{t_a} \right\} \\ &\quad+ (K-m) {\bf 1} \left\{ \max_{t+1-s\leq u\leq t+1} S_u \leq S_{t_a} \right\} {\bf 1} \left\{ S_{t+1} \leq m \right\} , \end{aligned}$$

and we obtain the following proposition.

Proposition 5

We have

$$\begin{aligned} {\mathbb{E}}_t \left[ Z_{t+1} \right] &= K_t \Uppsi_\rho \left( -B, - A(m_t) \right) - S_t e^{\mu} \Uppsi_\rho \left( -B^{\prime}, - A^{\prime}(m_t) \right)\\ &\quad-\left( \frac{S_{t_a}}{S_t} \right)^{k_1 -1} m_t \Uppsi_{-\rho} \left( C, - D(m_t) \right) + \left( \frac{S_{t_a}}{S_t} \right)^{k_1 -1} S_t e^{\mu} \Uppsi_{-\rho} \left( C^{\prime}, - D^{\prime}(m_t) \right)\\ &\quad - (K_t- m_t )\left( \frac{S_{t_a}}{S_t} \right)^{k_1 -1} \left\{ \Upphi\left( -D(m_t) \right) - \Uppsi_\rho \left( -C , -D(m_t) \right) \right\}, \end{aligned}$$
(22)

where

  • \(A(m_t)=\frac{\ln(S_t /m_t) + \mu}{\sigma} - \frac{\sigma}{2}, A^{\prime}(m_t)=A(m_t)+\sigma\),

  • \(B=\frac{\ln(S_t /S_{t_a}) + \mu (1-s)}{\sigma \sqrt{(1-s)}} - \frac{\sigma\sqrt{(1-s)}}{2}, B^{\prime}=B+\sigma\sqrt{(1-s)}\),

  • \(C=\frac{\ln(S_{t_a}/S_t) + \mu (1-s)}{\sigma \sqrt{(1-s)}} - \frac{\sigma\sqrt{(1-s)}}{2}, C^{\prime}=C+\sigma\sqrt{(1-s)}\),

  • \(D(m_t)=\frac{\ln(S_{t_a}^2/S_t m_t) + \mu}{\sigma} - \frac{\sigma}{2}, D^{\prime}(m_t)=D(m_t)+\sigma\),

  • \(k_1 = \frac{2\mu}{\sigma^2}\),

and \(\rho=\sqrt{(1-s)}\).

Appendix 3: Sensitivities of impairment probability and expectation

The following partial derivatives can be obtained and are used in the analysis:

$$\frac{\partial}{\partial x} \Uppsi_\rho \left(x,y\right) = {\exp}\left( -\frac{x^2}{2\sqrt{1-\rho^2}} \right) \Upphi\left( \frac{y-\rho x}{\sqrt{1-\rho^2}} \right),$$
(23)
$$\frac{\partial}{\partial y} \Uppsi_\rho \left(x,y\right) = {\exp}\left( -\frac{y^2}{2\sqrt{1-\rho^2}} \right) \Upphi\left( \frac{x-\rho y}{\sqrt{1-\rho^2}} \right),$$
(24)
$$\frac{\partial}{\partial a} \Uppsi_\rho \left(x(a) ,y\right) = {\exp}\left( -\frac{x(a)^2}{2\sqrt{1-\rho^2}} \right) \Upphi\left( \frac{y-\rho x(a)}{\sqrt{1-\rho^2}} \right) \times x^{\prime}(a),$$
(25)
$$\frac{\partial}{\partial a} \Uppsi_\rho \left(x ,y(a) \right) = {\exp}\left( -\frac{y(a)^2}{2\sqrt{1-\rho^2}} \right) \Upphi\left( \frac{x-\rho y(a)}{\sqrt{1-\rho^2}} \right) \times y^{\prime}(a),$$
(26)
$$\begin{aligned} \frac{\partial}{\partial a} \Uppsi_\rho \left(x(a) ,y(a) \right) = {\exp}\left( -\frac{x(a)^2}{2\sqrt{1-\rho^2}} \right) \Upphi\left( \frac{y(a) -\rho x(a)}{\sqrt{1-\rho^2}} \right) \times x^{\prime}(a)\\ &\quad+ {\exp}\left( -\frac{y(a)^2}{2\sqrt{1-\rho^2}} \right) \Upphi\left( \frac{x(a)-\rho y(a)}{\sqrt{1-\rho^2}} \right) \times y^{\prime}(a), \end{aligned}$$
(27)
$$\frac{\partial}{\partial \rho} \Uppsi_\rho \left(x,y\right) = \frac{\rho}{1-\rho^2} \left[ \Uppsi_\rho \left( x,y \right) -1 \right],$$
(28)

and

$$\begin{aligned} \frac{\partial}{\partial \rho} \Uppsi_\rho \left(x(\rho),y(\rho)\right) = \frac{\rho}{1-\rho^2} \left[ \Uppsi_\rho \left( x(\rho),y(\rho) \right) -1 \right]\\ &\quad+{\exp}\left( -\frac{x(\rho)^2}{2\sqrt{1-\rho^2}} \right) \Upphi\left( \frac{y(\rho) -\rho x(\rho)}{\sqrt{1-\rho^2}} \right) \times x^{\prime}(\rho)\\ &\quad+ {\exp}\left( -\frac{y(\rho)^2}{2\sqrt{1-\rho^2}} \right) \Upphi\left( \frac{x(\rho)-\rho y(\rho)}{\sqrt{1-\rho^2}} \right) \times y^{\prime}(\rho). \end{aligned}$$
(29)

Appendix 4: Proof of results about the distribution of next year impairment

We decompose the expression of the cumulative distribution function to obtain:

$$\begin{aligned} {\mathbb{P}}_t \left[ {\lambda_{t+1} \leq l} \right] = {\mathbb{P}}_t \left[ { \lambda_{t+1} \leq l , \lambda_{t+1}=0} \right] +{\mathbb{P}}_t \left[ { \lambda_{t+1} \leq l ,\lambda_{t+1} \neq 0} \right]\\ &={\mathbb{P}}_t \left[ {\lambda_{t+1} =0} \right] +{\mathbb{P}}_t \left[ { \lambda_{t+1} \leq l ,\lambda_{t+1} \neq 0} \right]. \end{aligned}$$
(30)

Then, we have, for K t  − l < m t ,

$${\mathbb{P}}_t \left[ { \lambda_{t+1} \leq l , \lambda_{t+1} \neq 0} \right] ={\mathbb{P}}_t \left[ { J_{t+1}} \right] - {\mathbb{P}}_t \left[ { S_{t+1} \leq K_t - l} \right].$$

Consequently, ∀ K t  − m t  < l ≤ K t , we have

$${\mathbb{P}}_t \left[ {\lambda_{t+1} \leq l} \right] = \Upphi \left( A(K_t - l) \right).$$
(31)

The second step is to study what happens when l ≤ K t  − m t . Obviously, if m t  = K t , then l can only be equal to zero, and then \({\mathbb{P}_t \left[ {\lambda_{t+1} \leq l} \right] = 1 - \mathbb{P}_t \left[ { J_{t+1}} \right]}\). Else, if m t  = (1 − α) \(S_{t_a}\), then we have

$$\left\{ K_t - l \leq S_{t+1} \right\} \cap \left\{ S_{t+1} \leq (1-\alpha) S_{t_a} \right\} = \emptyset$$

and

$$\begin{aligned} {\mathbb{P}}_t \left[ { \lambda_{t+1} \leq l , \lambda_{t+1} \neq 0} \right] &= {\mathbb{P}}_t \left[ { K_t- l \leq S_{t+1} \leq K_t , \max_{t+1-s \leq u\leq t+1} S_u \leq S_{t_a} } \right]\\ &= {\mathbb{P}}_t \left[ { S_{t+1} \leq K_t } \right] - {\mathbb{P}}_t \left[ { S_{t+1} \leq K_t - l} \right]\\ &\quad+ {\mathbb{P}}_t \left[ { \max_{t+1-s \leq u\leq t+1} S_u > S_{t_a} , S_{t+1} \leq K_t - l} \right]\\ &\quad- {\mathbb{P}}_t \left[ { \max_{t+1-s \leq u\leq t+1} S_u > S_{t_a} , S_{t+1} \leq K_t } \right]. \end{aligned}$$

We then use previous results about exotic options to conclude:

$$\begin{aligned} {\mathbb{P}}_t \left[ { \lambda_{t+1} \leq l , \lambda_{t+1} \neq 0} \right] &= \Upphi \left( A( K_t - l ) \right) - \Upphi \left( A( K_t ) \right)\\ &\quad+ \left( \frac{S_{t_a}}{S_t} \right)^{k_1 - 1} \left[ \Uppsi_\rho \left( C , D(K_t) \right) - \Uppsi_\rho \left( C, D(K_t - l) \right) \right]\\ &\quad+ \Uppsi_\rho \left( B , A( K_t) \right) - \Uppsi_\rho \left( B, A(K_t - l) \right). \end{aligned}$$

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Azzaz, J., Loisel, S. & Thérond, PE. Some characteristics of an equity security next-year impairment. Rev Quant Finan Acc 45, 111–135 (2015). https://doi.org/10.1007/s11156-014-0432-x

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Keywords

  • Equity
  • Impairment
  • IFRS
  • IAS 39
  • Available-for-sale (AFS)
  • Rear-end up-and-out put option
  • Barrier option