The Impact of Downward Rating Momentum

Abstract

Rating downgrades are known to make subsequent downgrades more likely. We analyze the impact of this “downward momentum” on credit portfolio risk and bond portfolio management. Using Standard&Poor’s ratings from 1996 to 2005, we apply a novel approach to estimate a transition matrix that is sensitive to previous downgrades and contrast it with an insensitive benchmark matrix. First, we find that, under representative economic conditions, investors who rely on insensitive transition matrices underestimate the momentum-sensitive Value-at-Risk (VaR), on average, by 107 basis points. Second, we show that bond portfolio managers should use our downgrade-sensitive probabilities of default as they seem to be better calibrated than momentum-insensitive estimates.

Appendices

Appendix I: General modeling framework

Let $\left\{ {1,...,N} \right\}$ be the set of ratings where 1 denotes the best rating and N the default state, which we formally consider a rating. Time t is measured in discrete periods, which are 5 days long in our application. For a given time t (in periods), let τ _t be the time evolved since the last rating change if that was a downgrade and if it is not more than ε periods ago. If more time has elapsed since or if the last transition was an upgrade or if the rating was stable throughout, set ${\tau_t} \equiv {\text{nex}}$, where the abstract element nex represents the non-excited state. To reconcile rating downward momentum with the Markov property, we formally extend the state space from $\left\{ {1,...,N} \right\}$ to $\left\{ {1,...,N} \right\} \times \left\{ {1,...,\varepsilon, {\text{nex}}} \right\}$, where $\left\{ {1,...,\varepsilon, {\text{nex}}} \right\}$ denotes the set of possible values for τ _t . On the extended state space, we observe the process (R _t , τ _t ). We call a realization at time t excited if τ _t  ≤ ε and non-excited otherwise. To specify the distribution of the process, we assume the Markov property

$$ \Pr \left( {\left( {{R_{t + 1}},{\tau_{t + 1}}} \right) \in {A_{t + 1}}\left| {\begin{array}{*{20}{c}} {\left( {{R_t},{\tau_t}} \right) \in {A_t},} \hfill \\ {\left( {{R_{t - 1}},{\tau_{t - 1}}} \right) \in {A_{t - 1}},...} \hfill \\ \end{array} } \right.} \right) = \Pr \left( {\left( {{R_{t + 1}},{\tau_{t + 1}}} \right) \in {A_{t + 1}}|\left( {{R_t},{\tau_t}} \right) \in {A_t}} \right), $$

(6)

which has to hold for deliberate sets ${A_{t + 1}},{A_t},{A_{t - 1}},... \subset S$, and define rating transition probabilities separately for excited and non-excited ratings.²⁰ In addition to rating transitions, there are transitions of τ _t . This variable can be thought as a timer that starts to tick after a downgrade and rings when the excited rating is to be set back to normal state. This mechanism is implemented by the following formal steps. Let $\lambda_{i,j}^{nex}$ and $\lambda_{i,j}^{ex}$ be real numbers between 0 and 1 for i ≠ j such that $\lambda_{i,i}^{ex} \equiv \sum\limits_{j \ne i} {\lambda_{i,j}^{ex}}$ and $\lambda_{i,i}^{nex} \equiv \sum\limits_{j \ne i} {\lambda_{i,j}^{nex}}$ lie between 0 and 1, too. These numbers are one-period transition probabilities. Their precise meaning is as follows. We set

$$ \Pr \left( {{R_{t + 1}} = j,{\tau_{t + 1}} = 1|{R_t} = i,{\tau_t} = {\text{nex}}} \right) = \lambda_{i,j}^{nex}\;{\text{for}}\;j > i, $$

(7)

which defines the downgrade probability of a non-excited rating (the timer τ _t is reset and switched on);

$$ \Pr \left( {{R_{t + 1}} = j,{\tau_{t + 1}} = 1|{R_t} = i,{\tau_t}\kern1.5pt<\kern1.5pt\varepsilon } \right) = \lambda_{i,j}^{ex}\;{\text{for}}\;j > i $$

(8)

(downgrade of an excited rating; the—already ticking—timer is reset to 1 and ticks on);

$$ \Pr \left( {{R_{t + 1}} = j,{\tau_{t + 1}} = {\text{nex}}|{R_t} = i,{\tau_t}\kern1.5pt<\kern1.5pt\varepsilon } \right) = \lambda_{i,j}^{ex}\;{\text{for}}\;j\kern1.5pt<\kern1.5pti $$

(9)

(upgrade of an excited rating; timer is switched off);

$$ \Pr \left( {{R_{t + 1}} = j,{\tau_{t + 1}} = {\text{nex}}|{R_t} = i,{\tau_t} = {\text{nex}}} \right) = \lambda_{i,j}^{nex}\;{\text{for}}\;j\kern1.5pt<\kern1.5pti $$

(10)

(upgrade of a non-excited rating; timer remains off). Furthermore, expiration after ε excited periods (timer rings) is implemented by

$$ \Pr \left( {{R_{t + 1}} = i,{\tau_{t + 1}} = {\text{nex}}|{R_t} = i,{\tau_t} = \varepsilon } \right) = 1 - \lambda_{i,i}^{ex}. $$

(11)

The probability of persistence in an excited rating (timer is ticking), is given by

$$ \Pr \left( {{R_{t + 1}} = i,{\tau_{t + 1}} = {\tau_t} + 1|{R_t} = i,{\tau_t}\kern1.5pt<\kern1.5pt\varepsilon } \right) = 1 - \lambda_{i,i}^{ex} $$

(12)

and, finally, for a non-excited rating (timer off throughout), by

$$ \Pr \left( {{R_{t + 1}} = i,{\tau_{t + 1}} = {\text{nex}}|{R_t} = i,{\tau_t} = {\text{nex}}} \right) = 1 - \lambda_{i,i}^{nex}. $$

(13)

Equations 11, 12, 13 do not follow from the preceding equations but ensure that all other formally possible transition probabilities are zero.

The maximum-likelihood estimator for one-period transition probabilities has been defined by (2) in the main text.

Appendix II: Efficient calculation of long-term transition matrices

The goal of this exercise is the calculation of momentum-sensitive transition matrices over a horizon of M periods. We first present how the distribution of $\left( {{R_{t + 1}},{\tau_{t + 1}}} \right)$ is efficiently obtained from the distribution of $\left( {{R_t},{\tau_t}} \right)$. Let $p_{i,s}^t$ denote this time-t distribution, where 1 ≤ i ≤ N and $s \in \left\{ {1,...,\varepsilon, {\text{nex}}} \right\}$. Define, furthermore,

$$ p_i^{t,ex} \equiv \sum\limits_{i = 1}^\varepsilon {p_{i,s}^t} . $$

(14)

The distribution of the next period is then obtained from (7) through (13), which can be simplified to

$$ p_{i,1}^{t + 1} = \sum\limits_{j\kern1.5pt<\kern1.5pti} {\left( {p_j^{t,ex}\lambda_{j,i}^{ex} + p_{j,{\text{nex}}}^t\lambda_{j,i}^{nex}} \right)\;{\text{for}}\;{\text{2}} \leqslant i \leqslant N,} $$

(15)

which covers downgrades;

$$ p_{i,{\text{nex}}}^{t + 1} = \sum\limits_{j > i} {\left( {p_j^{t,ex}\lambda_{j,i}^{ex} + p_{j,{\text{nex}}}^t\lambda_{j,i}^{nex}} \right)} + p_{i,{\text{nex}}}^t\left( {1 - \lambda_{i,i}^{nex}} \right) + p_{i,\varepsilon }^t\left( {1 - \lambda_{i,i}^{ex}} \right)\;{\text{for}}\;{\text{1}} \leqslant i \leqslant N, $$

(16)

which covers upgrades (left-hand sum), plus persistence of non-excited ratings (middle term) and expiration (right-hand term), and, finally,

$$ p_{i,s}^{t + 1} = p_{i,s - 1}^t\left( {1 - \lambda_{i,i}^{ex}} \right)\;{\text{for}}\;1 \leqslant i \leqslant N\;{\text{and}}\;2 \leqslant s \leqslant \varepsilon, $$

(17)

which covers persistence in an excited rating while τ _t is shifted.

For the transition over M periods, this calculation is repeated M times. To obtain a full M-period matrix, the whole procedure has to be performed for each initial distribution concentrated on a single initial state.

In total, the calculation requires floating point operations at the order of N ² ε ² M, given that N should be substantially lower than ε. In contrast, a brute-force matrix multiplication would require floating point operations at the order of N ³ ε ³ M.