A short paper arguing how to evaluate CLO equity performance. As the equity tranche of a CLO has a maturity date, we can consider that as a bond with indeterministic coupon. So it is natural to use IRR as a measure for equity. IRR is the solution of \(r\) in

\[\sum_j \frac{p_j}{(1+r)^{t_j}} = 0\]where \(p_j\) is the payment (negative if from investor) and \(t_j\) is the time measured in years. IRR measures the yield amounts to break-even PV of cash flows.

Problems of IRR:

- different IRR for different compounding period
- market rate affects rate of return but IRR can’t distinguish the attribution of yields
- In case of zero or negative profit: IRR give more negative \(r\) for shorter term of recovery
- e.g., zero profit, recover whole investment in 1 yr vs 10 yr are both IRR 0%
- e.g., loss, recover 50% of investment in 1 yr vs 10 yr. 10 yr has less negative IRR

Alternative measure is “return of exposure” (ROX). The paper claims that it is similar to discount margin

\[ROX = \sum_{j=0}^N z_j\frac{p_j}{\phi_0 D}\]where \(z_j\) are zero coupon discount factors, \(\phi_0\) is exposure par amount, \(D\) is *spread duration*,

with \(\phi_j\) the remaining par of the investment at time \(t_j\). ROX is the PV of all payments received (\(j=1,\cdots,N\)) minus the PV of all funded investments (\(j=0\)), divided by the sum-product of exposure par amount and duration. If the investment return if LIBOR-flat (i.e., LIBOR+0), ROX is zero.

The metric this paper proposed is “CLO Sharpe” (CLO#). Defined as follows: Excess return \(\xi_j\) (for \(j=1,\cdots,N\)) is actual payments \(p_j\) minus the LIBOR interest on outstanding residual par

\[\xi_j = p_j - \psi_{j-1}L_{j-1}(t_j-t_{j-1})\]with \(\psi_0=\phi_0\), \(\phi_j=\phi_{j-1}-\xi_j\) for \(j=1,\cdots,N\)

The proposed volatility measure:

\[CLO\# = \frac{\textrm{mean}(\xi_j)}{\textrm{StdDev}(\xi_j)}\]## Bibliographic data

```
@article{
title = "Better measurements for CLO equity performance",
author = "Joseph M. Pimbley",
journal = "The Journal of Structured Finance",
volume = "Summer",
year = "2016",
pages = "24--30",
}
```