Residual Income Valuation with Term Rates

Article Quant Q&A · Author: Mild_Thornberry

Summary

The document asks how the residual income valuation model should change when the cost of equity varies over time. It begins from the constant-rate formulation, where discounting and the residual income adjustment use the same rate, and considers why the familiar telescoping identity may not hold when that rate becomes time dependent.

The proposed adjustment uses the period’s forward rate in the residual income numerator and the corresponding spot rate in the discount denominator. The author states that this formulation preserves the telescoping series that sums to zero, allowing residual income valuation, and the related abnormal earnings growth approach, to accommodate a term structure. The document presents the result as a mathematical resolution but supplies no worked numerical example or independent validation, so implementation details and assumptions about rate conventions are not explored.

Key ideas

A time-varying cost of equity changes the residual income valuation identity.
The proposed numerator uses the relevant forward rate for the period.
The denominator discounts each period using its spot rate.
The author claims this pairing preserves the telescoping sum for residual income valuation.
The discussion gives no numerical example or broader validation of the formulation.

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# Residual Income Valuation with Term Structure


# Residual Income Valuation with Term Structure












I'm implementing a residual income model (RIM) to value stocks as described by Ohlson.

https://pdfs.semanticscholar.org/c0a5/4ef41311951fe406d15cd7d7ce19502cdc7c.pdf

The key to this model is described on page 330:

"Equating RIV to PVED depends only on a simple scheme that requires no reference to economic or accounting concepts...

$$0 = y_0 + R^{-1}(y_1-Ry_0) +R^{-2}(y_2 - Ry_1)+...$$

I've seen this implemented with time-varying rates once before in a recent article (note that they use $coe_t$ (cost of equity at time t):

https://www.fondazioneoiv.it/wp-content/uploads/2019/06/Residual-Income-Model.pdf

However, I wonder if the scheme presented in this work requires an adjustment, since introducing a term structure of rates turns $R$ into $R_t$, and the telescoping equation above does not necessarily equal 0. Does anyone have any insight on what the adjustment term (if necessary) should be? If it's not necessary, then what exactly does the remainder represent?

EDIT:

I figured out the math. It is still possible to use RIM (and AEG for that matter) with time-varying rates. This is the classic presentation of RIM

$$V_0 = \sum\limits_{t=0}^{\infty} \frac{D_t + BV_t - (1+k)BV_{t-1}}{(1+k)^t} $$

But in the modified version, $k$ in the numerator needs to be the forward rate $f$ at year t while the $k$ in the denominator needs to be the spot rate $s$ for year t.

$$V_0 = \sum\limits_{t=0}^{\infty} \frac{D_t + BV_t - (1+f_{t,1})BV_{t-1}}{(1+s_t)^t} $$

This still yields the telescoping time series that is equal to 0.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Using Forward and Spot Rates in Residual Income Valuation

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