Single-Stage and Multistage Residual Income Valuation

Say you need to evaluate two different companies: Springfield Electronics, a mature company with relatively stable cash flows, and Interception Security Company, a more recent startup with volatile cash flows. Two pretty different situations, wouldn't you say? Do you think Springfield and Interception should be forecast the same way?

Yes! Well actually, those could both be right. It really depends on how residual income develops. In some cases, Springfield and Interception might actually use the same valuation model, especially if residual income grows consistently. But under some other circumstances, Interception and Springfield could use different models.

In fact, a single-stage residual income model might actually be better for Interception than Springfield because Interception might achieve more consistent growth over time. The single residual income model equation is $$\displaystyle V_0 = B_0 + \frac{ROE - r}{r-g}B_0$$. And the inputs are straightforward. It's book value, ROE, the required return, the required return, and the growth rate. $$\displaystyle \frac{ROE - r}{r-g}B_0$$ So that equation represents the additional value expected because Interception can generate returns in excess of its cost of capital.

But the single-stage model has more than just one use: it can also calculate the growth rate. If you know Interception's book value and the required rate of return and ROE, you could solve for the growth rate. So it's a multipurpose equation. Interception's growth rate could be pretty high because it's a new company, so you'd expect it to continue growing at a sustainable rate over time. But say that Interception's ROE equals its required rate of return. What would the value be compared to the book value?

No. A higher growth rate would decrease the denominator, so the excess returns could be higher.

No, actually. That's a natural assumption, but in this case the required rate of return equals ROE.

Exactly! If ROE equals the required rate of return, then the excess returns over Interception's cost of equity are zero. So book value would equal the intrinsic value. And that's a natural drawback of the single-stage model—it assumes that ROE will grow indefinitely. But think back to Springfield and Interception. Most likely, Interception would have the more consistently growing ROE, while Springfield would have a ROE that reverts to its required rate of return. Why do you think that's the case?

No.

Bingo!

Higher returns on equity will bring in more competition. So for a mature company like Springfield, you'd expect that residual income would fade over time. And that's where a multistage residual income model comes in.

Just like the DDM model and free cash flow models, the multistage residual income valuation model forecasts residual income for a certain time period and then estimates a terminal value based on continuing residual income. __Continuing residual income__ is residual income after the forecast horizon. However, in many cases, a mature company's ROE will approach its cost of equity. What do you think makes up most of Springfield's value in a multistage residual model?

That's not it. Dividends are a use of funds and aren't a part of the residual income calculation.

No. The terminal value typically gets closer to 0 as ROE decreases towards the cost of equity.

You got it! Typically, Springfield's book value will make up most of the residual income valuation. So that's different than the DDM and free cash flow models, where the terminal value makes up a large portion of the total valuation. But other similarities exist between all the valuation models, namely assumptions. Some of the key estimates include how residual income will change over time, from indefinitely positive to 0. Or if residual income will gradually decrease as ROE decreases to a mean level.

Say that Springfield retains a premium over book value for a short-term valuation period. In this case, if the valuation period were longer, the premium would most likely be lower due to the ROE convergence. But in this case, say the valuation period is five years. If so, the intrinsic value calculation is $$\displaystyle V = B_{0} + \sum^{T}_{t=1}\frac{(ROE_t - r)B_{t-1}}{(1+r)^t} + \frac{P_T - B_T}{(1+r)^T}$$. Where the final component equals the premium (P) over book value (B) at the end of the forecast.

But say that Springfield's ROE fades over time, eventually hitting the cost of equity. In this case, the formula would use a persistence factor, which ranges between 0 and 1, where 1 indicates that ROE will continue indefinitely. $$\displaystyle V_0 = B_0 + \sum^{T-1}_{t=1}\frac{(E_t - rB_{t-1})}{(1+r)^t} + \frac{E_T - rB_{T-1}}{(1+r - \omega)(1+r)^{T-1}}$$ And the persistence factor is represented by ω. If the persistence factor were 0, how would that impact the initial residual income forecast?

No. The persistence factor of 0 won't lower the initial forecast value.

Not quite. A factor of 0 won't increase the initial value.

Bingo! A persistence factor of 0 wouldn't impact the initial forecast value at all. Remember that the persistence factor is included in the second phase, where ROE declines to the required rate of return. So for Springfield, it probably runs closer to 0 than 1, while Interception's would probably be closer to 1 since the company is trying to grow. Simply put: persistence factors can vary between companies.

In summary: [[summary]]

No way!

Of course!

Equal

Lower

Higher

If ROE is higher, more companies will exit the industry

If ROE is higher, more companies will enter the industry

Book value

Terminal value

Dividend cash flow value

It would be lower

It would be higher

It would be the same

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