# Multi-Stage Residual Income

Hello,
Can some one explain what am I doing wrong in the question. I have written the equation to solve this question in the end:

Question: Castovan’s final assignment is to determine the intrinsic value of TTCI using both a single-stage and a multistage RI model. Selected data and assumptions for TTCI are presented in Exhibit 2.

Exhibit 2. Tantechi Ltd. (TTCI) Selected Financial Data and Assumptions

|Book value per share|€45.25|
|Market price per share|€126.05|
|Constant long-term ROE|12.00%|
|Constant long-term earnings growth rate|4.50%|
|Cost of equity|8.70%|

For the multistage model, Castovan forecasts TTCI’s ROE to be higher than its long-term ROE for the first three years. Forecasted earnings per share and dividends per share for TTCI are presented in Exhibit 3. Starting in Year 4, Castovan forecasts TTCI’s ROE to revert to the constant long-term ROE of 12% annually. The terminal value is based on an assumption that residual income per share will be constant from Year 3 into perpetuity.

Exhibit 3. Tantechi Ltd. (TTCI) Forecasts of Earnings and Dividends

                  ||**Year 1**|**Year 2**|**Year 3**|


|Earnings per share (€)|7.82|8.17|8.54|
|Dividends per share (€)|1.46|1.53|1.59|

Beckworth questions Castovan’s assumption regarding the implied persistence factor used in the multistage RI valuation. She tells Castovan that she believes that a persistence factor of 0.10 is appropriate for TTCI.

Question:

Based on Exhibits 2 and 3 and the multistage RI model, Castovan should estimate the intrinsic value of TTCI to be closest to:

1. €54.88.

2. €83.01.

3. €85.71.

Answer: I got Book Value as B0 = 45.25 , B1 = 51.61 B2 = 58.25 and B3 = 65.2

Then I did

Bo + Σ (EPS – rBt)/(1+r) + (ROE- r)Bt4/r*(1+r)^4… assumed w(persistence as 1)

=45.25 +3.57 +3.11+2.7 + (0.12-0.087)65.2/(1.087)^4(0.087)

What am I doing wrong??

CFAI book EOC question 33 pg 541.

Your residual income from Year 1 to 3 is wrong (however, your computed book values are correct).

RI(1) = 7.82 - (8.7\% \times 45.25) = 3.88325
RI(2) = 8.17 - (8.7\% \times 51.61) = 3.67993
RI(3) = 8.54 - (8.7\% \times 58.25) = 3.47225

From Year 4 and onwards, the residual income will be 3.47225 (same as Year 3), so the terminal value at Year 3 is:

TV(3) = \frac{RI(4)}{r} = \frac{3.47225}{0.087} = 39.91092

Intrinsic value
= 45.25 + \frac{3.88325}{1.087^1} + \frac{3.67993}{1.087^2} + \frac{(3.47225 + 39.91092)}{1.087^3}
= 85.71

Alternative:
Since the perpetuity starts from Year 3 and onwards, you can calculate the TV at Year 2 as:

TV(2) = \frac{RI(3)}{r} = \frac{3.47225}{0.087} = 39.91092

Intrinsic value
= 45.25 + \frac{3.88325}{1.087^1} + \frac{(3.67993 + 39.91092)}{1.087^2}
= 85.71

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Thanks for the reply. This makes sense.

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Mmm, if the terminal value starts from Year 3, why is the denominator required rate of return raised to the power 2 and not 3. Starting from Year 3 implies at the end of Yr 3.