From Ron Longhofer
I asked Perplexity to create a formula for RI (residual income) valuation that incorporated a terminal growth rate that declined over time. I asked it to use a persistence factor, and alternatively a fade rate. Here is what it came up with:
Using a Persistence Factor
Certainly! Let’s calculate the intrinsic value using a combination of a terminal growth rate and a persistence factor without using coding format.
### Example Calculation
Assume: Book value at time 0 (B0) = $12 – Cost of equity (r) = 10% – Terminal growth rate (g) = 3% – Persistence factor (ω) = 0.4
Residual incomes for years 1-5 are as follows:
RI1 = $1.92, RI2 = $2.15, RI3 = $2.40, RI4 = $2.70, RI5 = $3.02
- **Calculate the present value of residual incomes for years 1-5**:
Year 1: $1.92 / (1.1) = $1.745 – Year 2: $2.15 / (1.1)^2 = $1.779 – Year 3: $2.40 / (1.1)^3 = $1.804 – Year 4: $2.70 / (1.1)^4 = $1.842 – Year 5: $3.02 / (1.1)^5 = $1.875
Sum of present values of residual incomes for years 1-5 = $1.745 + $1.779 + $1.804 + $1.842 + $1.875 = $9.045
- **Calculate the terminal value at the end of year 5**:
Terminal Value = RI5 * (1 + g) / (r – g + ω) – RI5 = $3.02 – g = 3% or 0.03 – r = 10% or 0.10 – ω = 0.4 Terminal Value = $3.02 * (1 + 0.03) / (0.10 – 0.03 + 0.4) = $3.1106 / 0.07 = $44.437
- **Discount the terminal value back to present value**:
PV of Terminal Value = $44.437 / (1.1)^5 = $44.437 / 1.61051 = $27.59
- **Calculate the intrinsic value**:
Intrinsic Value (V0) = B0 + Sum of present values of residual incomes for years 1-5 + PV of Terminal Value – B0 = $12 – Sum of present values of residual incomes = $9.045 – PV of Terminal Value = $27.59 Intrinsic Value = $12 + $9.045 + $27.59 = $48.635
Thus, the intrinsic value of the company is approximately $48.64.
Using a Fade Rate
Certainly! I’ll modify the calculation to use a fade rate instead of a persistence factor to reach the same intrinsic value. We’ll use the same initial data but introduce a fade rate for the terminal growth.
Given: – Book value at time 0 (B0) = $12 – Cost of equity (r) = 10% – Residual incomes for years 1-5: RI1 = $1.92, RI2 = $2.15, RI3 = $2.40, RI4 = $2.70, RI5 = $3.02
Instead of using a persistence factor, we’ll use: – Initial terminal growth rate (g) = 5% – Fade rate (f) = 0.5% per year
- Present value of residual incomes for years 1-5:
Sum = $1.745 + $1.779 + $1.804 + $1.842 + $1.875 = $9.045 (This remains the same as in the previous calculation)
- Calculate the terminal value using the fade rate:
Terminal Value = RI5 * (1 + g) / (r – g + f) = $3.02 * (1 + 0.05) / (0.10 – 0.05 + 0.005) = $3.171 / 0.055 = $57.655
- Discount the terminal value to present value:
PV of Terminal Value = $57.655 / (1.1)^5 = $35.79 4. Calculate the intrinsic value: Intrinsic Value = B0 + PV of RIs + PV of Terminal Value = $12 + $9.045 + $35.79 = $56.835
The intrinsic value using the fade rate approach is $56.835, which is higher than the $48.635 we calculated using the persistence factor. This difference is due to the different assumptions about long-term growth embedded in each method. To reach the same value as before ($48.635), we would need to adjust the initial growth rate or fade rate.
Let’s try adjusting the initial growth rate: If we use an initial growth rate (g) of 3% instead of 5%, keeping the fade rate at 0.5%: Terminal Value = $3.02 * (1 + 0.03) / (0.10 – 0.03 + 0.005) = $3.1106 / 0.075 = $41.475 PV of Terminal Value = $41.475 / (1.1)^5 = $25.75 Intrinsic Value = $12 + $9.045 + $25.75 = $46.795
This value is much closer to our original calculation. The remaining small difference is due to the different ways the two methods (persistence factor vs. fade rate) model the decline in residual income over time.
Does Perplexity have it right?
Well, Perplexity can follow a formula! But as the book maintains: Beware of formulas; formulas are for fools. Or, in this chapter: Valuation models are not for valuation. We cannot expect a formula to deliver the “true” intrinsic value. Perplexity does it by assumption: Assume a persistence factor and/or assume a fade rate for residual income. But where does that come from: speculation, guessing? We can get any valuation by making the assumption that fits, as Perplexity does in reconciling the two valuations. Persistence and fade rate (with non-persistence) are the same thing, just different assumptions.
Valuation formulas are to direct our thinking. And Perplexity has some of that thinking: Value in the future will be determined by the persistence or fade of terminal residual income. However, that has to be evaluated. The approach in Chapter 4 does not assume a fade rate but rather first understands the market’s assessment of future residual income (growth) and then understands if that pricing is appropriate, bringing fundamental analysis to the task (in subsequent chapters). The market’s fade rate can be imputed but note that this is the fade in the growth rate of residual income, not the fade in residual income.
Perplexity’s valuation raises the question: Can AI really do valuation? Or must it have the prop of a formula to plug into?