Perhaps the analyst benchmark is not a good one? Analysts are known to be sometimes biased in their forecasts. Perhaps a better benchmark might be forecasts from a focused financial statement analysis. That would be interesting to see. As would the ability of AI to forecast the amount of earnings and not just the direction.

See Kim, A., M. Muhn, and V. Nilolaev. 2024. Financial Statement Analysis with Large Language Models. Chicago Booth Research Paper. At https://ssrn.com/abstract=4835311.

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

1. **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

2. **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

3. **Discount the terminal value back to present value**:

PV of Terminal Value = $44.437 / (1.1)^5 = $44.437 / 1.61051 = $27.59

4. **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

1. 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)

2. 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

3. 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?

Here is a question asked of ChatGPT in May 2024:

Let’s say hypothetically that ROE is 12% and the cost of equity is 9%. The difference is 3%. Let’s say the retention rate is 75%. 75% of 3% is 2.25%. Is 2.25% the sustainable growth rate of residual income?

Here is ChatGPT’s answer:

In the scenario you’ve described, the 2.25% is not the Sustainable Growth Rate (SGR) of residual income; rather, it’s the growth rate of the residual income itself. Here’s why:

The SGR is calculated using the formula: SGR=ROE × Retention Ratio

In your hypothetical case:

The ROE is 12%. The Retention Ratio is 75%.

So the SGR would be:

SGR = 12%×75% = 9%

The SGR indicates how much a company can grow sustainably without needing additional financing, assuming it reinvests its retained earnings at the current ROE.

Now, the 3% you’ve calculated is the excess return over the cost of equity, which represents the additional value created over and above the required return by shareholders. When you multiply this excess return by the retention ratio (75%), you get 2.25%, which represents the growth rate of the residual income, not the SGR.

The growth rate of residual income is essentially how much the residual income can grow each year, given the company’s ability to generate earnings above the cost of equity and the proportion of those earnings it chooses to retain and reinvest. So in your example, the residual income can grow at a rate of 2.25% per year, based on the excess return and the retention ratio.

Does ChatGPT have it right?

ChatGPT does quite a good job but not get it quite right. Retention does not necessarily add to residual earnings. It’s a question of how the retained earnings are invested. If, for example, the firm can only find investments to earn at the cost of capital, there is no growth in residual income (value added). The answer assumes that reinvestment is at the same ROE as is currently being earned. Further, for a growth rate for the future, it assumes that the firm will perpetually earn at the current ROE and that all the firm must do is retain so it will automatically grow residual earnings.

To be fair, ChatGDP states this assumption…“assuming it reinvests its retained earnings at the current ROE.” But this a very special case which almost never applies; ROE changes over time; often new investment earns at a lower rate than existing investment. The appropriate analysis involves a forecast of investment opportunities and the ROE that will be earned from that investment. It is a common mistake to assume ROE is a permanent feature of firms.

There are parallels in scientific endeavor.

Think of cosmology. Scientists are interested in the course of our universe in the future, indeed many universes, and can speculate about that…the existential question, maybe. But what they look at is what is happening now and how it is changing. That guides forecasting of future changes. And so, conducting fundamental analysis of firms provides indications of what they might look like in the future.

Think of forecasting the effect of climate change in the future. Observation of what is happening currently is a starting point. Or go back to the past when there was a significant increase in temperatures (50-55 million years ago) and examine the fossilized sea life in the ocean sediments. Scientists find that that changed the atomic composition (oxygen isotopes) of sea life: A basis for prediction of the future.

Here is a report on investing in China using a residual income approach. It is quite interesting as it challenges market prices with a no-growth valuation. That was the approach of Benjamin Graham who refused to pay for growth, rather investing in stocks where the price was less than a no-growth valuation. In mature stock markets, such stocks are rare these days so one has to deal with buying growth⸺the key focus of the book. The Chinese stock market has been subject to big swings, potentially with mispricing. Could it be that a no-growth valuation can identify mispricing there? The report suggests so.

In the study, the no-growth valuation is:

is calculated with forward consensus sell-side analysts’ earnings forecasts provided by Tushare (tushare.pro) which has additional accounting and trading data necessary for the calculation. This value is compared to price in a V/P ratio after each quarterly report date. The strategy then invests in the top 10% of V/P ratios from all Chinese listed stocks for which earnings forecasts are available, equally weighted, with the portfolio rebalanced at each report date.

The cumulative returns from this strategy (before transaction costs) from 2010 to mid-2024 are in the figure below. The benchmark return is that for the HS300 index for the largest stocks on Shanghai Stock Exchange and the Shenzhen Stock Exchange.

Source for all exhibits: Yan, Oliver. Does Value Investing Work in China? Masters thesis, Columbia Business School, 2024. With permission.

Here are the calendar-year returns from the strategy compared to the benchmark:

While the strategy beat the benchmark in most years, it did not in the three years, 2018‒2020. 2020 was a bull market, perhaps a period with price momentum where value strategies are typically challenged. However, with the subsequent drop in the market in 2021, the strategy performed relatively well. And the comparative returns in 2014 indicates resilience of the strategy in up markets with the subsequent drop in 2015 less than the market. There was a significant drawdown in 2011 and 2015 but considerably less than that for the benchmark.

The following plots returns for V/P ratios with a regression line drawn through the observations. For V/P greater than 1.5, there are some very high returns though some with negative returns. This calls for further analysis to sort out these firms.

Value investors invest with are margin of safety, so only invest in stocks with a V/P greater than 1.2 and apply some fundamental features: the firm has positive net income and positive net income growth and exclude small firms with a market cap less than $2 billion RMB. Here are the returns:

There is some improvement in the returns. This shows promise of improving returns by pressing on further with the fundamental analysis in subsequent chapters.

To get us going, we have the building block diagram in February 2024 on this webpage to add to that in June 2023 in chapter 4. And to that we have added the pricing in mid-December 2024. We maintain the 10% required return and base the no-growth valuation on the trailing EPS at the time. With a 10 for 1 stock split in 2024, the per-share numbers are now post-split. Fiscal-year end is January of each year. (One can also work this with the no-growth value based on the forward EPS).

A few points to note from a Financial Times Lex piece, December 11, 2024:

• EPS for 2025 is forecasted by analysts to be 20.2% below that for 2024.
• Revenue for 2025 was 114% above that for 2024.
• Operating profit margin (before tax) for 2024 was 62.4%, more than Apple, Microsoft, Alphabet (Google) and Intel have achieved in the last 25 years.
• China announced an antitrust investigation into the firm in early December 2024
• In January 2025, DeepSeek AI arrived with less expensive chips, challenging U.S. AI and NVIDIA
• U.S. is restricting sales of NVDIA’s chip to China
• Bank of America is forecasting that NVIDIA will account for $223 billion of the $300 billion chip market by 2027
• NVIDIA’s customers, including Amazon, Google, and Meta Platforms are developing their own chips for AI.

These are only points to direct you to dig deeper. The moat seems deep and wide at this point, but let’s see how things work out.

There is another feature to add to the tracking: The accounting for the no-growth component of the market price. Based on anchoring information, this is an accounting without speculation about growth. If the no-growth valuation is maintained over time, but the market’s pricing of growth declines, that is worth investigating. On the other hand, if the market’s pricing of growth increases but the no-growth valuation declines, that waves a red flag. Referring to the building block diagram, the ratio of the no-growth component to market’s growth component is a metric that can be employed.

In January 23, 2024, on the announcement of fiscal-year 2023 results, the stock price stood at $492.19, edging back to the January 2022 price and an increase of 180% over the June 2022 price. Nice pickings. The firm had added an estimated 13.1 million subscribers in the 4^th quarter of 2023 and 8.8 million in the 3^rd quarter, ending the year with 260.28 million subscribers. On January 27, 2025 when the 2024 10-K was filed, the stock price stood at $971.89. Too pricey?

You might want to track it, getting the market’s pricing of growth and implied growth rate.

In valuation models, growth is incorporated by capitalizing the valuation attribute (for example, residual earnings) by r – g. However, if growth is risky, r (the required return for risk) is related to g (expected growth) so, if g is higher, r is also likely to be higher. Never treat r and g as independent inputs in valuation.

You see this relationship operating in chapter 4, but in a different way. To finesse the fact that we don’t know r, we advised to carry out the reverse engineering exercise with a relatively high r. That builds in a margin of safety. You will have noticed that the higher the r, the higher the g that comes out of the reverse engineering. So, you see the r and g (growth and risk) relationship. In this context, a higher hurdle rate, r, requires higher growth to meet the hurdle. If you deem that growth can be achieved, you have a margin of safety.

From Penman, S., R. Gutsche, L. Gorkun-Voevoda, and A. Rif. Buying Risky Growth and an Explanation of the Returns to Trading on P/E Ratios. Journal of Portfolio Management, April 2025. © 2025 PMR. All rights reserved.

Be careful in interpreting this growth pricing: The returns could be due to mispricing growth but not so fast: The returns could also be due to the risk of buying growth. It looks like low SG is an underpricing that yields higher returns and high SG is overpricing that yields lower returns, but this could be due to risk that requires a higher return. The standard risk metrics—the return standard deviations and betas—indicate that the returns are not due to risk, but a value investor remains skeptical of these metrics. More fundamental analysis is needed to challenge the market’s pricing of growth. See particularly chapters 7 and 8.

Note that SG is related to E/P and chapter 2 informs that E/P (and P/E) is determined by both growth and risk. So, the higher returns to buying high E/P (low P/E) might be returns to mispricing but might also be due to risk: Price in the E/P ratio is discounted for risk to yield a higher return as reward for that risk.