## Motivation: why work out valuations vs buying the index

Passive indexes provide investment benchmarks that are simple to replicate (the largest position being in the companies with the largest float) and limit rebalancing needs (as a market capitalization rule implies never needing to trade). However, such a passive approach implies that the market weights are best, which is a strong efficient market assumption.

We saw such market capitalization rules being egregiously wrong in the past:

- the Japanese stock market in the late 80s had more value than the US market, it is now 10x smaller than the US market. The valuation did not make sense based on fundamentals, and with hindsight, we can ascribe the inefficiency to the availability of leverage.
- TSLA, a stock that never made money and produces fewer cars than incumbent car producers, is priced in 2020 as if there was any possibility of maintaining a moat in the automobile industry. The jury is still out on this one, but it seems clear that shorting TESLA is much more dangerous than just omitting it from a portfolio.

We will review some fundamental pricing considerations aiming to put a reasonable price on growth stock, and will apply this to the stock market. You can read more about value investing in this section.

## Fundamental Pricing Methodology

If the annual discount rate required by stock investors is $y_a$, a company that can produce a steady flow of annual dividend $D$ coming from earning per shares $E$ with payout ratio $f$ is valued as:

$P = D ((1+y_a)^{-1} + (1+y_a)^{-2} + ... )= D /y_a = f E / y_a$

An alternative continuous paying dividend version justifies using a continuous discount $y$ rather than an annual one.

$P = \int_0^{+\infty} {D \exp(-yt) dt} = D /y = f E / y$

In practice, the continuous yield is related to the annual one by $y = \ln(1+y_a)$. The taylor rule ensures that the two values are similar for small enough $y$. In the domain of interest, we see $y_a=6\%$, $y=5.83\%$ and $y_a=10\%$, $y=9.53\%$, so there is little difference compared to the uncertainty in stock returns. We point this out to illustrate the robustness of the formula and the suitability of continuous rates.

The adequate level for the return $y$ and its range is discussed here. We will use for $y$ a yield of $5\%$ above inflation.

The payout ratio $f$ is such an important component that it deserves its own discussion there , let's say that an optimist will set it at 70% and a pessimist at 30%, a perfectionist (like Buffet or Graham) not only will compute $f$ but also will compute their own earnings measure. The main objection to the above formula is that earnings per share change. They are subject to revenue growth, margin changes, and corporate share issuance policy. To determine the likely evolution of the earnings per share, we point out that for a stable business, revenue $R$ grows with some volatility and a margin $m$ which depends on the business type can be extracted from the revenue. The number of shares $S$ is typically constant, but may be increased if the managers are extracting value from the company by granting themselves stocks or stock options. The earning per share is time dependent, given by:

$$E_t = f R_t m_t / S_t$$

We then can compute the mean and standard deviation of :

- annual revenue growth rate $g=\ln(R_{i+1}/R_i))$
- net margin rate $m_i$
- shares dilution rate $d=\ln(S_{i+1}/S_i))$

Following Lindy Law, we'll posit that the company will have the same growth, margin and dilution for the next 5 years as in the previous 5 years, after that, the earnings are assumed to mean revert for the next 5 year to a no growth. Assuming growth for 7.5 year and suddenly no growth after that is a much simpler but similar hypothesis.

For international comparison, and when inflation fluctuates, revenue growth $g$ can to be tempered by fx and CPI variation. One simplest way to do this is to adjust the $g$ by the $\ln \textrm{fx}$ variation for the corresponding period, and by the $\ln \textrm{CPI}$ variation if we want to $y$ to be a premium above inflation. This corresponds to the simple assumption that fx and CPI move by steps between statement periods, though the assumption not very realistic if FX and CPI are very volatile. Including CPI and FX helps determine which companies have pricing power and are able to grow above inflation.

We use log returns because it allows compounding rates by averaging them. We assume that the company keeps same distribution and buyback policy until date $T$ as we keep $d$ constant. The payout ratio $f_d$ corresponds solely to dividend yield (reduced by the applicable withholding tax), and there is no growth and dilution after the horizon date $T$, and a discount rate $y$ from 0 to $T$. We introduce the rate net of growth and dilution $r:= y-g-d\ln fx-d\ln cpi+d$.

- the future fair price is given by $P_T = f m R_0 \exp((g+dlnfx+dlncpi-d)T) / y $.
- the value of the dividends distributed between 0 and the terminal valuation date $T$ is given by $\int_0^T f_d m R_0 \exp(-r t) dt$.

The present fair price to sales ratio is given by adding the two terms above:

$$P_0/R_0 = f m \frac{\exp(-r T)}{y} + f_d m \frac{1-\exp(-r T)}{r}$$

Therefore, the current price to revenue ratio $P/R$ (aka price to sales) can be fundamentally judged against an assumption about payout ratio $f$, after tax dividend ratio $f_d$, margin $m$, growth $g$, dilution $d$ and discount yield $y$.

Using a shorter time $T$ growth period will lead to our favoring value stocks, whereas a longer time will favor growth stocks, provided dilution is low.

We can also compute a stressed value by shifting $g,d,m$ by a certain number $a$ of standard deviations, this does away with much of the handwaving present in the fundamental litterature, that earnings should demonstrate some stability. The payout fraction $f$ can be either be taken for what it is or optimistically assumed to converge with the market standard of 28%

This valuation formula can be further complicated by taking into account the uncertainty of $r$, resulting in a positive adjustment for uncertainty of $r$, whereas the formula is linear w.r. to on $m$ or $f$, and uncertainty will average out.

## Some real world application in May 2022

For this first study, we look at 1800 stocks from US, Japan, Europe and Australia, use last 5 years of account based on interactive brokers financial data. We used $f=1$ and $y=6$%.

## Update in April 2023

Given the strength of the USD in the last 5 years, adjusting by the fx leads to an adjustment down of revenue growth by 2% to 3% for stocks in EUR and JPY, and higher( 7% for stocks with accounts denominated in less strong currencies such as Colombian pesos). The CPI adjustment due to US is 3.5% over the 5Y as US inflation was 2% most of the year and recently grew to 8% and 5%. The US CPI number is "official" which means that it is subject to manipuation, and is probably to low compared to reality.

### Results by Sector

We observe the following:

- tech has high growth and low earning yield as predicted with low capital intensity business
- utilities and basic materials have high earning yield and low growth
- energy is highly cyclical
- financials have high earning yield

sector | count | earning_yield | growth | margin | dilution | ratio | sratio |

Basic Materials | 125 | 6.46% | 4.93% | 7.30% | -0.14% | 164.14% | 5.61% |

Consumer Cyclicals | 270 | 5.09% | 3.89% | 5.55% | -0.32% | 109.78% | 9.44% |

Consumer Non-Cyclicals | 152 | 4.04% | 2.15% | 5.20% | -0.15% | 66.80% | 18.57% |

Education | 3 | 2.45% | -0.08% | 10.16% | 0.04% | 23.37% | 0.02% |

Energy | 59 | 5.02% | 12.38% | 1.39% | 2.03% | 6.67% | -1.42% |

Financials | 217 | 8.48% | 3.38% | 22.35% | -0.36% | 212.59% | 32.24% |

Healthcare | 175 | 3.33% | 8.64% | 9.87% | 0.54% | 67.74% | 12.08% |

Industrials | 306 | 4.25% | 5.26% | 6.66% | -0.10% | 96.05% | 19.80% |

Real Estate | 130 | 3.39% | 5.40% | 29.39% | 2.99% | 36.93% | 1.28% |

Technology | 301 | 2.83% | 10.02% | 7.73% | 0.27% | 67.04% | 5.65% |

Utilities | 68 | 4.26% | 4.07% | 10.31% | 1.12% | 70.47% | 7.91% |

### Results by Region

Looking at markets by region:

- eu and aus seem to have a growth problem, they are expensive
- us has highest growth, jp has great bargains

region | count | earning_yield | growth | margin | dilution | ratio | sratio |

aus | 59 | 2.16% | -12.14% | 10.86% | 0.57% | 9.69% | 0.08% |

eu | 285 | 3.66% | -6.69% | 8.04% | 0.00% | 20.94% | 0.21% |

jp | 493 | 5.86% | 2.60% | 5.80% | -0.08% | 124.63% | 34.38% |

us | 966 | 4.03% | 8.84% | 9.14% | 0.25% | 86.50% | 9.42% |

### Results by Region and Sector

It is harder to make conclusion on region and sector, but the gist of it is that the trends seen above seem confirmed: eu and aus are expensive due to growth concerns, japan has best value.

The full table of results by region and sector can be found here

### Example Portfolio

The following portfolio was constructed by after filtering for each sector from lower than median dilution, better than median consensus ranking and price appreciation target and selecting the best 10 stressed fundamental valuation ratio amongst those. This is not investment advice of course, just for research purpose:

The portfolio selected following the criteria can be found here.

## Conclusions and Next Steps

The algorithm can find cheap companies relative to the market except in the energy and utility sector. One might decide to underweight utilities, which have government-regulated output costs and still get some exposure to energy extraction and mining given the inflationary outlook.

The following information is relevant to better understand the structure of a company's finances:

- whether margin is high
- whether R&D is high
- high debt low margin is a typical low moat pattern
- LT Debt history as indicia of serial aquirer
- current ratio = current asset/current liability needs to be high and not moving down
- turnover = sales / total asset is better if high

### Tax Effect on Dividends

After reading more books on value investing, it appears that I should refine earning estimates to be owner earnings rather than whatever management threw at the GAAP or IFRS.

For foreign investors in US and Australian stock with no tax treaty, the dividends are worth 70%. WHT information can be found on this pwc site. For Hong Kong residents:

- NL, UK, MX, SG, HK 0 WHT
- IT, FR, SP, NE, JP 10% WHT
- CAN: 15%
- GER: 25%
- AUS, US 30%

Arguably, there are two payout ratios: one that compute the fundamental value of the stock to one's tax situation through holding it, and another one for the average market participant, in which case he is computing a fair average market value. Buying a stock on the Canadian exchange halves the withholding tax compared to buying it on a US exchange.

### Transaction Costs

Transaction costs also need be amortized over the trade life, here are the costs for USD10,000 transactions with Interactive Brokers:

- UK: 50 dollar per trade (0.50%), levied on any transaction size in GBP
- France: 5 dollar + 32 dollar of tax (0.38%)
- Hong Kong: 156HKD for HKD75000 (0.22%)
- Mexico: 115 persofor 115000 (0.10%)
- Japan: 800 yen per million (0.08%)
- Swiss, Germany, NL: 5 dollar per trade (0.05%)
- Can: 3.39 dollar per trade (0.03%)
- US: 1 dollar per trade (0.01%)

We see that UK, France, Hong Kong transaction costs charge a Tobin-like tax.

The tax situation of the owner needs to be considered, the US has much better liquidity and lower transaction but for a resident in a non tax treaty foreign country, the dividend withholding is 30% and 40% of the stock value is taxed away in the unfortunate event of death.

### Update 20230305 Some criterias for stock selection

- analyst avg consensus < 3 - a consensus of 1 is buy, 2.5 is hold and 5 is sell.
- analyst avg target price > 1.2 market price, analysts are optimist, this is an easy critera
- model target price > 1.4 market price, model uses actual price ratio
- growth > 0, only companies growing revenue, which is a low bar given inflation
- dilutiuon < 5%, only companies that do not significantly dilute shareholders
- stressed margin > 0, company that manage to remain profitable is a show of moat
- payout ratio > 0.25, this is a marker of shareholder alignment, ratio is computed over several years
- cfmargin > 1/2 margin, cashflow from financing needs should be equal to earning in the long run if the company is stable. We tolerate a 50% difference in case the company invests to expand
- stressed cf margin > -1.05, stressed cf margin is a metric checking whether the company embarks on sudden large investment of a size similar to its revenue.
- growth > asset growth - 1%, make sure company revenue growth is better than the balance sheet growth.
- turnover > 0, ensure revenue is positive.

### Psychotic Allocation?

In the above study, we used an equity hurdle rate of 6%, the analytics penalized financials volatility with a factor $a=-1$ and assumed a payout ratio $f=0.6$ in buyback only. In fact, different assumptions can be used to obtain different allocation depending on the level of optimism underlying the assumptions.

For instance, a portfolio of high growth tech can be obtained by using a positive growth factor $a$ and a high value $f$ later. This would lead to a subportfolio that can own Google or Facebook.

A value portfolio can be obtained using a conservative portfolio $a<0$ with $f$ using actual real world after tax values.