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# T-model

In finance, the T-model is a formula that states the returns earned by holders of a company's stock in terms of accounting variables obtainable from its financial statements.[1] The T-model connects fundamentals with investment return, allowing an analyst to make projections of financial performance and turn those projections into a required return that can be used in investment selection. Mathematically the model is as follows:

${\displaystyle {\mathit {T}}={\mathit {g}}+{\frac {{\mathit {R}}OE-{\mathit {g}}}{{\mathit {P}}B}}+{\frac {\Delta PB}{PB}}{\mathit {(}}1+g)}$
where ${\displaystyle T}$ = total return from the stock over a period (appreciation + "distribution yield" — see below);
${\displaystyle g}$ = the growth rate of the company's book value during the period;
${\displaystyle PB}$ = the ratio of price / book value at the beginning of the period.
${\displaystyle ROE}$ = the company's return on equity, i.e. earnings during the period / book value;

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#### Transcription

You may have heard of Alan Turing, father of computer science and the mathematical genius who cracked the German Enigma Code during World War II -- a feat that Winston Churchill once remarked was “the secret weapon that won the war.” But what you probably didn’t know was that during his short lifetime, Alan Turing also proposed a theory to help explain a series of patterns that recur over and over again throughout the natural world -- like the spherical organization of cells in an embryo, the spiral arrangement of petals on a flower, the whorled tentacle pattern of a Hydra, the waves on a sand dune, the spots on a leopard and even the stripes on a zebra. Turing’s desire to understand nature’s recurring patterns stemmed from his fascination with Embryology. Turing wanted to know how the small, uniform ball of cells in an embryo could differentiate and morph into a fully formed, complex being. His hunch was that there had to be some mathematical principle underpinning the recurring patterns in an embryo’s development. So in 1952, Turing published a paper called “The Chemical Basis of Morphogenesis”. Within it, he proposed that the diversity of patterns we see in nature can be explained by a mathematical model, called the “reaction-diffusion system”. Essentially the system can be broken down like this: let’s say we have two identical cells within an embryo. In the mix are two chemicals that can either activate or inhibit a specific reaction within an embryo’s cells. Turing called these “morphogens.” As these morphogens diffuse through the embryo they causing the cells around them to transform, ultimately creating patterns like spots, stripes, spirals, hexagons and whorls. But this is just one example of the “reaction-diffusion system” at work. Morphogens can really be any two opposing components that work together to stop and start a reaction, like chemicals, genes or proteins. Morphogens can really be any two chemical substances that work together to stop and start a reaction, like hormones, proteins or acids. And changing the rate at which these components interact, diffuse and decay determines the way those elemental patterns like waves, spots and stripes appear. Today, many theoretical biologists and mathematicians believe that Turing's system could also be applied to the patterns found in the vegetation on a landscape, weather systems, and even to the formation of galaxies. Sadly, Turing never found out whether his theory was right. In an age of intolerance, he took his own life in 1954, following a conviction for “gross indecency”, the charge for being openly gay. And for a long time afterward his ambitious model was forgotten. But in the sixty years that have passed, some experimental data has to emerged to prove that Turing really was onto something. Perhaps the biggest breakthrough to date is a 2012 study that applied Turing’s model to the formation of digits in the paws of mouse embryos. It turns out that of digits are, on a fundamental level, a series of stripes. During the early stages of development within an embryo, the paw or hand is a continuous plate of tissue. But over time, the cells change to either form digits, or to die and create the spaces between them. What researchers found was that the entire process fit the reaction-diffusion system proposed by Turing. actually conducted by three morphogens - which in this case were three genes. In this case, two genes controlled the production of morphogens that formed digits, while a third gene controlled the production of morphogens that caused cell death, forming the gaps between digits. In other words, those three genetic pathways, working in opposition to each other, create the “stripes” that are digits and gaps. What researchers called, the perfect example of Turing’s “reaction-diffusion system”. Of course, more research is needed to determine whether Turing’s model could really be applied to all the patterns that we see in the natural world. But at the very least, Alan Turing should be remembered as a mathematically visionary who changed the way we see our world. Another trailblazing mathematician called Ada Lovelace, became the world’s first computer programmer. Watch this episode to learn more about how Ada is partly responsible for all the amazing computer tech we rely on today. Thanks for watching and don’t forget to subscribe.

## Use

When ex post values for growth, price/book, etc. are plugged in, the T-Model gives a close approximation of actually realized stock returns.[2] Unlike some proposed valuation formulas, it has the advantage of being correct in a mathematical sense (see derivation); however, this by no means guarantees that it will be a successful stock-picking tool.[3]

Still, it has advantages over commonly used fundamental valuation techniques such as price–earnings or the simplified dividend discount model: it is mathematically complete, and each connection between company fundamentals and stock performance is explicit, so that the user can see where simplifying assumptions have been made.

Some of the practical difficulties involved with financial forecasts stem from the many vicissitudes possible in the calculation of earnings, the numerator in the ROE term. With an eye toward making forecasting more robust, in 2003 Estep published a version of the T-Model driven by cash items: cash flow, gross assets and total liabilities.

Note that all "fundamental valuation methods" differ from economic models such as the capital asset pricing model and its various descendants; fundamental models attempt to forecast return from a company's expected future financial performance, whereas CAPM-type models regard expected return as the sum of a risk-free rate plus a premium for exposure to return variability.

## Derivation

The return a shareholder receives from owning a stock is:

${\displaystyle (2){\mathit {T}}={\frac {\mathit {D}}{\mathit {P}}}+{\frac {\Delta P}{P}}}$

Where ${\displaystyle {\mathit {P}}}$ = beginning stock price, ${\displaystyle \Delta P}$ = price appreciation or decline, and ${\displaystyle {\mathit {D}}}$ = distributions, i.e. dividends plus or minus the cash effect of company share issuance/buybacks. Consider a company whose sales and profits are growing at rate g. The company funds its growth by investing in plant and equipment and working capital so that its asset base also grows at g, and debt/equity ratio is held constant, so that net worth grows at g. Then the amount of earnings retained for reinvestment will have to be gBV. After paying dividends, there may be an excess:

${\displaystyle {\mathit {X}}CF={\mathit {E}}-{\mathit {D}}iv-{\mathit {g}}BV\,}$

where XCF = excess cash flow, E = earnings, Div = dividends, and BV = book value. The company may have money left over after paying dividends and financing growth, or it may have a shortfall. In other words, XCF may be positive (company has money with which it can repurchase shares) or negative (company must issue shares).

Assume that the company buys or sells shares in accordance with its XCF, and that a shareholder sells or buys enough shares to maintain her proportionate holding of the company's stock. Then the portion of total return due to distributions can be written as ${\displaystyle {\frac {{\mathit {D}}iv}{\mathit {P}}}+{\frac {{\mathit {X}}CF}{\mathit {P}}}}$. Since ${\displaystyle {\mathit {R}}OE={\frac {\mathit {E}}{{\mathit {B}}V}}}$ and ${\displaystyle {\mathit {P}}B={\frac {\mathit {P}}{{\mathit {B}}V}}}$ this simplifies to:

${\displaystyle (3){\frac {\mathit {D}}{\mathit {P}}}={\frac {{\mathit {R}}OE-{\mathit {g}}}{{\mathit {P}}B}}}$

Now we need a way to write the other portion of return, that due to price change, in terms of PB. For notational clarity, temporarily replace PB with A and BV with B. Then P ${\displaystyle \equiv }$ AB.

We can write changes in P as:

${\displaystyle {\mathit {P}}+\Delta {\mathit {P}}=({\mathit {A}}+\Delta {\mathit {A}})({\mathit {B}}+\Delta {\mathit {B}})\,={\mathit {A}}B+{\mathit {B}}\Delta {\mathit {A}}+{\mathit {A}}\Delta {\mathit {B}}+\Delta {\mathit {A}}\Delta {\mathit {B}}\,}$

Subtracting P ${\displaystyle \equiv }$ AB from both sides and then dividing by P ${\displaystyle \equiv }$ AB, we get:

${\displaystyle {\frac {\Delta P}{P}}={\frac {\Delta {\mathit {B}}}{\mathit {B}}}+{\frac {\Delta {\mathit {A}}}{\mathit {A}}}\left({\mathit {1}}+{\frac {\Delta {\mathit {B}}}{\mathit {B}}}\right)}$

A is PB; moreover, we recognize that ${\displaystyle {\frac {\Delta {\mathit {B}}}{\mathit {B}}}={\mathit {g}}}$, so it turns out that:

${\displaystyle (4){\frac {\Delta P}{P}}={\mathit {g}}+{\frac {\Delta PB}{PB}}{\mathit {(}}1+g)}$

Substituting (3) and (4) into (2) gives (1), the T-Model.

## The cash-flow T-model

In 2003 Estep published a version of the T-model that does not rely on estimates of return on equity, but rather is driven by cash items: cash flow from the income statement, and asset and liability accounts from the balance sheet. The cash-flow T-model is:

${\displaystyle {\mathit {T}}={\frac {{\mathit {C}}F}{\mathit {P}}}+{\boldsymbol {\Phi }}g+{\frac {\Delta PB}{PB}}{\mathit {(}}1+g)}$

where

${\displaystyle {\mathit {C}}F=cashflow\,}$ ${\displaystyle {\mbox{(net income + depreciation + all other non-cash charges),}}\,}$

and

${\displaystyle {\boldsymbol {\Phi }}={\frac {{\mathit {M}}ktCap-grossassets+totalliabilities}{{\mathit {M}}ktCap}}}$

He provided a proof [4] that this model is mathematically identical to the original T-model, and gives identical results under certain simplifying assumptions about the accounting used. In practice, when used as a practical forecasting tool it may be preferable to the standard T-model, because the specific accounting items used as input values are generally more robust (that is, less susceptible to variation due to differences in accounting methods), hence possibly easier to estimate.

## Relationship to other valuation models

Some familiar valuation formulas and techniques can be understood as simplified cases of the T-model. For example, consider the case of a stock selling exactly at book value (PB = 1) at the beginning and end of the holding period. The third term of the T-Model becomes zero, and the remaining terms simplify to: ${\displaystyle {\mathit {T}}={\mathit {g}}+{\frac {{\mathit {R}}OE-{\mathit {g}}}{1}}=ROE}$

Since ${\displaystyle {\mathit {R}}OE={\frac {\mathit {E}}{{\mathit {B}}V}}}$ and we are assuming in this case that ${\displaystyle {\mathit {B}}V={\mathit {P}}\,}$, ${\displaystyle {\mathit {T}}={\frac {\mathit {E}}{\mathit {P}}}}$, the familiar earnings yield. In other words, earnings yield would be a correct estimate of expected return for a stock that always sells at its book value; in that case, the expected return would also equal the company's ROE.

Consider the case of a company that pays the portion of earnings not required to finance growth, or put another way, growth equals the reinvestment rate 1 – D/E. Then if PB doesn't change:

${\displaystyle {\mathit {T}}={\mathit {g}}+{\frac {{\mathit {R}}OE-{\mathit {R}}OE(1-D/E)}{{\mathit {P}}B}}}$

Substituting E/BV for ROE, this turns into:

${\displaystyle {\mathit {T}}={\mathit {g}}+{\frac {D}{\mathit {P}}}}$

This is the standard Gordon "yield plus growth" model. It will be a correct estimate of T if PB does not change and the company grows at its reinvestment rate.

If PB is constant, the familiar price–earnings ratio can be written as:

${\displaystyle {\frac {\mathit {P}}{\mathit {E}}}={\frac {{\mathit {R}}OE-{\mathit {g}}}{{\mathit {R}}OE({\mathit {T}}-{\mathit {g}})}}}$

From this relationship we recognize immediately that P–E cannot be related to growth by a simple rule of thumb such as the so-called "PEG ratio" ${\displaystyle {\frac {{\mathit {P}}/E}{g}}}$; it also depends on ROE and the required return, T.

The T-model is also closely related to the P/B-ROE model of Wilcox[5]