Re: Mental Models - Valuation/Risk Drivers of Financial Assets As Partial Derivatives.

Don't let the title scare you. This Mental Model from Calculus is extremely powerful in understanding the Drivers for Valuation and Risk for ALL Financial Assets across the Capital Structure.

I know the title is scary, but if you bear with me, I think this is the most powerful Mental Model I’ve introduced yet, because it ties together so many equivalent concepts from not just different fields but different Asset Classes from within Finance/Economics.

But first, the motivation behind finding good Mental Models…

How Mental Models Help Me

One of my earliest mentors, my girlfriend’s boss at the time (a Senior Managing Director of Institutional Equities at Bear Stearns in the early 90’s), gave me invaluable advice at the beginning of my career:

“Jobs that entail arriving at predictable, precise answers pay the least; jobs that entail distilling imperfect information to make probabilistic bets/predictions pay the most.”

This was akin to “preaching to the converted,” as I had just transitioned from a one-year role as a software programmer for the J. Aron Currency Options team at Goldman Sachs to a junior analyst/trader on the Metals/GSCI Trading Desk at Goldman/J. Aron.

This adage then shaped my own corollary that has served as a guiding principle throughout my career:

“It is better to be GENERALLY ACCURATE than to be PRECISELY WRONG.”

It is a reminder that the world of Investing/Trading cannot be modeled with precision and that having the right Conceptual Frameworks and Mental Models can help you avoid age-old Errors of Conflation and the seduction of analyses that give you the illusion of precision (staring at the Tree Roots) while risking broad inaccuracy (missing the whole Forest).

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Errors of Conflation

One of the biggest Errors of Conflation I see being made today is the mixing up of Interest-Rate/Duration Risk and Sovereign/Credit Risk. By the way, even the Ratings Agencies are culpable of this conflation, in my opinion.

Between the aggressive Fed hiking and the recent Bear Steepening leading to Long-Term USTs yielding close to 5%, there are endless comparisons of US Sovereign Credit Risk to that of EM countries.

These comparisons are just plain wrong, and I wrote a piece back in March addressing this:

This is where having the right Mental Model really helps.

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Why Calculus Matters in Financial/Economic Analysis

It’s beyond the scope of this post to teach Calculus 101, but for the purposes of what I want to convey, a brief overview of Mathematical Derivatives (vs. Financial Derivatives, which we talk about later) is in order.

A Derivative, according to the aforementioned link, can be described as:

“A derivative in calculus is the rate of change of a quantity y with respect to another quantity x…It also represents the instantaneous rate of change at a point on the function.”

This is just a hugely important concept with broad applications to Science/Engineering as well as Finance/Economics.

Application 1: Calculus

The simplest and purest application for a derivative is to calculate the Instantaneous Slope of a Quadratic Function that generates a Parabola, where:

The First Derivative of this function is denoted as:

I found this wonderful GIF from this site https://infinityisreallybig.com/ that graphically depicts what the notion of Instantaneous Slope means:

The Second Derivative is denoted as:

This constant value of the Second Derivative tells us that the Curvature of this function is the same everywhere, and since 2 is a positive number, the graph is always Concave Up (shaped like a U).

Leave it to Finance people to confuse terminology and relabel this Positive Concavity with the opposite-meaning term Positive Convexity (more on this later).

Application 2: Physics

In Physics, the simplest application of Derivatives is determining Velocity and Acceleration of an object.

For instance, if you represent the Position function of an object moving through time as s(t), the First Derivative of s(t) represents the Velocity v(t) of the object. To translate the following mathematical formula to layman’s terms, “Velocity is the First Derivative with respect to Time, or the rate of change with respect to Time, of the Position of the object”:

Acceleration, represented by a(t), is simply the rate of change of the Velocity v(t). You can either think of it as the First Derivative of Velocity:

or as the Second Derivative of Position:

So now, we see that the Mathematical concept of Slope is like the Physics concept of Velocity and that the Mathematical concept of Curvature is like the Physics concept of Acceleration.

As we will see later, these concepts are exactly analogous to the concepts of Delta/Duration vs. Gamma/Convexity when talking about Options and Bonds.

Application 3: Inflation

If you’re with me so far, we are now going to apply this concept to Finance / Economics and specifically to the topic du jour of Inflation.

Instead of a Position function, let’s talk about a Price function, s(t).

Now, what we think of as CPI, or rate of change of Price over time, is simply the First Derivative of Price, just like Velocity is the rate of change of Position over time.

When folks talk about the Rate of Change of CPI over time, that is simply the First Derivative of CPI or Second Derivative of Price over time, just like Acceleration is the First Derivative of Velocity or the Second Derivative of Price over time.

If you didn’t know what I meant by this Tweet from last year, now you do:

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Mental Model: Risk Components of Asset Prices as Partial Derivatives

Extending the Concept to Partial Derivatives, I can finally explain my Mental Model in how I think about breaking down the disparate Risk Components of Asset Prices.

So far in my examples, we have been living in a world with just one Independent variable of Time (t), in both the Position and Price functions I described.

What about a more complex example where a particular Asset might be modeled as a function of three Independent Variables, f(x,y,z)?

That’s where the concept of Partial Derivatives from Multivariable Calculus come in.

Partial Derivatives work the same way as regular Derivatives work in a single variable function, except now the other variables are held constant.

For example, the “First Partial Derivative of f(x,y,z) with respect to x” is denoted as

and simply treats the other variables y and z as constants. It is the mathematical equivalent of the term ceteris paribus, which means “all other things being held constant.”

Similarly, the “First Partial Derivative of f(x,y,z) with respect to y” is denoted as

with x and z held constant.

Applications of the Partial Derivative Mental Model in Finance

It is way beyond the scope of this post to teach Multivariable Calculus, Bond Math or Option Math, nor am I trying to be mathematically precise in the illustrations of my Mental Model (because actual pricing models are far more complex and often combine multiple esoteric techniques when you consider the likes of Vasicek Interest Rate Models, Black-Scholes, Binomial Trees, Monte Carlo Simulations, etc.).

To me, it doesn’t matter that these complex Assets might not be mathematically differentiable (ability to calculate a derivative) functions; it’s the Mental Model that is valuable. My purpose here is to simplify these frameworks into digestible Mental Models that can serve as powerful guiding principles in how to think about isolating various Risk Components of a particular Asset.

In truth, almost no Finance professionals build their own models from scratch anymore, given the plethora of off-the-shelf software packages that do the hard math, but it is critical that folks understand the underlying conceptual frameworks and variables that DRIVE these models.

Two years ago, I wrote this post to explain how the concept of Duration can be modeled as Levers.

The Mental Model I’m introducing today is more abstract and mathematical, but it is more practically useful, in my opinion.

Application 4: Modeling a Bond’s Risk Components

In my post above, I showed that the mathematical formulas for valuing a Bond or a Stock are basically the same:

Here is a Bond:

If you think of the math as a “Black Box” function, you can just express it simply as:

Value of a Bond Vb = f(C,P,r,T) = f(C,P,Rf,CS,T)

Where:

• C = Annual Coupon

• P = Principal Value

• r = Risky Discount Rate (where r = Risk-Free Rate Rf + Credit Spread CS)

• T = Valuation Period (“Maturity”)

The Partial Derivative Mental Model tells you that you can tease out the “sensitivity” or “Rate of Change” of the Bond (Vb) with respect to any of the underlying variables by taking the First Partial Derivative with respect to whatever variable you are studying while holding others constant.

For instance, the concept of Bond Duration can be conceptualized as the First Partial Derivative of Bond (Vb) with respect to the Risk-Free Rate (Rf). It is the “Velocity” of how a Bond Price changes with respect to changes in the Risk-Free Rate.

The concept of Bond Convexity can be conceptualized as the First Partial Derivative of Bond Duration with respect to Rf or the Second Partial Derivative of Bond Price with respect to Rf. It is the “Acceleration” of how a Bond Price changes with respect to changes in the Risk-Free Rate — the rate of change of the rate of change.

Notice that I have not addressed Credit Spreads yet, but in theory you can just extend the thought process to Partial Derivatives with respect to Credit Spread (CS) to gain insights into “Credit Duration” and “Credit Convexity.”

I started this missive by noting that one of the biggest Errors of Conflation I see right now in FinTwit/MSM is the Conflation of Interest-Rate Risk with Sovereign Credit Risk. It’s easy to see why people do it, because people are just focused on the Risky Discount Rate (r) when they should be looking at the specific contributions of Risk-Free Rate (Rf) and Credit Spread (CS)!

I will further add that in the current Credit Cycle, by far the biggest component shift in the overall Risky Discount Rate (r) has been the Risk-Free Rate (Rf) component and not yet the Credit Spread (CS) component.

Credit Spreads will adjust, but it takes time because of the following reason:

Application 5: Modeling a Stock’s Risk Components

Here is a Stock:

Similarly, if you abstract the math into a “Black Box,” you can express it as:

Value of a Stock Vs = f(CF,TV,i,t) = f(CF,TV,Rf,ERP,t)

Where:

• CF = Annual Cash Flow

• TV = Terminal Value

• i = Risky Discount Rate (where i = Risk-Free Rate Rf + Equity Risk Premium ERP)

• t = Valuation Period (“Maturity”)

Note how this formula looks almost identical to the one for Bonds above, and note these specific parallels:

Coupon (C) ~ Cash Flow (CF)

Principal Value (P) ~ Terminal Value (TV)

Discount Rates are in principle the same, but note that both Risky Discount Rates (r) and (i) can be further broken down into Risk-Free Rate (Rf) components + an additional spread or premium to account for the additional risk of a Bond or Stock. For a Bond, this additional premium is called a Credit Spread (CS), and for a Stock this additional premium is called an Equity Risk Premium (ERP).

Folks who are in the business of Fundamental Valuation of Stocks generally follow a far less rigorous framework than Bond Investors. I’m not going to get into it here, but I wrote a whole summary about this in my “Levers” thread above.

Here is the relevant snippet:

I think that Equity Valuation “laziness” in relying on heuristics stems from the difficulty of forecasting the Cash Flow (CF) and especially the Terminal Value (TV) components for Equities, whereas the Coupon (C) and Principal (P) component for Bonds is contractually stipulated.

Nevertheless, after reading this thread, I hope you will better understand this recent exchange I had with Bob Elliott about his comment that “there are bonds in stocks”:

Bonds and Stocks are purely artificial delineations in what I consider to be a CONTINUUM of increasingly hard-to-determine Risk Components to Valuation, where “Senior Secured” Bonds/Loans have contractually stipulated Coupons and Principal amounts and relatively small Credit Spread components whereas “Junior” Equities can have volatile Cash Flow components not to mention indeterminate Terminal Value components, both discounted by equally hard-to-estimate Equity Risk Premiums on top of the Risk-Free Rate.

In between these Capital Structure Book Ends are a whole range of “Mezzanine” securities ranging from Senior Unsecured High Yield Bonds to Convertible Bonds to Preferreds.

Again, think of the Capital Structure as a smooth Continuum of bundles of cash flows ranging from contractually defined cash flow components to “squishier and squishier” components as you descend to the junior tranches, with Equities having the “squishiest” components of all.

When I talk about “High Duration Assets” or “Short Duration Assets,” understand that I often use the terms interchangeably between Bonds and Equities — specifically because of this Continuum Framework I espouse. Just because people don’t normally talk about Duration and Convexity concepts with respect to Equities doesn’t mean they don’t exist!

Seniority does not necessarily impact Duration either. For instance, a Senior Long-Dated Zero-Coupon Bond can have a very High Interest-Rate and/or Credit Duration, whereas an Equity of a stable, unlevered, cash-flowing business might be considered to be a Short Duration Asset.

In fact, my Long-Term Oil Private Equity is such a Short Duration Asset:

When I say that I’m positioning my book to focus on being long Short Duration Assets, I want assets whose Partial Derivatives (and thus sensitivities) with respect to Risk-Free Rates are LOW. Similarly, I favor being short High Duration assets because I think these assets have high Partial Derivatives with respect to Risk-Free Rates.

Because I believe that we are in a Higher For Longer (H4L) Regime, I expect value to flow from High Duration to Low Duration Assets.

As for Asset Classes like Gold, BTC, and Crypto, the reason why I call these “Greater Fool Trades” is that they are 100% reliant on the Greater Fool for an Exit Strategy with ZERO Intrinsic Valuation Framework whatsoever.

From my “Levers” post:

I go deep on the whole topic of Exit Strategies here if you’re interested:

Application 6: Modeling an Option’s Risk Components

This Mental Model is so rich that I’m going to extend it some more.

I’m not getting into the complexities of Black-Scholes or Binomial Pricing Models, but the key point is that there are even more additional variables being introduced to the “Black Box” function:

Value of an Option Vo = f(S,K,V,Rf,D,T)

Where:

• S = Current Stock Price

• K = Strike Price of Option

• V = Volatility of Stock

• Rf = Risk-Free Rate

• D = Dividend Yield of Stock

• T = Valuation Period (“Maturity”)

If you’ve ever been mystified by the Greek Alphabet Soup that Options Traders like to spout, let me help demystify what all the “Greeks” mean, using my Partial Derivatives Mental Model:

Option Delta = First Partial Derivative of Option Value Vo with respect to Stock Price (S)

Think of “Delta” as the “Velocity” of an Option to relative to movements in the Stock Price (S). It is exactly analogous to the concept of “Bond Duration” of a Bond relative to movements in Risk-Free Rates.

Option Gamma = First Partial Derivative of Option Delta with respect to Stock Price (S) = Second Partial Derivative of Option Value Vo with respect to Stock Price (S)

Think of “Gamma” as the “Acceleration” of an Option relative to movements in the Stock Price (S). It is exactly analogous to the concept of “Bond Convexity” of a Bond relative to movements in Risk-Free Rates.

The concepts of Delta/Duration and Gamma/Convexity are so analogous that Option Traders sometimes use the terms interchangeably, which sometimes further confuses people.

If you go back up to the top of this post and reread the Math and Physics sections, you will begin to understand why these concepts are so similar.

Option “Vega” = First Partial Derivative of Option Value Vo with respect to Volatility (V)

Option Rho = First Partial Derivative of Option Value Vo with respect to Risk-Free Rate (Rf)

Application 7: Modeling a Convertible Bond’s Risk Components

Now I get to the security within the Capital Structure upon which I built my hedge fund career — the Convertible Bond.

This diagram is taken from an old presentation from my old fund and shows diagrammatically how a Convertible Bond in its simplest form resembles a Bond + Option, although the truth is far more complex when you layer in Put/Call Features, Make-Whole Features, Change-of-Control Features, and a whole panoply of bells and whistles that often get thrown into the mix:

Again, it’s useful just to “Black Box” the Valuation if only to think about the DRIVERS of Valuation:

Value of a Convertible Bond Vcb = f(C,P,S,K,V,Rf,CS,D,T)

Where:

• C = Annual Coupon

• P = Principal Value

• S = Current Stock Price

• K = Conversion Price (“Strike”)

• V = Volatility of Stock

• Rf = Risk-Free Rate

• CS = Credit Spread

• D = Dividend Yield of Stock

• T = Valuation Period (“Maturity”)

It is precisely my involvement with Convertible Bonds that led me to this Partial Derivative Mental Model that I have now shared with you.

I gravitated to Convertible Bonds as an Asset Class and made them the centerpiece of my Capital Structure Arbitrage trades for nearly two decades because their innate complexity often led to mispricings across one or multiple Risk Component dimensions.

Because they touch upon every corner of the Capital Structure Continuum (from Bonds to Stocks to Options on Stocks), I loved to identify corners of hidden value where I could isolate and potentially hedge out certain Risk Components while keeping exposure to others.

Again, the mathematicians and quants among you may quibble with how I sometimes mix terminology across Asset Classes, but it all stems from my mantra of seeking to be “Generally Accurate than Precisely Wrong.” I find that having Mental Models from other disciplines allows me to attack problems from a First Principles Perspective and think out-of-the-box.

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Conclusion

My goal for this piece was to demystify a lot of Financial Jargon and explain how many different fields and even Asset Classes within the same field of Finance have conceptual commonality and equivalency.

First Derivatives are used to describe Rates of Change (“Velocity”), and Second Derivatives are used to describe Rates of Change of Rates of Change (“Acceleration”) as well as “Curvature” and “Convexity.”

Partial Derivatives just refer to the same applications of Derivatives — except on more complex functions and Assets that are driven by multiple variables.

This Partial Derivatives Mental Model forces me to think about the underlying DRIVERS of Value across the Capital Structure and position myself according to how my Macro views favor or disfavor certain Risk Components.

Hopefully, you can see its utility and application to almost every corner of Finance and Economic Analysis even if you don’t remember Calculus!