# Asset Space We start with **asset space**, $\mathbb{A} = \mathbb{R}^n$, with basis vectors $e_1, \dots, e_n$ being *assets*. Suppose we have a vector $v$ in this space. The $i$-th coordinate of $v$ tells us "how much along asset $is axis". There are two objects that live in this space: *returns* and *weights*. However, they each have their own specific *geometry*. Remember, [Geometry](Geometry.md) is just a rule for measuring lengths, angles and distances on a space. Formally, this is just an [Inner Product](Inner%20Product.md) ([Metric](Metric.md)) that induces a norm and angle formula. # The Geometry of Returns, $R$ Returns, $R$, is just a random vector in asset space. We can draw individual points $r \sim R$. These are our observations. We keep the asset space fixed—the coordinates and basis vectors don’t change—but we can change the geometry we put on it. In particular, the return geometry can be isotropic or anisotropic depending on the covariance structure. Anything living in asset space must be given a geometry (often we are unaware we are doing this as it is implicit). We could give $R$ a euclidean geometry. But there are problems with that. To show why we need to introduce the quantity *risk*. We can define risk as follows. We start with a point cloud of returns $r \sim R$. Then, pick a direction $u \in \mathbb{A}$. This direction defines a line through the entire space—a linear combination of assets multiplied by a scalar. The risk along the direction $u$ is the variance of the projected $rs onto the line defined by $u$. We can see that risk is [Anisotropic](Anisotropic.md)—it varies by direction. Different directions have different risk.  This presents a problem for euclidean geometry. Why? Because in an anisotropic space, equivalent euclidean distances do not equal equivalent risk. Because euclidean geometry implicitly assumes that all directions are [Isotropic](Isotropic.md) with respect to the quantities you care about. In our plot above, moving $1$ unit along the $u_1$ direction will have the same euclidean distance as moving $1$ unit along $u_3$. But these $1$ unit euclidean steps don't represent $1$ unit steps in risk! Given the nature of our problem, it would be very useful to give $R$ a different geometry: a *risk adjusted geometry* where a $1$ unit step in any direction represents a $1$ unit change in *risk*. In other words, a geometry that was [Isotropic](Isotropic.md) in risk. But how can we do that? It is helpful to think about our target (an isotropic geometry), and how we'd get from that to the anisotropic geometry we find ourselves in. If we know how to from isotropic to anisotropic, we can inverse that transformation to move in the opposite direction. Below we can this in action. There is some matrix $M$ that takes us from isotropic to anisotropic, and $M^{-1}$ takes us back. But what are $M$ and $M^{-1}$?  ###### Isotropic $\to$ Anisotropic The matrix $M$ is the [Linear Transformation](Linear%20Transformations.md) that takes points in asset space from an isotropic return geometry (where covariance is $I$ and "risk units" are equivalent to euclidean units) into our actual anisotropic return geometry (where our covariance is $\Sigma$). Mathematically, we start with $z \in \mathbb{R}^n$ drawn from an isotropic distribution: $\mathbb{E}[z] = 0, \quad \mathrm{Cov}(z) = I$ We want to produce $r \in \mathbb{R}^n$ whose covariance is $\Sigma$: $\mathrm{Cov}(r) = \Sigma$ If we let $r = Mz$, then: $\mathrm{Cov}(r) = M \, \mathrm{Cov}(z) \, M^\top = M M^\top$ We can set the previous two equations equal to each other: $MM^\top = \Sigma$ From which we can show: $M = \Sigma^{1/2}$ Here $\Sigma^{1/2}$ means the *matrix square root*—the unique symmetric positive-definite matrix such that $\Sigma^{1/2}(\Sigma^{1/2})^\top = \Sigma$. It is computed from the eigen decomposition $\Sigma = Q\Lambda Q^\top$ by replacing each eigenvalue $\lambda_i$ with its scalar square root $\sqrt{\lambda_i}$. The key idea is this: > In order to move from an isotropic geometry with covariance $I$ to an anisotropic geometry with covariance $\Sigma$, you transform your isotropic geometry by $\Sigma^{1/2}$. ###### Anisotropic $\to$ Isotropic How do get out of this anisotropic geometry and into the isotropic one? For that we just need to apply $M^{-1}$: $z = M^{-1}r$ For us, $M^{-1} = \Sigma^{-1/2}$. And thus, applying $M^{-1}$ to our anisotropic returns will *whiten* them: $\mathrm{Cov}(z) = \Sigma^{-1/2} \, \Sigma \, \Sigma^{-1/2} = I$ The key idea is this: > In order to move from an anisotropic geometry with covariance $\Sigma$ to an isotropic geometry with covariance $I$, you transform the anisotropic geometry by $\Sigma^{-1/2}$. ###### Transforming $R$ from Anisotropic to Isotropic via $\Sigma^{-1/2}$ We now have a way to take our anisotropic geometry and make it isotropic: transforming it by $\Sigma^{-1/2}$. We can see this below. We transform our points to be isotropic. We then project them to any direction and see that the variance of the projection is the same. This change of coordinates doesn’t just rescale axes — it _changes the underlying geometry_ of the space so that Euclidean distances now measure risk directly, in “1 risk unit = 1 Euclidean unit” form.  We can summarize what we have done so far as follows. We transformed our points into a risk adjusted, isotropic geometry via the $\Sigma^{-1/2}$ transform. # The Geometry of weights, $w$ Weights, $w$, are deterministic vectors in $\mathbb{A}$. Although $w$ lives in the same asset space as $R$, it has a different geometry. For $w$, the natural notion of length is *portfolio risk*—the standard deviation of $w^\top R$. We can derive this geometry as follows. Given $R$ with a covariance matrix $\Sigma$, the portfolio return is: $r_p = w^\top R$ With variance: $\mathrm{Var}(r_p) = w^\top \Sigma w$ This variance defines an inner product on asset space: $\langle u, v \rangle_{\Sigma} := u^\top \Sigma v$ And the induced risk norm: $\|w\|_{\Sigma} := \sqrt{w^\top \Sigma w}$ is exactly the portfolio’s volatility (it's standard deviation). ###### This Geometry is Anisotropic Just like in return geometry, this risk geometry is generally anisotropic. The unit ball $\{w : w^\top \Sigma w = 1\}$ is an ellipsoid in asset coordinates. Directions in which $\Sigma$ has large eigenvalues are _high-variance_: the ellipsoid is short in those directions — small changes in w produce large changes in risk. Directions with small eigenvalues are _low-variance_: the ellipsoid is long in those directions — larger weight changes are needed to reach the same risk. Let's put this another way. Changing the direction of $w$ is anisotropic with respect to the function $\text{Var}(r_p)$. Different directions of $w$ will have different *risk*. We'd like each direction of $w$ be isotropic with respect to risk. MISSING LINK! The ellipses are with respect to variance of portfolio returns, but wrt different geometries / objects in the space (R and it's geometry vs w and it's geometry). But variance of portfolio returns is the common link - The R-geometry ellipse uses \Sigma^{-1} because we’re measuring risk distance **between return vectors**. - The w-geometry ellipse uses \Sigma because we’re measuring **risk produced by a weight vector**. - They’re **duals**: same eigenvectors, reciprocal axis lengths. - If return risk is large along some axis, the weight risk contour is small there (and vice versa). --- # TODO * Does the optimizer look for w in isotopic (sigma -1/2) geometry, or the fully inverse sigma (anisotropic ) geometry? * We’d like to be able to say: move in any direction towards a “mu”- risk is the same in all directions, we’re istropic to risk. So we need to get our mu target in an isotropic space, then find w there * Picture of all 3 geometries laid out at once --- ### Understanding Point Clouds and Projections We have the concept of a **point cloud** and selecting directions within that space. By projecting our points onto these directions, we can observe the variance of points along the projected direction. #### Importance of Projections Why is this important? Essentially, when we project our points, we get a variance of portfolios along that direction. This projection forms an anisotropic space, which often needs to be made isotropic. #### Visualizing Isotropic Space This visualization helps us understand why making our space isotropic is useful. The purpose is to measure distances in a way that corresponds to units of risk or portfolio variance. In an anisotropic space, projecting onto different directions results in varying portfolio variances. Different directions in asset space have different degrees of risk, meaning the space is not isotropic with respect to risk. #### Benefits of Isotropic Space Working in an isotropic space ensures that any vector we pick represents the same movement in terms of risk. One unit in any direction equates to the same risk adjustment, making geometric quantities like distance, angles, and lengths meaningful and consistent in terms of risk. These plots illustrate why moving to an isotropic space is beneficial. They show that our notions of distance and geometry are risk-adjusted and consistent across all directions. This consistency ensures one step in any direction represents a one standard deviation step in terms of risk, which is the motivating factor behind these visualizations. --- I think the return is more clear right now, particularly the contour in return space, the ellipse. Essentially, we're saying that contour is defined as the set of points that are a constant 1 from the origin in an isotropic space. In our space, measured via Mahalanobis distance equipped with this sigma inverse metric, these points are 1 Mahalanobis distance from the origin. --- # Geometry Issue in Asset Space Let's work through this geometry issue. Everything stems from sigma and the point cloud, so let's start there. Our initial point is the point cloud. ## Key Questions A good way to approach this is by asking questions: - We have sigma and our point cloud. - The point cloud exists in asset space. - If we look at different directions in that space and examine the variance of assets along those directions, we can project our points onto any direction and calculate the variance. ## Point Cloud Analysis From the point cloud, we can compute a matrix sigma. This leads us to ask: - What is the variance of return for a single fixed vector? However, fixing W isn't practical. Instead, we should consider the set of points that are all one unit of risk away in terms of variance. ## Ellipses and Level Sets Ellipses define a level set where some function is constant. We need to differentiate between using directions and weights: - One approach projects onto rays and examines the spread. - The other considers weights. By building this up from the ground, we see two views: 1. **Weight Space View**: Associated with an ellipse. 2. **Return Space View**: Also associated with an ellipse. These ellipsoids help understand what happens as you move spaces. ## Variance of Returns The key link is examining the variance of returns based on varying objects: - In portfolio space, an ellipse defines a set of portfolios with constant variance. - In return space, an ellipse defines a set of returns with constant variance. Understanding these concepts helps clarify how risk is computed from portfolios. Points in return space represent different sigma events: - One sigma event - Two sigma event - And so on --- When you say "Define the return geometry" what does that mean, intuitively? We start with returns in asset space. We need to define a way to measure distance between them. We could say use euclidean. But we don't. We instead define the geometry you laid out. Why? What problem does it solve? Okay first round of questions: The R-geometry allows us to measure return vectors based on their risk. But the contours are ellipses. In other words: Sigma^-1 hasn't whitened our space. It just allows us to see the equidistant risk contours, that are ellipses. Put yet another way: we haven't transformed our space or points (the point cloud is still its original, raw shape)? We just have a metric that allows us to compute and see points of equidistant risk? But these equidistant points (contours) will be *ellipses*, not circles. Our space is still anisotropic. # Key ideas to introduce 1. R-geometry: asking "what is the risk of observing this point"? Analogous to asking with a 1d distribution: "what was the probability of observing this event?" 2. w-geometry: asking "what w's all have the same risk?" # New understanding ### R-Geometry * Space remains unchanged * Metric is changed (we've picked a new ruler) * Contours are ellipses (because we are plotting in original space) * We can plot—in this anisotropic space—what the mahal distance is ### Whitened-Geometry * So whitening is: _“change coordinates so that the Euclidean metric now expresses risk directly.”_ * this is a change of basis * - After whitening, we are still in \mathbb{R}^n. * But the axes are no longer “assets” — they’re new directions (linear combos of assets) chosen to make risk isotropic. * Need to ensure this is reflected in x and y axes of whitened plots * R-geometry is: _“leave coordinates alone, but change the metric so that distances mean risk.”_ * ### R vs W-geometry * **R-geometry (**\Sigma^{-1}**)**: metric tells you how risky a return vector is. * **w-geometry (**\Sigma**)**: metric tells you how risky a weight vector is. - Both describe the _same underlying variance structure_, but seen from dual perspectives (returns vs portfolios). Because different returns have different risk, different portfolios must vary inversely in order to keep the contour equal risk (variance) The ellipses are all the same variance contours—just defined differently