Abstract
One of our principal interests for any portfolio is its return. The return is of interest from both backward and forward-looking perspectives. Not only are the size and magnitude of historical returns interesting, but our interest also extends to the factors contributing to this return. How, for example, did our portfolio manager generate his or her return? This is termed performance attribution. We also have an natural interest in future returns. Predicting specific future returns is extremely difficult, but understanding the uncertainty inherent in future returns is less difficult and can be very helpful.
It is better to be roughly right than precisely wrong.
John Maynard Keynes
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Notes
- 1.
In measuring risk, we are essentially interested in the statistical distribution of our portfolio returns. With this distribution in hand, we may compute its moments, its tails, or its risk relative to a pre-defined strategic benchmark.
- 2.
Note, of course, that any sovereign yield curve is a relatively complicated object that itself requires a few factors to adequately describe it. The underlying factors driving the yield curve likely include macroeconomic variables such as output, inflation, and monetary policy. Statistical factors such as level, slope, and curvature may also be employed.
- 3.
The credit spread is a classic example. Is it sufficient to consider the spread as a risk factor or does one wish to identify the underlying macroeconomic elements driving the credit spread? The answer depends on what one is trying to accomplish. Be aware, however, that the latter approach involves significantly more complexity.
- 4.
An alternative definition involves the log differences,
$$\displaystyle\begin{array}{rcl} r(t_{0},t_{1}) =\ln \left (\frac{V (t_{1})} {V (t_{0})}\right ).& & {}\end{array}$$(3.2)This continuously compounded return definition is commonly used by economists, but sadly not by market practitioners.
- 5.
We will see in the following development that the results from the previous chapter make a startling reappearance.
- 6.
- 7.
The same applies for the multivariate result provided in a few pages.
- 8.
The theorem indicates that if this remainder term is essentially well-behaved, then the infinite Taylor series converges to f. The term R k+1(a) is a remainder term. A number of results, such as Lagrange’s formula, provide descriptions of the magnitude of the remainder function, R k+1(a).
- 9.
More precisely, the result permits us to expand the value of a one-dimensional function around a given point an infinite sum of polynomials of increasing order. The coefficient of each term in this series depends on the function’s derivatives evaluated at the known value, a.
- 10.
One can loosely think of the remainder term in Eq. (3.10) as the distance between the true and estimated value of the function f(x 1) that could not be explained by the first six terms of the Taylor series expansion.
- 11.
Note that there is nothing terribly special about this function, it was really just (mostly) randomly selected to demonstrate Theorem 3.1.
- 12.
This is a typical result: almost all continuous functions are locally linear and, thus, small movements are generally quite well described by a linear approximation. There are, of course, some exceptions to this rule. Brownian motion is one classic example.
- 13.
As indicated in Theorem 3.1, adding additional terms in the Taylor-series expansion permits us to converge to the true value, f(x 1).
- 14.
It is unfortunately a bit ugly. In mathematics textbooks, it looks a bit better since it is typically written in vector notation.
- 15.
Note that there are two cross terms in the second-order expansion: \(\frac{\partial ^{2}V (t,y)} {\partial t\partial y}\) and \(\frac{\partial ^{2}V (t,y)} {\partial y\partial t}\). Since, under normal conditions that we assumed to be fulfilled, second partial derivatives are symmetric, we may collect them into a single term.
- 16.
Note that in some cases, such as convexity, we need to raise the change in the factor to a power.
- 17.
Such an expansion is always possible. One merely requires a mathematical description, or model, of the underlying treasury yield curve.
- 18.
We have enriched our notation for an exchange rate to E(i, t), where i denotes the currency and t continues to denote the point in time for which the exchange rate applies.
References
M. Abramovitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1965)
T. Apostol, Calculus Volume II: Multi Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability (Wiley, New York, 1969)
D.J. Bolder, T. Rubin, Optimization in a simulation setting: Use of function approximation in debt-strategy analysis. Bank of Canada: Working Paper 2007–13 (2007)
C.J.C. Burges, Dimension reduction: A guided tour. Microsoft Research Tech Report MSR-TR-2009-2013 (2013)
T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning: Data Mining, Inference and Prediction, 1st edn. (Springer, New York, 2001)
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Bolder, D.J. (2015). A Useful Approximation. In: Fixed-Income Portfolio Analytics. Springer, Cham. https://doi.org/10.1007/978-3-319-12667-8_3
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DOI: https://doi.org/10.1007/978-3-319-12667-8_3
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