 Quadratic transformations2 were pioneered by a number of researchers. The first complete published solution was Rouvinez (1997). Consider a portfolio (0p, 1P) with quadratic portfolio mapping

[10.8]

where 1R Nn(1|0μ, 1|0Σ), c is a symmetric n matrix, b is an n-dimensional row vector, and a is a scalar. We assume 1|0Σ is positive definite.

As described in Section 3.13, we may express 1P as a linear polynomial of independent chi-squared and normal random variables. Based upon this representation, we can value various PMMRs. To calculate 0std(1L), we apply the techniques of Section 3.13 to calculate conditional moments 0E(1P) and 0E(1P2) of 1P. Then (see Exercise 3.15):

[10.9]

Related metrics such as 0var(1P) are calculated similarly.

Value-at-risk metrics are more difficult to calculate. Various solutions have been proposed. Zangari (1996b) approximates a solution using Johnson (1949) curves. Fallon (1996) and Pichler and Selitsch (2000) recommend approximate solutions based on the Cornish-Fisher expansion. Rouvinez (1997) uses the trapezoidal rule to invert the characteristic function. Britten-Jones and Schaefer (1999) use an approximation due to Solomon and Stephens (1977). Cárdenas et al. (1997) use the fast Fourier transform.

We shall focus on the Cornish-Fisher expansion, as described in Section 3.14, and on applying the trapezoidal rule to invert the characteristic function, as described in Section 3.17.

###### 10.3.1 Example: Platinum Derivatives

Measure time in trading days. A Japanese metals trading firm has exposure to forward and options positions in platinum. Some of the positions are USD-denominated. Treat interest rates and lease rates as constant. Model three key factors

[10.10]

and assume 1R N3(1|0μ,1|0Σ) where

[10.11]

A primary portfolio mapping 1P = θ(1R) is constructed based upon applicable forward and options pricing formulas. This is quadratically remapped as

[10.12]

where:

[10.13]

[10.14]

[10.15]

We set = 0p = JPY 12.79MM. Let’s value the following two PMMRs:

• 1-day standard deviation of portfolio JPY market value, and
• 1-day 90% JPYvalue-at-risk.

For the purpose of illustration, we shall evaluate the latter metric twice, first using the Cornish-Fisher expansion and then by inverting the characteristic function.

###### 10.3.2 Example Continued: Linear Polynomial Representation

Paralleling the techniques of Section 3.13, we construct the Cholesky matrix z of 1|0Σ

[10.16]

and a matrix u with rows equal to orthonormal eigenvectors of zc z,

[10.17]

Define a mapping

[10.18]

where ~ N3(0, I). We obtain

[10.19]

with

[10.20]

[10.21]

[10.22]

Multiplying [10.19] out

[10.23]

We “complete the squares” to obtain

[10.24]

We have expressed conditionally as a linear polynomial of three independent random variables:

• ( 1 + 7.353)2 ~ χ2(1, 54.06),
• ( 2 + 93.81)2 ~ χ2(1, 8800),
• ( 3 + 120.4)2 ~ χ2(1, 14490).
###### 10.3.3 Example Continued: Standard Deviation of Portfolio Value

We calculate conditional values g[k] for , as defined in Section 3.13. These are indicated in Exhibit 10.2. Exhibit 10.2: Using formula [3.167], conditional values g[k] are calculated for the remapped portfolio’s value. Inputs for the calculations are obtained from [10.11], [10.13], [10.14], [10.15], [10.20], and [10.21].

From these, we calculate the conditional moments and central moments of and the conditional central moments and cumulants of the normalization * of . Results are indicated in Exhibit 10.3. Exhibit 10.3: Conditional moments and central moments of the remapped portfolio’s value as well as conditional central moments and cumulants of the normalization of the remapped portfolio’s value are indicated. Computations are performed according to the discussions of Section 3.13. Inputs are the values of Exhibit 10.2.

Based upon its first and second moments, we calculate the conditional standard deviation of :

[10.25]

The portfolio’s 1-day standard deviation of JPY market value is JPY 51.06MM.

###### 10.3.4 Example Continued: Cornish-Fisher Expansion

To approximate the 1-day 90% JPYvalue-at-risk using the Cornish-Fisher expansion [3.206], we apply the expansion to the normalization * of based upon the conditional cumulants from Exhibit 10.3. We obtain the approximate .10-quantile of * as

[10.26]

[10.27]

From [3.207],

[10.28]

[10.29]

The .10-quantile of is JPY 150.8MM, and the remapped portfolio has approximate 1-day 90% JPYvalue-at-risk of

[10.30]

[10.31]

[10.32]

###### 10.3.5 Example Continued: Inverting the Characteristic Function

As an alternative, we can obtain an exact result by numerically inverting the characteristic function of . Based upon representation [10.24], as well as inversion theorem [3.226], the conditional CDF for is

[10.33]

for

[10.34]

[10.35]

[10.36]

[10.37]

where α = –4.752×107; γ1, γ2, and γ3 are 3.432×106, –21880, and 18277; and , , and are 54.06, 8800 and 14489. For this example, we employ approximation [3.233] with u = 5×10–7. We partition the interval [0,u] into 100 subintervals to apply the trapezoidal rule.

To calculate 90%value-at-risk, we require the conditional .10-quantile of . This is that value such that 1|0Φ( ) = .10. To find it, we employ the secant method. This requires two seed values—initial “guesses”—for the desired value for . If a portfolio has an established value-at-risk limit, the seed values might reasonably be set equal to and minus .01 times the value-at-risk limit. For this example, let’s use seed values

• = 200MM,
• = 190MM.

Subsequent values   , … are obtained with the recursive formula

[10.38]

The resulting sequence of values converges to the desired value for . Results are indicated in Exhibit 10.4. Exhibit 10.4: Results of applying the secant method to obtain the .10-quantile of the remapped portfolio.

The .10-quantile of is JPY 150.8MM. This result matches, at least to the number of decimal places indicated, the approximate result we already obtained using the Cornish-Fisher expansion. Based upon calculations identical to [10.30] through [10.32], we obtain the same 61.9MM 1-day 90% JPYvalue-at-risk.

A shortcoming of this approach—inverting the characteristic function using the trapezoidal rule and approximation [3.233]—is the fact that both upper bound of integration u and the number of subintervals in which to partition [0, u] must be specified. There is no systematic method for sspecifying either, other than by trial-and-error. If any readers have addressed this problem in their own value-at-risk work, please share your experiences with others by commenting at the bottom of this page.

###### Exercises
10.2

Measure time in trading days and employ USD as a base currency. Consider a quadratic portfolio (53600, 1P)3 that depends upon a single key factor 1R3 N(25,16):

[10.39]

In this exercise, you will use a quadratic transformation to evaluate the portfolio’s standard deviation of 1-day loss and 1-day 90% USDvalue-at-risk. You will calculate the 90%value-at-risk twice: approximately using the Cornish-Fisher expansion and then exactly by inverting the characteristic function.

1. Express [10.39] as a quadratic polynomial 1P of a new risk factor 1 N(0,1). What are your values for , , and ?
2. Is what you did in part (a) a mapping or a remapping? Draw a schematic indicating the mappings and/or remappings that relate 1P, 1R1 and 1.
3. Based upon your results from part (a) and the discussion of Section 3.13.3, determine conditional values g through g for 1P. (Hint: Formula [3.181] for the g[k] is multidimensional, but you can easily interpret it for this one-dimensional problem.)
4. Based upon your results from parts (c) and the discussion of Section 3.13.3, determine conditional moments 0E(1P) through 0E(1P5).
5. Use results from part (d) to calculate the portfolio’s standard deviation of 1-day loss, 0std(1L).
6. Based upon your moments from part (d), calculate conditional central moments of 1P as well as the conditional central moments and cumulants of the normalization 1P* of 1P.
7. Use the Cornish-Fisher expansion to approximate the .10-quantile of 1P*. Approximate the 1-day 90% USDvalue-at-risk for our portfolio (53600, 1P).
8. Complete the squares in your expression for 1P from part (a). Doing so will express 1P as a linear polynomial of a single chi-squared random variable.
9. What are the values of the parameters ν and of the chi-squared random variable from part (h)?
10. Based upon your result from part (i), construct the characteristic function of 1P. (Note: We perform this calculation as a formality. We won’t need the result for subsequent calculations.)
11. Based upon your representation from parts (h) and (i), calculate the .10-quantile of 1P by inverting its characteristic function. Use seed values of 1p = 25,000 and 1p = 50,000 with the secant method. Approximate the 1-day 90% USDvalue-at-risk for our portfolio (53600, 1P).

10.3

Measure time in trading days and employ EUR as a base currency. A European investor has a portfolio with holdings in three United States stocks whose USD accumulated values we denote 1R1,1R2, and 1R3. Holdings are, respectively, 1000, 3000, and –2000 shares. Denote the EUR/USD exchange rate 1R4. Consider the key vector

[10.40]

Suppose:

[10.41]

and assume 1R N4(0r,1|0Σ).

1. Calculate 0p.
2. Specify a mapping 1P = θ(1R). Is your mapping function θ a quadratic polynomial?
3. Express your mapping from part (b) in the form

[10.42]

Specify the symmetric matrix c, row vector b, and scalar a.

4. Construct the Cholesky matrix z of 1|0Σ.
5. Construct the matrix zc z. Calculate its orthogonal eigenvectors. Normalize them (scale them so they each have unit length), and construct a 4×4 matrix u with rows equal to those normalized eigenvectors. (Hint: zc z has 0 as a repeated eigenvalue, so its eigenvectors need not be orthogonal. Make sure the eigenvectors you select are orthogonal.)
6. Specify a mapping 1R = φ( ) such that  N4(0, I). (Note: We perform this calculation as a formality. We won’t need the result for subsequent calculations.)
7. Specify a mapping 1P = θ φ( ) of the form

[10.43]

Indicate the symmetric matrix , the row vector , and the scalar . (Hint: Work directly with formulas [3.161], [3.162], and [3.163].)

8. Based upon your results from parts (c) and (g) and the discussion of Section 3.13.3, determine conditional values g through g for 1P.
9. Based upon your results from part (h) and the discussion of Section 3.13.3, determine conditional moments 0E(1P) through 0E(1P 5).
10. Use results from part (i) to calculate the portfolio’s standard deviation of 1-day EUR loss, 0std(1L).
11. Based upon your conditional moments from part (i), calculate conditional central moments of 1P as well as the conditional central moments and cumulants of the normalization 1P* of 1P.
12. Use the Cornish-Fisher expansion to approximate the conditional .05-quantile of 1P*. From this, approximate the 1-day 95% EURvalue-at-risk of portfolio (0p, 1P).
13. Multiply out your expression from part (g) and complete the squares to express 1P as a linear polynomial of independent chi-squared random variables. What are the degrees of freedom and noncentrality parameters of the chi-squared random variables?
14. Based upon your representation from part (m), calculate the conditional .05-quantile of 1P by inverting its characteristic function. Use seed values of 1p = 280,000 and 1p = 290,000 with the secant method. Calculate the 1-day 95% EURvalue-at-risk of portfolio (0p1P).
15. Retain your results, as we will refer back to them in Exercise 10.5.