# 10.3 Quadratic Transformation Procedures

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

[10.8]

where ^{1}*R**N _{n}*

**(**

^{1|0}

**μ,**

^{ 1|0}

**Σ**),

**is a symmetric**

*c**n×*

*n*matrix,

**is an**

*b**n*-dimensional row vector, and

*a*is a scalar. We assume

^{1|0}

**is positive definite.**

**Σ**As described in Section 3.13, we may express ^{1}*P* as a linear polynomial of independent chi-squared and normal random variables. Based upon this representation, we can value various PMMRs. To calculate ^{0}*std*(^{1}*L*), we apply the techniques of Section 3.13 to calculate conditional moments ^{0}*E*(^{1}*P*) and ^{0}*E*(^{1}*P*^{2}) of ^{1}*P*. Then (see Exercise 3.15):

[10.9]

Related metrics such as ^{0}*var*(^{1}*P*) 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 ^{1}*R**N*_{3}**(**^{1|0}**μ**,^{1|0}**Σ**) where

[10.11]

A primary portfolio mapping ^{1}*P* = θ(^{1}** R**) 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 = ^{0}*p* = 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|0}

**Σ**

[10.16]

and a matrix ** u** with rows equal to orthonormal eigenvectors of

**′**

*z***,**

*c z*[10.17]

Define a mapping

[10.18]

where ~ *N*_{3}(**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}~ χ^{2}(1, 54.06), - (
_{2}+ 93.81)^{2}~ χ^{2}(1, 8800), - (
_{3}+ 120.4)^{2}~ χ^{2}(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.

*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.

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 = –4.752×10^{7}; γ_{1}, γ_{2}, and γ_{3} are 3.432×10^{6}, –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|0}Φ() = .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

^{[1]}= 200MM,^{[2]}= 190MM.

Subsequent values ^{[3]}, ^{[4]}, ^{[5]}, … 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.

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

Measure time in trading days and employ USD as a base currency. Consider a quadratic portfolio (53600, ^{1}*P*)3 that depends upon a single key factor ^{1}*R*_{3} *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.

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

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 ^{1}*R*_{1}*,*^{1}*R*_{2}, and ^{1}*R*_{3}. Holdings are, respectively, 1000, 3000, and –2000 shares. Denote the EUR/USD exchange rate ^{1}*R*_{4}. Consider the key vector

[10.40]

Suppose:

[10.41]

and assume ^{1}*R**N*_{4}(^{0}** r**,

^{1|0}

**Σ**).

- Calculate
^{0}*p.* - Specify a mapping
^{1}*P*= θ(^{1}). Is your mapping function θ a quadratic polynomial?*R* - Express your mapping from part (b) in the form
[10.42]

Specify the symmetric matrix

, row vector*c*, and scalar*b**a*. - Construct the Cholesky matrix
of*z*^{1|0}**Σ**. - Construct the matrix
′*z*. Calculate its orthogonal eigenvectors. Normalize them (scale them so they each have unit length), and construct a 4×4 matrix*c z*with rows equal to those normalized eigenvectors. (Hint:*u*′*z*has 0 as a repeated eigenvalue, so its eigenvectors need not be orthogonal. Make sure the eigenvectors you select are orthogonal.)*c z* - Specify a mapping
^{1}= φ() such that*R**N*_{4}(**0**,). (Note: We perform this calculation as a formality. We won’t need the result for subsequent calculations.)**I** - Specify a mapping
^{1}*P*= θφ() 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].)

- Based upon your results from parts (c) and (g) and the discussion of Section 3.13.3, determine conditional values
*g*^{[0]}through*g*^{[4]}for^{1}*P*. - Based upon your results from part (h) and the discussion of Section 3.13.3, determine conditional moments
^{0}*E*(^{1}*P*) through^{0}*E*(^{1}*P*^{ 5}). - Use results from part (i) to calculate the portfolio’s standard deviation of 1-day EUR loss,
^{0}*std*(^{1}*L*). - Based upon your conditional moments from part (i), calculate conditional central moments of
^{1}*P*as well as the conditional central moments and cumulants of the normalization^{1}*P** of^{1}*P*. - Use the Cornish-Fisher expansion to approximate the conditional .05-quantile of
^{1}*P**. From this, approximate the 1-day 95% EURvalue-at-risk of portfolio (^{0}*p*,^{1}*P*). - Multiply out your expression from part (g) and complete the squares to express
^{1}*P*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? - Based upon your representation from part (m), calculate the conditional .05-quantile of
^{1}*P*by inverting its characteristic function. Use seed values of^{1}*p*^{[1]}= 280,000 and^{1}*p*^{[2]}= 290,000 with the secant method. Calculate the 1-day 95% EURvalue-at-risk of portfolio (^{0}*p*,^{1}*P*). - Retain your results, as we will refer back to them in Exercise 10.5.