Mapping arises as a result of fundamental \(VaR\) nature which is the highest level of portfolio measurement. After a portfolio has been mapped on the risk factors, any method of \(VaR\) can be used to build the distribution of profits and losses. Mapping processes for various financial instruments are illustrated in this chapter. When characteristics of instruments change over time aping is the only solution, options must be mapped on their primary risk factor.

**Mapping as a Solution to Data Problems.**

Consider a mutual fund with an investing strategy in IPOs of common stocks with no history. The risk manager will replace these positions with exposures on similar risk factors that are already in the system. The time at which prices are recorded is another problem with the global market. Stale prices are problematic to risk managers as they are not synchronous and low daily correlations thereby affecting the measure of portfolios at risk. The solution to this problem is mapping. Therefore, it is paramount for the risk manager to ensure that the data collection process will lead to meaningful risk estimates.

**The Mapping Process**

Making all positions to market in current dollars (or whatever reference currency is used) should be the first analytical step. You then allocate the risk factors to market value for each instrument.Let’s take, instrument \(1,2,3…,n\) to have a market value \({ V }_{ 1 }{ V }_{ 2 }{ \quad …V }_{ n }\) allocated to three exposures \({ x }_{ 11 }{ x }_{ 12\quad … }{ x }_{ 1n.. }\) The system allocates positions for instruments 2 and so on. Therefore, for the risk factor, the dollar exposure is given by:

$$ { X }_{ 1 }=\sum _{ i=1 }^{ n }{ { x }_{ i1 } } $$

A vector \(x\) is created for the three risk factor exposures which can be fed to the risk measurement system. Mapping provides an exact exposure allocation for the risk factors or alternatively estimates the exposures.

**General and Specific Risks**

The choice for the set of primitive risk factors is supposed to reflect the trade-off between better quality approximation and faster processing. It also influences the size of specific risk which is the risk due to issuer-specific price movements after accounting for general market factors.

Consider a portfolio of \(N\) stocks. The return on a stock \({ R }_{ i }\) is regressed on the stock market index \({ R }_{ m }\), that is:

$$ { R }_{ i }={ \alpha }_{ i }+{ \beta }_{ i }{ R }_{ m }+{ \epsilon }_{ i } $$

The relative weight for each portfolio is given by \({ w }_{ i }\). The portfolio return is:

$$ { R }_{ p }=\sum _{ i=1 }^{ N }{ w_{ i }{ R }_{ i } } =\sum _{ i=1 }^{ N }{ { w }_{ i }{ \beta }_{ i }{ R }_{ m } } +\sum _{ i=1 }^{ N }{ { w }_{ i }{ \epsilon }_{ i } } $$

These exposures are aggregated across all the stocks in the portfolio thereby giving:

$$ { \beta }_{ p }=\sum _{ i=1 }^{ n }{ { w }_{ i }{ \beta }_{ i } } $$

He mapping index for a portfolio with value \(W\) is \(x=W{ \beta }_{ p }\)

Decomposing the variance \({ R }_{ p }\), we find:

$$ V\left( { R }_{ p } \right) =\left( { \beta }_{ p }^{ 2 } \right) V\left( { R }_{ m } \right) +\sum _{ i=1 }^{ N }{ { w }_{ i }^{ 2 }{ \sigma }_{ \epsilon i }^{ 2 } } $$

From this decomposition, we observe that with more descriptions on primitive and general-market risk factors, the specific risk for a fixed amount of total risk \(V\left( { R }_{ p } \right)\) is less. The primitive risk factors could be movements in \(J\) government bond yields \({ z }_{ j }\) and in a set of \(K\) credit spreads \({ s }_{ k }\) sorted by credit rating.

The movement in value \(W\) then is:

$$ dW=\sum _{ i=1 }^{ N }{ DVB{ P }_{ i }{ dy }_{ i }=\sum _{ j=1 }^{ J }{ DVB{ P }_{ j }+\sum _{ k=1 }^{ K }{ DVB{ P }_{ k }d{ s }_{ k }+ } } } \sum _{ i=1 }^{ N }{ DVB{ P }_{ i }{ d }\epsilon _{ i } } $$

Where \(DVBP\) is the total dollar value of a basis point for the associated risk factor. This leads to a total risk decomposition of:

$$ V\left( dW \right) =general\quad risk+\sum _{ i=1 }^{ N }{ DVB{ P }_{ i }^{ 2 } } V\left( d{ \epsilon }_{ i } \right) $$

**Mapping Fixed-Income Portfolios**

**Mapping Approaches**

For fixed income portfolios, we can differentiate three mapping systems: principal, duration and cash flows. For principle mapping, one risk factor corresponding to the average portfolio maturity is chosen. With duration mapping, the risk factor chosen corresponds to portfolio duration. The portfolio cash flows are grouped into maturity buckets for cash flow mapping.

Principal mapping considers the timing of redemption payments only. The simplicity of this method is its only positive aspect.In duration mapping, we replace the portfolio with a zero-coupon bond with maturity equal to the duration of the portfolio. For the cash flow mapping method all cash flows with term-structure corresponding to maturities with which volatilities are provided, are grouped.

$$ VaR=\alpha \sqrt { { X }^{ \prime }\Sigma X } =\sqrt { { \left( xxv \right) }^{ \prime }R\left( xxV \right) S } $$

Where \(V=\alpha \sigma \) is the vector \(VaR\) for zero-coupon bond returns and \(R\) is the correlation matrix.

With perfect correlation across all zeros, the \(VaR\) of the portfolio is:

Undervisified \(VaR=\sum _{ i=1 }^{ N }{ |{ x }_{ i }|{ V }_{ i } } \)

**Stress Test**

Assuming that all zeroes from movement in zero values are perfectly correlated, then all zero values could be decreased by their \(VaR\). If all the zeroes are perfectly correlated, then they should fall by their respective \(VaR\) values. This demonstrates the connection between \(VaR\) computed through matrix multiplication and through underlying prices movements.

**Benchmarking**

Let’s assume a portfolio of $100 million is benchmarked. Over a monthly horizon using a 95% confidence level, the \(VaR\) index $1.99 million. We then match the index with two bonds.

The \(VaR\) of the deviation relative to the benchmark is:

Tracking Error \(VaR=\alpha \sqrt { { \left( x-{ x }_{ 0 } \right) }^{ \prime }\Sigma \left( x-{ x }_{ 0 } \right) } \)

Relative to the original index, The tracking index can be measured in terms of Variance reduction, similar to an \({ R }^{ 2 }\) in a regression.

**Mapping Linear Derivatives**

**Forward Contracts**

Assume that we are dealing with a forward contract on a foreign currency, then the basic valuation formula can be derived from an arbitrage argument.Let’s define the following notations:

- \( { S }_{ t }\) =spot price of a unit of the underlying cash asset.
- \(K\) = contracted forward price.
- \(R\) = domestic risk-free rate.
- \(Y\) = income flow of the asset
- \(\tau \) = time to maturity.

We want to establish the value of a forward contract \({ f }_{ t }\) to acquire a unit of foreign currency at \(K\) after timeτ. We consider two alternatives that are economically equivalent. First, buy \({ e }^{ -yt }\) units at price \({ s }_{ t }\) and hold for a period. Alternatively, a forward contract is entered to buy one unit of the asset in one period. Contract costs \({ f }_{ t }\) upfront and we ensure enough money to cater for \(K\) in the future which is \(K{ e }^{ -rt }\).

Therefore, we are led to the following evaluation formula for outstanding forward contracts:

$$ { f }_{ t }={ s }_{ t }{ e }^{ -y\tau }-K{ e }^{ -r\tau }\quad \quad \quad \quad \quad \left( a \right) $$

Setting \(K={ F }_{ t }\) and \({ f }_{ e }=0\), where \({ F }_{ t }\) is the current forward rate, in the above equation we have:

$$ { F }_{ t }=\left( { s }_{ t }{ e }^{ -y\tau }{ e }^{ -r\tau } \right) \quad \quad \quad \quad \quad \quad \left( b \right) $$

And therefore:

$$ { f }_{ t }=F{ e }^{ -r\tau }-K{ e }^{ -r\tau }=\left( { F }_{ t }-K \right) e^{ -r\tau }\quad \quad \quad \quad \left( c \right) $$

We compute the positions of \(x\) on each of the three building blocks of the contract. Finding the partial derivative of equation \(\left( a \right) \) with respect to the risk factors we have:

$$ df=\frac { \partial f }{ \partial s } df+\frac { \partial f }{ \partial { r }^{ \ast } } d{ r }^{ \ast }+\frac { \partial f }{ \partial r } dr $$

$$ ={ e }^{ { -r }^{ \ast }\tau }-s{ e }^{ { -r }^{ \ast }\tau }ds -s{ e }^{ { -r }^{ \ast }\tau }\tau d{ r }^{ \ast }+K{ e }^{ -r\tau }\tau dr $$

The interest rates can be replaced by the price bills. Defining:

$$ P={ e }^{ -rt }\quad and\quad { p }^{ \ast }={ e }^{ { -r }^{ \ast }t .} $$

We then replace dr with dp by:

$$ dp=\left( -\tau \right) { e }^{ -rt }dr $$

and:

$$ d{ p }^{ \ast }=\left( -\tau \right) { e }^{ { -r }^{ \ast }t }d{ r }^{ \ast } $$

Therefore, the risk forward contract changes to :

$$ df=\left( s{ e }^{ { -r }^{ \ast }t } \right) \frac { dS }{ S } +\left( s{ e }^{ { -r }^{ \ast }t } \right) \frac { d{ p }^{ \ast } }{ { p }^{ \ast } } -\left( K{ e }^{ -rt } \right) \frac { dp }{ P } $$

A position in the forward contract has three building blocks:

Long forward contract = Long foreign currency spot + Long foreign currency bill + U.S. dollar bill

The same method can be used more generally for long-term currency swaps that are equivalent to the portfolios of forward contracts.

**Commodity Forwards**

Valuation of forward or futures contracts in commodities is substantially more complex as compared to financial assets like currencies, bonds or stocks. For base metals, the monthly \(VaR\) measures are very high hitting 29% for near contracts. This implies that commodities are more volatile compared to typical financial assets.Volatilities for financial markets are primarily driven by spot prices.

Assuming we wish to compute 12-month forward position \(VaR\) on a specified number of oil barrels of known price per barrel.

Differentiating Equation (c) gives:

$$ df=\frac { \partial f }{ \partial F } dF={ e }^{ -rt }dF=\left( { e }^{ -rt }F \right) \frac { dF }{ F } $$

**Forward Rate Agreements (FRAs)**

These are forward contracts allowing users to lock in an interest rate at some future date. This implies that the long partyreceivesa payment if the spot rate is above the forward rate.The forward rate id defined as the implied rate that equalizes the return on a \({ \tau }_{ 2 }\) – period investment with a \({ \tau }_{ 1 }\) – period investment rolled over, ie:

$$ \left( 1+{ R }_{ 1 }{ T }_{ 1 } \right) =\left( 1+{ R }_{ 1 }{ \tau }_{ 1 } \right) \left[ 1+{ F }_{ 1,2 }\left( { \tau }_{ 2 }-{ \tau }_{ 1 } \right) \right] $$

Where \({ \tau }_{ 1 }\) Is the timing of the short Leg and \({ \tau }_{ 2 }\) is the timing of the long leg.

**Interest Rate Swaps**

Interest rate swaps can be decomposed into two legs, a fixed and floating leg, and are the most actively used derivatives. The fixed rate is priced as the coupon-paying bond while the floating is the floating rate note (FRN). Interest rate swaps can either be observed as a combined position in fixed-rate bond or a portfolio of forward contracts in a floating-rate bond

**Mapping Options**

The BS model for European options assumes that the underlying spot price follows a Brownian motion that is continuously geometric with a constant volatility \(\sigma \left( { dS }/{ S } \right) \). Based on these assumptions, the value of a European call is:

$$ c=c\left( S,K,\tau ,r,{ r }^{ \ast },\sigma \right) =s{ e }^{ { -r }^{ \ast }t }N\left( { d }_{ 1 } \right) -K{ e }^{ { -rt } }N\left( { d }_{ 2 } \right) $$

Wher \(N\left( d \right) \) is the cumulative normal distribution function of arguments:

$$ { d }_{ 1 }=\frac { ln\left( { S{ e }^{ { -r }^{ \ast }\tau } }/{ K }{ e }^{ -r\tau } \right) }{ \sigma \sqrt { \tau } } +\frac { \sigma \sqrt { \tau } }{ 2 } , $$

$${ d }_{ 2 }={ d }_{ 1 }-\sigma \sqrt { \tau } $$

We approximate changes in the values of the option by taking partial derivatives:

$$ dc=\frac { \partial c }{ \partial s } ds+\frac { 1 }{ 2 } \frac { { \partial }^{ 2 }S }{ \partial { S }^{ 2 } } d{ s }^{ 2 }+\frac { \partial c }{ \partial { r }^{ \ast } } d{ r }^{ \ast }+\frac { \partial c }{ \partial r } dr+\frac { \partial c }{ \partial \sigma } d\sigma +\frac { \partial c }{ \partial t } dt $$

$$= \Delta ds+\frac { 1 }{ 2 }\Gamma d{ s }^{ 2 }+{ \rho }^{ \ast }d{ r }^{ \ast }+\rho dr+\dot { \wedge } d\sigma +\Theta dt $$

The first partial derivative in a European call is:

$$ \Delta ={ e }^{ { -r }^{ \ast }\tau }N\left( { d }_{ 1 } \right) $$

With such a small change, the linear effect will dominate the nonlinear effect. It is instinctive to consider only the linear effects of the spot rates and two interest rates that is:

$$dc= \Delta d+{ \rho }^{ \ast }d{ r }^{ \ast }+\rho dr$$

$$= \left[ { e }^{ { -r }^{ \ast }t }N\left( { d }_{ 1 } \right) \right] ds+\left[ -S{ e }^{ { -r }^{ \ast }t }\tau N\left( { d }_{ 1 } \right) \right] d{ r }^{ \ast }+\left[ K{ e }^{ -r\tau }\tau N\left( { d }_{ 2 } \right) \right] dr $$

$$= \left[ S{ e }^{ { -r }^{ \ast }\tau }N\left( { d }_{ 1 } \right) \right] \frac { dS }{ S } +\left[ S{ e }^{ { -r }^{ \ast }t }N\left( { d }_{ 1 } \right) \right] \frac { d{ P }^{ \ast } }{ { P }^{ \ast } } -\left[ K{ e }^{ -r\tau }N\left( { d }_{ 2 } \right) \right] \frac { dP }{ P } $$

$$= { x }_{ 1 }\frac { dS }{ S } +{ x }_{ 2 }\frac { d{ P }^{ \ast } }{ { P }^{ \ast } } +{ x }_{ 3 }\frac { dP }{ P }$$

This formula closely resembles the foreign currency forward formula. In this case, BS model changes to

$$ C=S{ e }^{ { -r }^{ \ast }\tau }-K{ e }^{ -r\tau }, $$

This is the valuation formula for a forward contract. In addition the position dollar bill \(K{ e }^{ -r\tau }N\left( { d }_{ 2 } \right) \) is equivalent to: \(S{ e }^{ { -r }^{ \ast }\tau }N\left( { d }_{ 1 } \right) -c=s\Delta -c\)

This, therefore, indicates that:

$$ Long\quad Option=Long\quad \Delta Asset+Short\quad \left( \Delta S+c \right) bill $$

The portfolio option will be characterized by its net vega, \(\dot { \wedge }\), which takes into account the second order derivatives using the net gamma,┌.

**Practice Questions**

1) Examine the risk for a 1-year forward contractor to purchase 100 million Euros in exchange for 122.59 million. It is estimated that his Euro spot risk factor is $1.4422 and long EUR bill is 2.34% and Short USD bill is 3.14%.

- 7.98%
- 22.06%
- 20.04%
- 43.97%

The correct answer is **b**.

The EUR forward risk factor is: \(\frac { 122.59 }{ 100 } =1.2259\)

Recall that:

$$ { f }_{ t }={ s }_{ t }{ e }^{ -y\tau }-K{ e }^{ -r\tau } $$

Therefore:

$$ { F }_{ t }=1.4422\frac { 1 }{ \left( 1+{ 2.34 }/{ 100 } \right) } -1.2259\frac { 1 }{ \left( 1+{ 3.14 }/{ 100 } \right) } =1.4092 – 1.1886 = 0.2206 = 22.06\% $$

2) A bank has a cash flow decomposition with a duration of 5 years. Given that the \(VaR\) of the index at 95% confidence level is $4.33 million, with a tracking error of $2.56 million, calculate the variance improvement relative to the original index.

- 86.09%
- 65.05%
- 34.95%
- 2.86%

The correct answer is **b**.

Recall that the variance improvement is given by:

$$ 1-{ \left( { \left( Tracking\quad Error \right) }/{ \left( Absolute\quad risk\quad index \right) } \right) }^{ 2 } $$

$$= 1-{ \left( \frac { 2.56 }{ 4.33 } \right) }^{ 2 }$$

$$= 0.6505$$

$$= 65.05\%$$

3) Calculate the current forward rate that will set the contract value to be zero if you are given that the spot price of 1 unit underlying cash asset is $5.9 million, with a domestic free rate of 0.025 and \(\tau =0.1\). The income flow rate y is $2.23 million

- 1.5249
- 52.4911
- 53.6855
- 1.4893

The correct answer is **d**.

Remember that the current forward rate is given by the equation:

$$ { F }_{ t }=\left( { S }_{ t }{ e }^{ -y\tau }{ e }^{ r\tau } \right) $$

We know that, \({ S }_{ t }=5.9\), \(r = 0.025\), \(y = 2.23\) and \(\tau =0.1\).

Applying the formula:

$$ { F }_{ t }=\left( { 5.9 }^{ 2.23\times 0.1 }\times { e }^{ 0.025\times 0.1 } \right) $$

$$= 1.4893$$