• The choice of primitive risk factors also influences the size of specific risks. Specific risk can   be defined as risk that is due to issuer-specific price movements, after accounting for general   market factors.
• The Basel rules have a separate charge for specific risk. Typically, the charge is 4 percent of   the position value for equities and unrated debt, assuming that the banks’ models do not   incorporate specific risks.
• To illustrate this decomposition, consider a portfolio of 𝑁 stocks. We are mapping each stock   on a position in the stock market index, which is our primitive risk factor. The return on a   stock 𝑅_i is regressed on the return on the stock market index 𝑅_m, i.e.,

α, which does not contribute to risk, is ignored. We assume that the specific risk owing to 𝜖 is not correlated across stocks or with the market. The relative weight of each stock in the portfolio is given by 𝑤_i. Thus the portfolio return is

where 𝛽_P = Beta of the portfolio, which is the weighted average of the individual betas

Now, let’s decompose the variance of the portfolio return

The first component is the general market risk. The second component is the aggregate of specific risk for the entire portfolio. This decomposition shows that with more detail on the primitive or general-market risk factors, there will be less specific risk for a fixed amount of total risk 𝑉𝑎𝑟(𝑅_p).

So if there are two identical portfolios of stocks, where one is mapped to only one general risk factor, and the other is mapped to many general risk factors, then the specific risk for the second portfolio will be lesser.

• We have to restrict the number of risk factors to a small set. For some portfolios, one risk factor may be sufficient. For others, many may be necessary. For option portfolios, we need to model movements in their implied volatilities as well, apart from yields

• For a portfolio of  bonds, the  primitive risk factors could be movements in a set of  𝐽 government bond yields 𝑧_j and in a set of 𝐾 credit spreads 𝑠_k sorted by credit rating. We model the movement in each corporate bond yield 𝑑𝑦_i, by a movement in z at the closest maturity and in 𝑠 for the same credit rating. The remaining component is s_i

The movement in the value, say 𝑊 then is

where 𝐷𝑉𝐵𝑃 is the total dollar value of a basis point for the associated risk factor.

This leads to a total risk decomposition of

A greater number of general risk factors should create less residual risk. In practice, there may not be sufficient history to measure the specific risk of individual bonds, which is why i is often assumed that all issuers within the same risk class have the same risk.

Three Approaches For Fixed Income Portfolios

• The three approaches to mapping portfolios of fixed income securities are:
  1. Principal mapping – One risk factor is chosen that corresponds to the average portfolio maturity. Coupon is stripped out and only principal repayment is considered at maturity.
  2. Duration mapping – One risk factor is chosen that corresponds to the portfolio duration. Coupons are incorporated along with the principal.
  3.  Cash flow mapping – Several risk factor are chosen and the bond risk is decomposed into risk of each of the bond cash flows. The portfolio cash flows are grouped into maturity buckets. Cash flow mapping is the most precise method because present values of the cash flows are considered, and inter-maturity correlations are included.

Mapping Fixed Income Portfolios

Mapping Fixed Income Portfolios-CF Mapping

Three Approaches For Fixed Income Portfolios

• Cash Flow mapping VaR is less than the duration VaR because of two reasons :
  1. First, risk measures are not perfectly linear with maturity
  2. Second, correlations are below unity, which reduces risk even further.

• Out of the total $130,000 difference ($2.70 million – $2.57 million) in these measures, $70,000 ($2.70-$2.63 million) is due to differences in yield volatility, and the remaining $60,000 is due to imperfect correlations.

Stress Testing

COMPUTING VAR FROM CHANGE IN PRICE OF ZEROS

Benchmarking

This table presents the cash-flow decomposition   of   the   J.P. Morgan U.S. bond index, which has  a  duration  of  4.62  years. Assume  that  we  are  trying  to benchmark a portfolio of  $100 million. Over a monthly horizon, the VaR of  the index at the 95 percent confidence level is $1.99 million. This is about equivalent to the risk of a 4-year note.

• We try to match the index with two bonds. The rightmost columns in the table display the   positions of two-bond portfolios with duration matched to that of the index. Since no zero-   coupon has a maturity of exactly 4.62 years, the closest portfolio consists of two positions,   each in a 4- and a 5-year zero. The respective weights for this portfolio are $38 million and $62 million.

If we define the new vector of positions for this portfolio as 𝑥 and for the index as 𝑥₀. The VaR of the deviation relative to the benchmark is

Mapping Forwards

• From Part 1, we know that if we assume continuous compounding, then the current value of the forward contract is

And for discrete compounding it can be written as

For currency forwards, the foreign interest rate will take the place of 𝑦 in the above equations.

Apart from that it can also be expressed as the is the present value of the difference between the current forward rate and the locked-in delivery rate, i.e.

So, if we are long a forward contract with contracted rate 𝐾, we can liquidate the contract by entering a new contract to sell at the current rate 𝐹_t. This will lock in a profit of (𝐹_t – 𝐾), which we need to discount to the present time to find 𝐹_t .

Risk and Correlations for Forward Contract Risk Factors (monthly VaR at 95 percent level)

This table examines the risk of a 1-year forward contract to purchase 100 million euros in exchange   for $130.086 million. The first step is to find the market value of the contract using the previous   equation for discrete compounding

Thus the initial value of the contract is zero. This value may change, creating market risk.

Computing VaR for a EUR 100 Million Forward Contract (monthly VaR at 95 percent level)

1. A long position in a forward contract that is used to purchase foreign currency with USD, one year from now, has three components:
   • Long position in foreign currency spot i.e. a long spot position in EUR, worth EUR 100   million = $130.09 million in a year, or ((𝑆𝑒^–rc) = $125.89 million now
   •  Long position in foreign currency bill i.e. a long position in a EUR investment, also worth $125.89 million now,
   • Short position in USD bill i.e. a short position in a USD investment, worth $130.09 million i   a year, or (𝐾𝑒^–rc) = $125.89 million now.

• • 
    Considering only the spot position, the VaR is $125.89 million times the risk of 4.538 percent,   which is $5.713 million. To compute the diversified VaR, we use the risk matrix from the data   in the previous table and pre- and post-multiply by the vector of positions (PV of flows   column in the table). The total VaR for the forward contract is $5.735 million. This number is   about the same size as that of the spot contract because exchange-rate volatility dominates   the volatility of 1-year bonds.
  • More generally, the same methodology can be used for long-term currency swaps, which are   equivalent to portfolios of forward contracts. For instance, a 10-year contract to pay dollars   and receive euros is equivalent to a series of 10 forward contracts to exchange a set amount   of  dollars into marks. To compute the VaR, the contract must be broken down into a   currency-risk component and a string of  USD and EUR fixed-income components. As   before, the total VaR will be driven primarily by the currency component.

Mapping Commodity Forwards

• The valuation of forward or futures contracts on commodities is substantially more complex   than for financial assets such as currencies, bonds, or stock indices. Most commodities do not   make monetary payments but instead are consumed, thus creating an implied benefit. This   flow of benefit, net of storage cost, is loosely called convenience yield to represent the   benefit from holding the cash product. This convenience yield, however, is not tied to another   financial variable, such as the foreign interest rate for currency futures. It is also highly   variable, creating its own source of risk.
• Monthly VaR measures for commodity futures are very high, in contrast to currency and equity market VaRs. Thus commodities are much more volatile than typical financial assets.
• Volatilities decrease with maturity of commodity futures contracts. The effect is strongest for less storable products such as energy products and less so for base metals. And this effect is very light for precious metals, which have low storage costs and no convenience yield. For financial assets, volatilities are driven primarily by spot prices, which implies basically constant volatilities across contract maturities.
• The forward rate agreement can be decomposed into two zero coupon components. If long position in a 6 x 12 FRA is considered, the components are:

Computing the VaR of a $100 FRA (monthly VaR at 95 percent level)

his table provides a worked-out example. If the 360-day spot rate is 5.8125 percent and the 180-day rate is 5.6250 percent, the forward rate must be such that



The present value of the notional $100 million in 6 months is



This amount is invested for 12 months.

Mapping Forward Rate Agreements

• The previous table also displays the computation of VaR for the FRA. The VaRs of 6- and   12-month zeroes are 0.1629 and 0.4696, respectively, with a correlation of 0.8738. Applied to   the principal of $97.26 million, the individual VaRs are $0.158 million and $0.457 million,   which gives an undiversified VaR of $0.615 million. Fortunately, the correlation substantially   lowers the FRA risk. The largest amount the position can lose over a month at the 95 percent level is $0.327 million.

Mapping Interest Rate Swap

• Interest-rate swaps are the most actively used derivatives. They create exchanges of interest-   rate flows from fixed to floating or vice versa. Swaps can be decomposed into two legs, a   fixed leg and a floating leg. The fixed leg can be priced as a coupon-paying bond; the floating   leg is equivalent to a floating-rate note (FRN).
• To illustrate, let us compute the VaR of a $100 million 5-year interest-rate swap. We enter a   dollar swap that pays 6.195 percent annually for 5 years in exchange for floating-rate payments   indexed to London Interbank Offer Rate (LIBOR). Initially, we consider a situation where the   floating-rate note is about to be reset. Just before the reset period, we know that the coupon   will be set at the prevailing market rate. Therefore, the note carries no market risk, and its   value can be mapped on cash only. Right after the reset, however, the note becomes similar to   a bill with maturity equal to the next reset period.
• Interest-rate swaps can be viewed in two different ways: as (1) a combined position in a fixed- rate bond and in a floating-rate bond or (2) a portfolio of forward contracts. We first value the swap as a position in two bonds using risk VaR data from the earlier table that we used to map fixed income securities. The analysis is detailed in table given in the next slide.

Computing the VaR of a $100 million Interest Rate Swap (monthly VaR at 95 percent level)


• The second and third columns lay out the payments on both legs. Just before reset, the long position in the FRN is worth $100 million, and the market value of the swap is zero. To clarify the allocation of current values, the FRN is allocated to cash, with a zero maturity. This has no risk. The next column lists the zero-coupon swap rates for maturities going from 1 to 5 years. The fifth column reports the present value of the net cash flows, fixed minus floating. The last column presents the component VaR, which adds up to a total diversified VaR of $2.152 million. The undiversified VaR is obtained from summing all individual VaRs. As usual, the $2.160 millio value somewhat overestimates risk.This swap can also be viewed as the sum of five forward contracts, as shown in this table. The 1-year contract promises payment of $100 million plus the coupon of 6.195 percent; discounted at the spot rate of 5.813 percent, this yields a present value of -$100.36 million. This is in exchange for $100 million now, which has no risk.

An Interest-Rate Swap Viewed as Forward Contracts (monthly VaR at 95 percent level)

• The next contract is a 1 x 2 forward contract that promises to pay the principal plus the fixed   coupon in 2 years, or -$106.195 million; discounted at the 2-year spot rate, this yields -$94.64   million. This is in exchange for $100 million in 1 year, which is also $94.50 million when   discounted at the 1-year spot rate. And so on until the fifth contract, a 4 x 5 forward contract.
• The previous table also shows the VaR of each contract. The undiversified VaR of $2.401   million is the result of a simple summation of the five VaRs. The fully diversified VaR is
• $2.152 million, exactly the same as in the preceding table. This demonstrates the equivalence of the two approaches.
• Finally, we examine the change in risk after the first payment has just been set on the floating-   rate leg. The FRN then becomes  a 1-year bond  initially valued at par but subject to   fluctuations in rates. The only change in the pattern of cash flows in Table 4-10 is to add $100 million to the position on year 1 (from -$5.855 to $94.145). The resulting VaR then decreases from $2.152 million to $1.763 million. More generally, the swap’s VaR will converge to zero as the swap matures, dipping each time a coupon is set.

Mapping Options



• Let’s consider the mapping process for nonlinear derivatives, or options. Obviously, this nonlinearity may create problems for risk measurement systems based on the delta-normal approach, which is fundamentally linear. To simplify, consider the Black-Scholes (BS) model for European options. From Part 1, we remember that





where 𝑁(𝑑) is the cumulative normal distribution function with arguments, and



where 𝐾 is now the exercise price at which the option holder can, but is not obligated to, buy the asset.



• Delta, is particularly important for options. For a European call, this is . This figure displays its behavior as a function of  the underlying spot price and for various maturities. The figure shows that delta is not a constant, which may make linear methods inappropriate for measuring the risk of options. Delta increases with the underlying spot price. The relationship becomes more nonlinear for short-term options, for example, with an option maturity of 10 days. Linear methods approximate delta by a constant value over the risk horizon. The quality of this approximation depends on parameter values.





For instance, if the risk horizon is 1 day, the worst down move in the spot price is



leading to a worst price of  $97.92. With a 90-day option, delta changes from 0.536 to 0.452 only. With such a small change, the linear effect will dominate the nonlinear effect. Thus linear approximations may be acceptable for options with long maturities when the risk horizon is short.

• In the extreme case, where the option is deep in the money, both 𝑁(𝑑₁) and 𝑁(𝑑₂) are equal to unity, and the option behaves exactly like a position in a forward contract. In this case, the BS model reduces to 𝑐 = 𝑆𝑒^–r∗c — 𝐾𝑒^–rc, which is indeed the valuation formula for a forward contract. Also note that the position on the dollar bill 𝐾𝑒^–rc𝑁 𝑑₂is equivalent to

this shows that the call option is equivalent to a position of Δ in the underlying asset plus a short position of  Δ𝑆 — 𝑐  in a dollar bill, that is

L𝑜𝑛𝑔 𝑜𝑝𝑡𝑖𝑜𝑛  =  𝑙𝑜𝑛𝑔 Δ 𝑎𝑠𝑠𝑒𝑡 + 𝑠ℎ𝑜𝑟𝑡  Δ𝑆 — 𝑐 𝑏𝑖𝑙𝑙

For example, assume that the delta for an at-the-money call option on an asset worth $100 is Δ =

0.536. The option itself is worth $4.20. This option is equivalent to a ΔS = $53.60 position in the underlying asset financed by a loan of ΔS – c = $53.60 – $4.20 = $49.40.

• The next step in the risk measurement process is the aggregation of exposures across the portfolio. Thus all options on the same underlying risk factor are decomposed into their delta equivalents, which are summed across the portfolio. This generalizes to movements in the implied volatility, if necessary. The option portfolio would be characterized by its net vega, or Λ. This decomposition also can take into account second-order derivatives using the net gamma, or Γ. These exposures can be combined with simulations of the underlying risk factors to generate a risk distribution.