Exposure Profiles by Product

Product-by-product differences in exposure shape matter to a credit officer deciding how much of a counterparty's limit a trade uses, and to a valuation specialist who builds counterparty credit risk into the price of trades.

How the shape differs by product

In counterparty credit risk, the amount you could lose if the counterparty defaults is called exposure. For derivatives, you have exposure when the market value of the trade (the replacement cost) is positive for you, and none when it is negative¹. The market value at a future date depends on how market prices move. So the average exposure at each date is expressed as expected exposure (EE), and the value at a high confidence level as potential future exposure (PFE). The series of EE and PFE across dates is called the profile, and its shape is compared product by product. The definitions and calculation of these measures are covered in Exposure Measures: CE, PFE, EE/EPE and EAD.

The shape depends on two things combined: how far market prices (interest rates, exchange rates) can move, and how much of the payments still remain at that date. The combination differs between an interest rate swap, which exchanges only the difference in interest, and an FX forward, which exchanges the full amount at maturity.

Two effects that set the shape, and an approximate formula

The market value of an interest rate swap at a future date is the difference between the market swap rate at that date and the contracted fixed rate, multiplied by the notional amount over the remaining term and discounted to present value⁸. As time passes, rates can move further, which widens the spread of possible market values. At the same time, each payment date reduces the remaining payments, which narrows that spread⁹. An FX forward has no interim payments, so the second effect does not apply before maturity.

You can compare the size of the two effects with approximate formulas that ignore discounting and mean reversion in rates. Let the notional amount be $N$, the term $T$ years, and the annual interest rate volatility (measured as the size of rate changes) $\sigma_r$. For a swap with annual payments, just after a payment date at time $t$ (years), $T - t$ payments remain. EE from the fixed-rate payer's side is approximately:

$$\mathrm{EE}(t) \approx 0.3989 \times N \times \sigma_r \sqrt{t} \times (T - t)$$

For a long FX forward (receiving foreign currency and paying yen), with an annual exchange rate volatility of $\sigma_{FX}$, it is:

$$\mathrm{EE}(t) \approx 0.3989 \times N \times \sigma_{FX} \sqrt{t}$$

The factor 0.3989 is the average of the positive part of a normally distributed value with mean zero, expressed as a multiple of its standard deviation. The swap formula grows with $\sqrt{t}$ and shrinks with $T - t$, so it forms a hump partway through. $\sqrt{t}(T - t)$ is largest at $t = T/3$, so in this approximation the hump peaks one-third of the way through the term (see the deep dive below for the derivation)¹⁰. The FX forward formula has no term that makes it shrink, so it keeps growing in proportion to $\sqrt{t}$ until maturity.

As a worked example, take a five-year pay-fixed swap with a notional of 10 billion yen and a rate volatility of 0.80% (80bp). The factor is 0.3989 × 10 billion yen × 0.008 = 31.91 million yen.

For an FX forward with the same notional of 10 billion yen and an FX volatility of 10%, the factor is 0.3989 × 10 billion yen × 0.10 = 398.9 million yen. The approximate EE is 399 million yen at one year and 398.9 × 1.3229 = 528 million yen at 1.75 years. An interest rate swap exchanges only the interest difference, while an FX forward exchanges the full amount, which corresponds to principal, so the FX forward has larger exposure for the same notional.

Figure description

Both trades have a notional of 10 billion yen, and the EE values come from the approximate formulas in the worked example above. The rate volatility is 0.80% and the FX volatility is 10%, and the points are joined by straight lines. The swap's EE rises from 128 million yen at one year to 135 million yen at two years, then falls to 111 million yen at three years and 64 million yen at four years, and reaches 0 at five years. The FX forward's EE keeps growing, from 399 million yen at one year to 564 million yen at two years, 691 million yen at three years, and 798 million yen at four years, and is 892 million yen just before settlement at five years (and 0 after settlement). The FX forward's values from two years on are the approximate formula extended in the same form.

Deep diveDeriving the approximate formulas

For a value $X$ that follows a normal distribution with mean zero and standard deviation $s$, the average of the positive part is:

$$E[\max(X, 0)] = \frac{s}{\sqrt{2\pi}} \approx 0.3989 \times s$$

Ignoring discounting, the market value at time $t$ of a pay-fixed swap is roughly $N (s_t - K)(T - t)$, where $s_t$ is the swap rate at that time and $K$ is the fixed rate. If rate changes are normally distributed with no mean reversion, the standard deviation of $s_t - K$ is $\sigma_r \sqrt{t}$. The standard deviation of the market value is then $N \sigma_r \sqrt{t}(T - t)$, and applying the formula above gives the approximate formula for the swap.

The market value of an FX forward is roughly $N x_t$, where $x_t$ is the percentage change in the exchange rate from the contracted rate. Treating the standard deviation of that change as $\sigma_{FX} \sqrt{t}$ gives the approximate formula for the FX forward. Neither formula includes discounting, mean reversion in rates, skew in the distribution of exchange rates, or the slope of the FX forward rate caused by the interest rate differential.

The swap's hump peaks where the derivative of $\sqrt{t}(T - t)$ with respect to $t$ is zero, which gives $T - t = 2t$, that is, $t = T/3$.

Interest rate swaps peak partway through

As the approximate formula shows, interest rate swap exposure rises to a hump partway through the term and then shrinks toward maturity². Because no principal is exchanged, the credit risk is much smaller than for transactions that exchange principal⁸. On each payment date, one payment drops out of the remaining payments, so exposure falls each time, and the fall is larger for payment dates closer to maturity⁹.

The fixed-rate payer gains a positive market value when rates rise, and the fixed-rate receiver when rates fall⁸. When the yield curve is flat and no rise or fall in future rates is expected, the exposures of the two sides are roughly symmetric, and neither side's exposure is larger than the other's¹¹. With an upward-sloping curve, later floating rates are expected to be higher. This tilts the fixed-rate payer's market value toward positive values, so the fixed-rate payer's exposure is larger¹²¹³.

FX forwards and cross-currency swaps

An FX forward has no payments between the trade date and maturity, and at maturity the full amounts of foreign currency and yen are exchanged. PFE keeps growing toward maturity, reaches its maximum there, and drops to zero once settlement is complete⁵.

In a cross-currency swap that re-exchanges principal at maturity, the principal payment carries counterparty credit risk in addition to the interest exchanges³. The exposure from the interest exchanges falls on each payment date. But because the final exchange of principal remains, the overall shape differs from an interest rate swap and stays large until just before maturity⁴.

Comparing a swap and an FX forward in the simulator

The simulator below starts with trade A (a five-year pay-fixed interest rate swap with a notional of 10 billion yen) combined with trade B, a two-year long FX forward with a notional of 10 billion yen. "Netting" and collateral are both off, so nothing is offset by close-out netting, and the EE and PFE on screen are the sum of the two trades' exposures. EE rises to 645 million yen at 1.75 years, just before the FX forward matures, and drops to 118 million yen at two years, once the FX forward has settled. Of the 645 million yen at 1.75 years, the approximate formula puts the FX forward's share at about 528 million yen.

After two years, only the swap in trade A remains. EE falls from 133 million yen at 2.75 years to 103 million yen on the payment date at three years, then from 110 million yen at 3.75 years to 62 million yen on the payment date at four years, and reaches zero at maturity at five years. The EE just after the payment dates at two, three, and four years (118, 103, and 62 million yen) is a little smaller than the approximations in the worked example (135, 111, and 64 million yen). The approximate formula leaves out discounting and mean reversion in rates, so it gives somewhat larger values.

In this setting, the average of EE up to maturity, expected positive exposure (EPE), is 222 million yen, and effective EPE, calculated over the first year, is 307 million yen. The 95% PFE reaches its maximum of 2,331 million yen at 1.75 years. If you replace trade B with a five-year receive-fixed swap and leave "Netting" off, EPE becomes 188 million yen. The simulator's curve is flat, so the pay-fixed swap has exposure on paths where rates rise and the receive-fixed swap on paths where rates fall. Without close-out netting, the two exposures add up instead of offsetting each other. How netting and collateral change the shape is covered in Risk Reduction through Netting and Collateral.

The simulator uses a flat curve, ignores correlation between FX and rates, and models the exchange rate with geometric Brownian motion. The shape of swap exposure changes with the level of rates, the slope of the curve, and rate volatility¹⁴, so actual trades will not match the figures from this setting. For long-dated trades, leaving out mean reversion in rates can make exposure at later dates too large¹⁵.

Options and CDS are one-sided

An option buyer pays the premium at the start of the trade and then simply holds the right, with nothing left to pay. So the market value of the option to the buyer starts at a positive value equal to the premium, and even if it falls as the market moves, it does not turn negative⁶. The buyer has exposure to the seller as long as the option has value, and a seller that has received the premium has no exposure. The seller does, however, bear settlement risk while waiting to receive the premium⁶. A long-dated option is an example of a trade where, unlike an interest rate swap, exposure is large for only one party¹³.

A CDS protection buyer receives a payment from the seller if a credit event occurs for the reference entity, so the buyer has exposure to the seller. In a bilateral trade, if the seller defaults, the buyer may not receive the payment⁷.

Use in credit limits and CVA

In practice around 2003, credit officers set credit limits on PFE at each date. Limits tended to be larger for shorter horizons and smaller for longer ones. When approving a new trade, they recalculated the PFE profile for the counterparty including that trade and compared it with the limits¹⁶. Interest rate swap exposure shrinks after its hump partway through². FX forward PFE, by contrast, grows toward maturity and peaks at maturity⁵. So an FX forward uses more of the limit at dates near maturity than an interest rate swap with the same notional and term.

For example, suppose Company H holds, with the same counterparty, a five-year pay-fixed swap and a long five-year FX forward, each with a notional of 10 billion yen. To see the shapes, add up the approximate EE from the section "Two effects that set the shape, and an approximate formula" (the table values for the swap, and 398.9 × √t million yen for the FX forward). At one year, that is 128 + 399 = 527 million yen. At four years, it is 64 + 398.9 × √4 = 64 + 798 = 862 million yen. The swap's share at four years has fallen to about half its one-year value, while the FX forward's share has roughly doubled, so the total rises. EE here is a simple sum with no close-out netting, and an actual limit is compared with PFE. On paths where rates fall, the market value of the pay-fixed swap is negative, and the exposure on those paths is 0. EE is the average over all paths, including these zeros.

When exposure can be treated as moving independently of the counterparty's credit quality, you can use the EE profile directly to calculate the credit valuation adjustment (CVA). CVA builds the counterparty's credit risk into the value of a trade¹⁶. So even with the same counterparty and the same notional, products with differently shaped EE have different amounts of CVA. However, exposure can also grow as the counterparty's credit quality worsens, which is called wrong-way risk. In that case, the expected loss can be larger than when there is no such relationship¹⁷.

Notes

1. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual", Section 2020.1 (September 1999), "Presettlement Risk", page 3. ↩

2. Ian Bond, Gareth Murphy, and Gary Robinson, "Potential Credit Exposure on Interest Rate Swaps" (Bank of England Working Paper No. 25, August 1994), the discussion of Figure 1, page 12. The figure uses the 95% quantile path for a 10-year swap. ↩ ↩² ↩³

3. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual", Section 4335.1, "Currency Swaps" (February 1998), "Credit Risk", page 4. ↩ ↩²

4. R. Sathis Kumar, "Understanding Credit Risk Profiles of Over-the-Counter Derivative Transactions", "PFE Profiles for Cross Currency Swaps", page 13. The figure uses the 99% PFE for a five-year cross-currency swap. ↩ ↩²

5. The same white paper, "PFE Profiles for Forward Contracts", page 6. ↩ ↩² ↩³

6. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual", Section 2020.1 (September 1999), "Presettlement Risk", page 3, the paragraph on options. ↩ ↩² ↩³ ↩⁴

7. 吉井一洋, "クレジット市場における検討課題" (Issues to consider in the credit market, in Japanese; Financial Research and Training Center, Financial Services Agency, 2009), "5.2.1 2008 年金融危機と集中清算機関設立の提言" (5.2.1 The 2008 financial crisis and the proposal to set up a central clearing house), page 47. ↩ ↩²

8. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual", Section 4325.1, "Interest-Rate Swaps", "Credit Risk", page 10 (April 2001) and page 11 (April 2003). ↩ ↩² ↩³

9. R. Sathis Kumar, "Understanding Credit Risk Profiles of Over-the-Counter Derivative Transactions", "PFE Profiles for Interest Rate Swaps", pages 9–10. ↩ ↩²

10. The approximate formulas are not taken from a source. They are derived from the average of the positive part of a normal distribution ($1/\sqrt{2\pi} \approx 0.3989$) and from treating the swap's market value as the rate difference multiplied by the remaining term (Section 4325.1). The EE values in the worked example are rounded to the nearest million yen. ↩

11. Ian Bond, Gareth Murphy, and Gary Robinson, "Potential Credit Exposure on Interest Rate Swaps", the discussion of Figure 5, page 21. ↩

12. The same paper, the discussion of Figure 7, page 25. ↩

13. Eduardo Canabarro and Darrell Duffie, "Measuring and Marking Counterparty Risk" (Euromoney Books, 2003), "Market valuation of credit exposures", page 128. ↩ ↩²

14. Ian Bond, Gareth Murphy, and Gary Robinson, "Potential Credit Exposure on Interest Rate Swaps", Abstract, page 1. ↩

15. Eduardo Canabarro and Darrell Duffie, "Measuring and Marking Counterparty Risk", "Simulation engine", page 125. ↩

16. The same book, "Applications of exposure modeling", page 127. ↩ ↩²

17. The same book, "Definitions", page 123. ↩

18. The same book, "PFE models: an overview", page 124, and "Trade pricing", page 126. ↩ ↩²