Future Value and Exposure: formula sheet

Key formulas

Exposure

Exposure = max(V, 0)

V is the value of the contract or netting set. Negative value is not an exposure.

Current exposure

CE = max(V₀, 0)

Uses today's value only. No simulation needed.

Expected exposure

EE(t) = E[max(V(t), 0)]

Average over scenarios at date t. Includes zeros from scenarios where V is negative.

Potential future exposure

PFE(t) = the α-percentile of max(V(t), 0)

Typically α = 95% or 99%. Usually above EE(t), but not guaranteed. It depends on the distribution.

Expected positive exposure

EPE = Σ EE(tᵢ) × Δtᵢ ÷ T

Time-weighted average of EE. With equal time steps it is the simple average of the EE values.

Effective EE

Effective EE(tₖ) = max(EE(tₖ), Effective EE(tₖ₋₁))

Makes the profile non-decreasing. Reflects replacement of maturing trades.

Effective EPE

Effective EPE = average of Effective EE over min(1 year, longest maturity)

Basel regulatory measure for counterparty credit risk exposure.

Negative exposure

NE = min(V, 0); ENE(t) = E[min(V(t), 0)]

Quoted as a negative number, or as its absolute value. Feeds DVA.

Exposure at time t

Exposure(t) = max(V(t), 0)

V is the contract's value to you. Exposure is never negative. Ignores netting and collateral.

Expected exposure (EE)

EE(t) = E[max(V(t), 0)]

The average positive value at future date t. Plotting EE against t gives the expected exposure profile.

Potential future exposure (PFE)

PFE(t) = the α-percentile of max(V(t), 0)

A high quantile, such as 95% or 97.5%, of the exposure distribution at t. It is typically well above EE.

Diffusion scaling

Standard deviation of a risk factor ∝ σ × √t

Uncertainty grows with the square root of time, so diffusion alone gives a rising, concave profile.

Forward exposure at the money

EE(t) ∝ σ × √t (for an at-the-money forward)

Rises until maturity. Exposure is largest at the end, since no amortization occurs.

Profile shapes by product

Loan or bond: flat if bullet, declining if amortizing; Forward: rising; IR swap: hump; Cross-currency swap: rising, late peak; Bought option: value-based, generally rising to expiry, then zero once the payoff is settled

Use this as a memory map for exam questions.

Exposure on a path and date

E(t) = max(V(t), 0)

V is the net value of the netting set after collateral terms are applied. Negative value is not a loss to you.

Expected exposure (EE)

EE(t) = (1 ÷ N) × Σ max(V_i(t), 0), summed over paths i = 1 to N

Average of positive exposures across simulated paths at date t.

Potential future exposure (PFE)

PFE_α(t) = the α-percentile of the distribution of E(t)

Typically 95% or 97.5%. Pick the percentile from the sorted simulated exposures.

Expected positive exposure (EPE)

EPE = time-weighted average of EE(t) over the horizon

A single-number average of the EE profile, used in capital and CVA contexts.

GBM step (risk-neutral FX or equity)

S(t+Δt) = S(t) × exp[(r − q − σ²÷2)Δt + σ√Δt × Z]

Z is a standard normal draw. Under the real-world measure, replace (r − q) with the expected return μ.

Monte Carlo standard error

SE ≈ s ÷ √N

s is the sample standard deviation. Quadrupling paths halves the error.

Uncollateralised exposure, one trade

E = max(V, 0)

V is the mark-to-market value to you. Negative value means no loss to you on default.

Exposure without netting

E(no netting) = Σ max(Vᵢ, 0)

Sum of positive trade values only.

Exposure with close-out netting

E(netting) = max(Σ Vᵢ, 0)

Always ≤ exposure without netting. Applies per netting set, not across counterparties.

Netting benefit

Benefit = E(no netting) − E(netting)

Zero when all trades have the same sign.

Collateralised exposure (static)

E(coll) = max(V − C, 0)

C is collateral held, V is net value of the netting set.

Collateral called with threshold and MTA

Call = max(V − Threshold − C, 0), made only if the amount ≥ MTA

Applies to collateral called from the counterparty. Ignores independent amount and rounding.

Exposure over the MPOR

E(coll) = max(V(t + MPOR) − C(t), 0)

C(t) is collateral held at the last margin call before default.

Maximum uncollateralised gap at the call date (rule of thumb)

Gap ≈ Threshold + MTA (at the call date only, before any MPOR move)

Right after a margin exchange, collateral can be short of the net value by roughly the threshold plus the MTA. This excludes moves over the MPOR and the independent amount, so it is not a bound on total exposure.

Independent CVA (discrete approximation)

CVA ≈ LGD × Σ EE(tᵢ) × PD(tᵢ₋₁, tᵢ)

Uses discounted expected exposure and assumes exposure and default are independent. WWR makes this too low; RWR makes it too high.

Wrong-way risk condition

E[Exposure | default] > E[Exposure]

Exposure conditional on default exceeds unconditional expected exposure.

Right-way risk condition

E[Exposure | default] < E[Exposure]

Exposure at default is lower than the unconditional expectation.

Dependence-adjusted CVA idea

CVA_WWR = LGD × Σ E[Exposure(tᵢ) | default in (tᵢ₋₁, tᵢ)] × PD(tᵢ₋₁, tᵢ)

Uses conditional exposure. This is the conceptual fix for dependence.

Alpha-style multiplier

CVA_adj = α × CVA_independent, with α > 1 for WWR

A simple conservative adjustment. The value of α is a modelling or regulatory choice, not a universal constant.

Exposure at a date

Exposure = max(V, 0), after netting and collateral

V is the netted portfolio value less collateral held. Exposure is never negative.

Expected exposure (EE)

EE(t) = E[max(V(t), 0)]

Average positive exposure at date t. Not the same as the average value.

Potential future exposure (PFE)

PFE(t) = the α-percentile of exposure at date t

Typically 95% or 99%. Higher confidence gives a higher PFE.

Maximum PFE

Max PFE = max over t of PFE(t)

The peak of the PFE profile. Used for limits.

Effective EE

Effective EE(t) = max(EE(t), Effective EE(t−1))

Makes the EE profile non-decreasing within the first year. Reflects rollover of short trades.

EEPE

EEPE = time-weighted average of Effective EE over the first year

Use the first year, or the longest maturity if shorter than one year. With equal intervals, the simple average of the interval values is the time-weighted average.

Regulatory EAD

EAD = alpha × EEPE, alpha = 1.4

Alpha is 1.4 under the Basel IMM unless a supervisor-approved estimate is used.

Quick revision

• Exposure = max(V, 0). You lose only when the contract has positive value to you.
• EE at a date is the average of the positive exposures across scenarios at that date.
• PFE is a chosen high percentile of exposure at a date, used for limits.
• EPE is the time average of the EE profile over a period.
• Swap exposure rises, peaks mid-life, then falls as payments are made and fewer cash flows remain.
• A forward's exposure typically rises with time as uncertainty grows (the standard deviation of value scales roughly with the square root of time), peaks at maturity, and drops to zero on settlement.
• The option buyer's exposure equals the option's value (never negative). The seller has no exposure to the buyer on the option, because the buyer has no further obligations once the premium is paid.
• Netting lets you offset positive and negative values only within a legally enforceable netting set.
• Netted exposure is never greater than the sum of the individual exposures.
• Collateral reduces exposure but leaves residual risk over the margin period of risk.
• Wrong-way risk is a positive dependence between exposure and the counterparty's default probability. Exposure tends to rise as its credit quality falls.
• Right-way risk is a negative dependence between exposure and the counterparty's default probability. Exposure tends to fall as its credit quality falls.

Common mistakes

• Leaving out zero-exposure scenarios when computing EE. Fix: EE is the expectation of max(V, 0) over all scenarios. Count the zeros in the denominator.
• Treating PFE as the average of the exposures above some percentile. Fix: PFE is the percentile itself, a VaR-like quantile of exposure. It is not a tail average.
• Drawing a swap profile that keeps rising until maturity. Fix: For a swap, include amortization. Exposure falls to zero at maturity because no cash flows remain.
• Treating a cross-currency swap like an interest rate swap. Fix: Remember the final notional exchange. It keeps FX exposure large until the end, so the peak is later and the profile does not fall as early.
• Using the risk-neutral measure for PFE. Fix: Remember that PFE is a real-world risk measure. Risk-neutral is for pricing and CVA.
• Averaging negative values into EE. Fix: Apply max(V, 0) to the netted value on each path first, then average.
• Netting across different counterparties Fix: Net only within one legally enforceable netting set. Positive values with another counterparty stay separate.
• Summing absolute values or including negative trades without netting Fix: Without netting use max(V, 0) per trade. With netting use max(ΣV, 0).
• Calling a trade wrong-way just because the counterparty is risky. Fix: WWR needs a link between exposure and default probability. A weak counterparty with independent exposure has no WWR.
• Mixing up general and specific WWR. Fix: Specific means a direct link through the trade or collateral, such as a CDS on the counterparty itself. General means a shared macro or market driver.

Exam tips

• Read whether the question asks for a single date (EE, PFE) or a time average (EPE). This decides the method.
• When a question gives a percentile, remember PFE is a quantile of floored exposure. Do not average above it.
• Watch for the word effective. It signals the non-decreasing step and the one-year window.
• Link metrics to use: PFE for limits, the discounted risk-neutral EE profile for CVA, effective EPE for regulatory capital, ENE for DVA.
• Always check that your answer is not negative for positive exposure. PFE at a high percentile is usually above EE, but this is not guaranteed, so use it only as a rough plausibility check.
• Questions are often conceptual: match the product to the shape, or name the effect that dominates early or late.
• Expect a trap on cross-currency versus single-currency swaps. Look for notional exchange.
• Check which party is the option buyer. Only the buyer has credit exposure.