Two currency exposures, their volatilities and one correlation: the variance-covariance calculation a treasurer can defend, and the hedge ratio it points to.
Cash flow at risk is the shortfall against forecast cash flow that will be exceeded only with a chosen probability: the z-score times the standard deviation of the combined exposure, with the correlation between the exposures inside that standard deviation. On an illustrative company with 80 million of net USD inflows and 45 million of net GBP outflows over the next year, the 95 per cent CFaR is 9.15 million, not the 14.97 million you get by adding the two currencies' risks together.
Worked in full in Treasury Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A eurozone company forecasts its foreign currency cash flows for the next twelve months. Its US subsidiary remits net inflows; its UK supply contracts are paid in sterling. Both are converted at today's rates into euros. All figures are illustrative, in EUR millions.
| Input | Value |
|---|---|
| Net USD flow, next 12 months | +80 |
| Net GBP flow, next 12 months | −45 |
| One-year volatility, EUR/USD | 8% |
| One-year volatility, EUR/GBP | 6% |
| Correlation of the two moves against the euro | 0.5 |
| z-score, 95% one-tailed | 1.645 |
| z-score, 99% one-tailed | 2.326 |
| Liquidity headroom above the policy minimum | 12.0 |
1. The standard deviation of each exposure. Exposure times volatility, keeping the sign: the company is long dollars and short sterling.
σUSD = 80 × 8% = 6.40. σGBP = −45 × 6% = −2.70.
2. Combine them with the correlation. The variance of a sum is the sum of the variances plus twice the covariance.
Variance = σUSD2 + σGBP2 + 2 × ρ × σUSD × σGBP
= 40.96 + 7.29 + 2 × 0.5 × 6.40 × (−2.70) = 40.96 + 7.29 − 17.28 = 30.97
Standard deviation = √30.97 = 5.565
Excel, with signed exposures in B2:B3, volatilities in C2:C3 and the correlation matrix in E2:F3: =SQRT(MMULT(MMULT(TRANSPOSE(B2:B3*C2:C3),E2:F3),B2:B3*C2:C3))
The cross term is negative because a weaker euro raises the value of the dollar inflow and, with a positive correlation, also raises the cost of the sterling outflow. The two exposures partly hedge each other.
3. Scale by the z-score.
CFaR95 = 1.645 × 5.565 = 9.15. CFaR99 = 2.326 × 5.565 = 12.94.
In one year out of twenty, currency moves should leave the company at least 9.15 million short of the euro cash flow it planned. Against 12.0 million of liquidity headroom that is 76.3 per cent of the cushion, which is the comparison that makes CFaR useful: a risk number is only meaningful beside the buffer it could consume. Taken alone, each exposure would show 10.53 and 4.44; the correlation removes 5.81 million, 38.8 per cent of the simple sum.
Selling forward 60 per cent of the dollar inflow leaves 32 million exposed: a dollar standard deviation of 2.56, a combined variance of 6.93, a combined standard deviation of 2.633 and a CFaR of 4.33, down 52.7 per cent and now 36.1 per cent of the headroom. The table shows how much the answer depends on the correlation.
| Correlation | Unhedged | 60% of USD hedged |
|---|---|---|
| −0.5 | 13.32 | 7.49 |
| 0.0 | 11.43 | 6.12 |
| 0.5 | 9.15 | 4.33 |
| 0.8 | 7.47 | 2.74 |
The hedge ratio has an optimum. Because the sterling outflow offsets part of the dollar risk, the variance is lowest when the dollar exposure left open is −ρ × σGBP ÷ volUSD = 16.87 million, a hedge of 78.9 per cent, where CFaR is 3.85. Hedge the whole dollar inflow and the sterling exposure stands alone again: CFaR rises to 4.44. Hedging 60 per cent of both currencies gives 3.66.
CFaR assumes the exposures are known and the moves normally distributed. Neither holds in the tail: forecast volumes are themselves uncertain, which is why the book sizes its liquidity reserve on the measured error of the cash forecast, as in how to set a minimum cash balance, and not on market moves alone.
The book works through one company's five cash balances and the hedging programme behind them, and the free workbooks for this case reproduce every figure it prints with a live formula.
Value at risk measures how much the market value of a portfolio could fall over a short horizon, often one or ten days. Cash flow at risk measures how far a company's cash flows over a budget horizon, usually a year, could fall short of plan. The arithmetic is the same: on the illustrative exposures, a 5.565 million standard deviation gives a 95 per cent CFaR of 9.15 million and a 99 per cent CFaR of 12.94 million.
No, when exposures offset. On the illustrative company the USD inflow and GBP outflow partly cancel because the two currencies move together. Hedging 78.9 per cent of the USD minimises CFaR at 3.85 million; hedging all of it leaves the GBP exposure on its own, and CFaR rises back to 4.44 million.
Measure it from the same history as the volatilities and test the answer at other values, because correlations break in stress. On the illustrative exposures, 95 per cent CFaR is 9.15 million at a correlation of 0.5, 11.43 million at zero and 13.32 million at minus 0.5. If the decision changes across that range, report the range.
This article is one calculation from Treasury Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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