A bank's five-year swap with a mid-sized client has expected positive exposure of Rs 8, 12, 14, 11 and 5 crore by year. Default probability is 2% a year, loss given default 60%. Compute the CVA and say how collateral would change it.

The CVA is about Rs 0.49 crore, with year 3 the largest contributor, and a Rs 3 crore threshold collateral agreement would cut it to roughly Rs 0.14 crore. Each year's contribution is loss given default times expected exposure times the chance of default in that year times the discount factor. The default chance in year t is 2% times the 98% survival of each earlier year. Collateral leaves the default odds alone and shrinks the exposure above the threshold.

If you lend a friend money that he repays only when you ask, your risk is how much he owes on the day he disappears, times the chance he disappears, times how much you would not get back. The credit valuation adjustmentThe price of the chance that a derivatives counterparty defaults while owing you money. It is deducted from the trade value and charged to the client. is that same product, summed year by year and discounted to today. The swap's value to the bank moves with rates, so the amount owed is an average over scenarios where the client owes the bank: the expected positive exposure.

Work the marginal default chance first: 2.00%, 1.96%, 1.92%, 1.88%, 1.84%. The chance falls slightly each year because the client must survive the earlier years to default in a later one. Then multiply across. Year 1 gives 0.6 times 8 times 2.00% times 0.94, Rs 0.090 crore. Year 3 gives 0.6 times 14 times 1.92% times 0.83, Rs 0.134 crore, the largest piece, because the exposure peaks there. The total is Rs 0.4856 crore, about Rs 0.49 crore. Using a flat 2% every year, ignoring survival, gives Rs 0.503 crore, a small overstatement worth naming.

Two forces pull against each other. Early on, rates have had little time to move, so the swap's value cannot have drifted far from zero. Late in the life, few payments are left, so even a large rate move changes the value by little. The peak sits where rates have had time to move and enough payments remain for the move to matter, typically a third to a half of the way through. A cross-currency swap with a final exchange of principal would not hump; its exposure would keep climbing to maturity.

Under a collateral agreement the client posts collateral for whatever it owes above Rs 3 crore. Collateral does not change the client's default chance; it shrinks the exposure bars, here to at most Rs 3 crore a year, which cuts the CVA from Rs 0.49 crore to roughly Rs 0.14 crore, about 70% less. The figure is approximate: capping the average at the threshold slightly overstates the remaining exposure, and it leaves out the gap risk, the days between the client's last margin payment and the bank closing out after a default, during which rates can move again. A desk would model that margin period of risk explicitly.

Close with what the client hears. The swap price carries a credit charge of about Rs 0.49 crore today, spread into the fixed rate. Signing the collateral agreement would cut that to about Rs 0.14 crore, in exchange for the client finding liquid collateral when rates move against it. For a mid-sized company that liquidity cost can be larger than the saving, which is why many such clients stay uncollateralised and pay the charge.