OIS Discounting

If you lend a friend $100 and they hand you a gold watch as collateral, the interest rate you'd charge them should look a lot like the safest rate you could otherwise earn on that $100 — because your risk is basically zero, the watch covers you. If they hand you nothing, you'd charge more, because now you're exposed to them simply not paying you back. Discounting a derivative's cash flows works on exactly this logic. A derivative backed by posted collateral — cash that sits in an account earning the safe overnight rate — should be discounted at that overnight rate. Before the 2008 crisis, banks discounted almost everything, collateralized or not, at LIBOR, the rate banks charge each other unsecured. The crisis exposed that as wrong for collateralized trades, and the industry switched to OIS discounting: value the cash flows using the overnight indexed swap rate, which tracks what the collateral itself actually earns.

Why the rate you discount with matters at all

A future cash flow is worth less than the same amount today, and how much less depends on the rate of return available on money between now and then. Discounting at the wrong rate doesn't just shift the price a little — it consistently misprices every long-dated swap in the same direction, because the compounding effect grows with time.

$$PV = \frac{CF}{(1 + r)^{T}} .$$

In plain English: $CF$ is the future cash flow, $T$ is how many years away it lands, and $r$ is the discount rate — the annual return you could otherwise earn on money held for that long. Pick a higher $r$ and the same future cash flow is worth less today, because you're implicitly saying "money grows faster elsewhere, so this promise is less valuable by comparison." The entire OIS-discounting question is: which $r$ is the right one for a derivative's cash flows, and the crisis-era answer was that it depends on whether the position is collateralized.

Worked example 1 — the basis on a single cash flow

A bank is owed $1,000,000 in one year from a fully collateralized swap. The OIS rate for one year is 3.00%; the unsecured term rate for the same maturity (the old LIBOR-style benchmark) is 3.30% — a 30 basis point gap, the OIS-LIBOR basis. Discounted at OIS: $PV = 1{,}000{,}000 / 1.03 =$ $970{,}874. Discounted at the unsecured rate: $PV = 1{,}000{,}000 / 1.033 =$ $968{,}054. The gap is $970{,}874 - 968{,}054 =$ $2{,}820 on a single $1 million cash flow one year out — small in percentage terms, but real money, and it scales with both the notional and the tenor. A swap desk running a book of billions in notional across many years cannot round this away; using the wrong curve here is a direct, quantifiable pricing error, not a technicality.

Worked example 2 — the gap widens with maturity

The basis effect compounds the longer the cash flow's horizon. Take a $5,000,000 cash flow due in five years. Using an OIS rate of 2.5%: $PV = 5{,}000{,}000 / (1.025)^{5} = 5{,}000{,}000 / 1.1314 =$ $4{,}419{,}600. Using an unsecured funding rate of 4.0% instead: $PV = 5{,}000{,}000 / (1.04)^{5} = 5{,}000{,}000 / 1.2167 =$ $4{,}109{,}600. The gap is now $4{,}419{,}600 - 4{,}109{,}600 =$ $310{,}000 — over $300,000 of value on a single cash flow, purely from the choice of discount curve, with the underlying cash flow amount and date completely unchanged. Compare this gap to worked example 1's $2,820 on a one-year flow of similar relative size: the same rate difference, applied over five years of compounding instead of one, produced over 100 times more dollar impact. That's why the discounting curve choice matters most for the long-dated end of a swap book.

The multi-curve framework

Modern derivatives desks don't use one curve for everything — they use (at least) two. One curve, built from OIS rates, is used purely for discounting: converting any future cash flow into today's money. A separate curve, built from the relevant floating-rate benchmark (SOFR-based term rates today), is used purely for projecting what the floating leg of a swap is expected to pay. These two curves can, and usually do, disagree slightly — that disagreement is the basis from worked example 1 — and keeping them separate is what lets a desk price both the discounting effect and the projected cash flow correctly, instead of conflating two different jobs into one curve the way pre-crisis practice did.

Adjust the level and slope on the curve above. A real derivatives desk holds two of these side by side — one for discounting collateral, one for projecting floating payments — and the vertical gap between them at any given tenor is exactly the basis from worked example 1.

What this means in practice

Every collateralized derivatives trade today is priced and risk-managed using OIS discounting as standard practice — it's built into every major pricing library and required by CSA (credit support annex) terms that specify collateral is remunerated at the overnight rate. The switch from LIBOR to OIS discounting after 2008 was one of the largest single re-pricings in derivatives history: banks had to revalue enormous swap books overnight, and the basis between the two curves became a tradable instrument in its own right (basis swaps).

Practice

1. A $2,000,000 cash flow is due in three years. Compute its present value using a 2.8% OIS rate and, separately, using a 3.5% unsecured rate. What's the dollar gap?
2. Why does the OIS-LIBOR (or OIS-SOFR term rate) basis widen during periods of banking-sector stress, and what does that widening say about the market's view of unsecured bank credit risk versus overnight collateralized risk?