Swap valuation formulas

Thought to share some formulas I am using when valuing various swaps (all formulas assume 1 unit notional and _ all rates are _ _ un-annualized _).

  1. Value of fixed leg = fixed rate*(Z1+Z2+Z3+Z4) + Z4
  2. Value of floating leg = (1+floating rate)*Z1
  3. Value of equity leg = (Index value today) / (Index value inception)
  4. Value of foreign currency fixed leg = [fixed rate*(Z1+Z2+Z3+Z4)+Z4] * (current FX rate/inception FX rate)
  5. Value of foreign currency floating leg = [(1+floating rate)*Z1] * (current FX rate/inception FX rate)
  6. Value of swaption = (strike rate-new fixed rate) * (Z1+Z2+Z3+Z4)

Notes: - Z1, Z2 etc are the discount rates based on the new LIBOR structure. - The formulas assume you know how to calculate Z factors, fixed rates and how to identify the floating rate.

This is exactly what i needed - thanks so much! I get swaps, i know what i want to do, just can never remember the formulae - this should definitely help.

thanks again!

Thanks man, this helps.

Regarding a Pay Fixed Equity and Pay Fixed swap, I noticed that they calculated the value of the swap as

Value of Swap: (New Equity Level/Old Equity Level) - Z4 - Fixed Rate(Z1 + Z2 + Z3 + Z4)

This is in Schweser. Any reason why it’s slightly different?

Nice summary.

Is this for receiver swaption only? So what’s the formula for payer swaption? Max(0, floating - strike rate) ?


That’s valuing the contract. I wrote down individual leg valuation. Obviously in a contract there will be two legs, but knowing how to value each leg separately helps to prepare for any combination.

This formula of yours is correct but you can perhaps see it better using brackets.

(New Equity Level/Old Equity Level) - [Z4 + Fixed Rate(Z1 + Z2 + Z3 + Z4)],

which is 3) - 1)

Right… maybe you can suggest? :slight_smile:

Correct. Payer is (market - strike) x (days/360) x notional.

It is important to note that a swaption contract specifies a fixed rate. Whether you want to be the payer or the receiver, this is your reference rate. Let’s say this rate is 4%.

On expiration, suppose the LIBOR rates dictate a fixed swap rate of 3.5%. If you had a payer swaption ( to pay 4%) you would pass this opportunity and let the option expire, because you can make a better swap agreement using the market rates.

If you had a receiver swap, you would exercise the swaption and make an agreement to receive 4% fixed.

If you remember, at inception a swap has no value. In this case it has a value, because you have an advantage of (4-3.5=0.5%) against other market participants. That’s what we value.

This mean ther is only one formula for swaptions, which essentially uses the absolute value difference between the fixed rates of the contract and the market. You just have to make sure it has intrinsic value (i.e. can be exercised) rather than value something worth zero…

Is the reason an interest rate swap has no current credit risk at the beginning because the value is 0 to both parties?

why does potential credit risk reduce towards teh end of swap (and why is it highest in middle?)

Curency credit risk is always substantially higher because there is no netting process right?

At inception the contract is created using market rates, which are transparent and fair. So no party has an outright advantage at t=0. Only time will tell who was right who was wrong.

Every predetermined period (quarter, semester etc) the contract is marked to market to see who was right (who won) and who was wrong (who lost). The loser pays the difference. Towards the end of the contract the cash flow periods (quarters, semester etc) become less and less and hence the loser has more chances to make to the finish before becoming bankrupt…

Why is the midpoint the most dangerous in terms of counterparty risk? Because by that time the picture becomes more clear who is winning and chances are trends will continue. In the beginning when the two parties came together they had pretty good financial positions (due diligence was done) so it was assumed credit risk is small (otherwise call off the deal, right?). Towards the end we said dust is clearing and the loser is making it to the finish, so the greatest uncertainty is in the midpoint.

Currency swap is different. There is a large cash flow towards the end (the exchange of notional) that is why the last period is the most crucial one.

Thanks, this was very helpful.

Also, for #4 and #5 regarding the foreign currency. I just tried this out and it worked. For anyone using this, it is the value of the foreign leg IN the domestic currency.

For example, you start off with $20 million and you apply the FX at inception to get to the foreign currency. Then you calculate the fixed or floating payment in that foreign currency. Then you apply the Current FX rate to get back to your payment in domestic currency.

Essentially, formula #4 and #5 does all of it for you and you just need to multiply this with the NOTIONAL $

Brilliant! Please continue to make these posts for other topics too to impart your wisdom.

I need to practice on 4 and 5 but good to know of this list as I only discovered 6 today myself!

Do you guys have any good practice examples of 4 and 5?

Done the topic test ones but they had added layers

Do you guys have any good practice examples of 4 and 5?

Done the topic test ones but they had added layers

Thanks for this.

Should it be the values at last settlement rather than inception, since the swaps reset each settlement date?