Financial Engineering – IC302
Autumn Term 2020/1
Seminar 3: Swaps Answers
1. Suppose that it is September and you are receiving fixed in a one-year swap denominated in USD. Floating payments in the swap are made quarterly and the 3- month USD LIBOR rate for the next floating payment (to be made in December) has already been fixed. You decide to hedge the swap by trading in CME Eurodollar futures contracts.
(a) Should you go long or short the futures contracts?
You should go short futures contracts. An increase in interest rates will create a loss on the swap position, since you are receiving fixed. But this loss will be offset by profits on the futures hedge, since higher rates will mean lower futures prices. You can close out the short futures position by buying back futures at these lower prices.
(b) In which futures expiry months are you most likely to trade?
December 2020, March 2021, and June 2021. The forward periods covered by these contracts correspond to the swap settlement periods for which we do not yet know the LIBOR rates. There is no risk to hedge with respect to the first floating payment to be made in December because the rate for that period has already been fixed.
2. Why is the expected loss to a bank from counterparty default in a plain vanilla fixed- for-floating swap less than the expected loss from default on a loan to the counterparty with the same principal amount?
The principal amount in a swap is purely notional and not exchanged between the counterparties, so it will not be lost in the event of default. In a loan, however, the principal amount may be lost in the event of default, which significantly increases the credit exposure to the counterparty.
Credit risk in a swap is also reciprocal, because it arises from market risk associated with interest rate movements. Changes in rates may make the MTM value of the swap positive or negative to either counterparty. We have credit risk exposure to our counterparty when the position has positive MTM value from our point of view, since we will suffer a loss if they default. Our counterparty has credit risk exposure to us when the position has positive MTM value from their point of view, since this exposes
them to loss if we default. The reciprocal nature of the risk reduces our expected exposure relative to what it would be in a loan, where only the lender has exposure to the borrower and not the reverse.
Finally, credit exposure in a swap is in many cases likely to be significantly mitigated by netting (in which the MTM values of positions between counterparties may be offset against one another in the event of default to determine a single net exposure) and collateralization (in which the counterparty for which the netted positions have a positive MTM value takes collateral from the counterparty for which the netted positions have a negative MTM value as protection against possible losses in the event of default). Netting and collateralization significantly reduce counterparty credit risk exposure. They may be managed multilaterally through central clearing with a central counterparty (CCP) or bilaterally through an ISDA Master Agreement and the Credit Support Annex (CSA) to that agreement (see Hull, ch. 2).
3. Under LIBOR discounting, if the one-year discount factor based on market swap rates is 0.980392, the two-year discount factor is 0.942319, and the three-year swap rate is 4%, what is the three-year discount factor?
0.887588.
The bootstrapping formula allows us to calculate the three-year discount factor from the one-year and two-year discount factors and the three-year swap rate as follows:
𝐷! = 1 − 𝑆!(𝐷” + 𝐷#) = 1 − 0.04(0.980392 + 0.942319) = 0.887588 1+𝑆! 1+0.04
The discount factors in this exercise correspond to market swaps rates of 2%, 3%, and 4% for one year, two years, and three years, respectively.
4. Based on the discount factors you calculated in the previous question, what is the projected forward LIBOR for the period between two years and three years from now under LIBOR discounting?
Approximately 6.17%.
Under LIBOR discounting, the projected forward LIBOR for the period between one year and two years from now is:
𝐷# 0.942319
𝐿#,! =𝐷! −1=0.887588−1≈6.17%
∗
You can confirm the answers to this question and the preceding question using the Excel workbook ‘Swap Valuation – LIBOR Discounting’.
5. Use the Excel workbook Swap Valuation – OIS Discounting to answer this question. Set up the Excel workbook Swap Valuation – OIS Discounting to reflect the following
initial market conditions:
(a) Use the Excel workbook to show that the mark-to-market value of a newly initiated 4-year swap with notional amount $10 million in which we pay fixed at a rate equal to the market 4-year swap rate is zero.
[Hint: After entering the interest rate data in the table above, make sure that the ‘Shift swap curve’ and ‘Shift OIS curve’ values in cells B12 and B28, respectively, are set equal to zero; that the ‘First LIBOR’ value in cell H20 is set equal to the 1-year swap rate; and that the ‘Swap rate’ value in cell H21 is set equal to the current market 4-year swap rate.]
(b) Use the Excel workbook to calculate the mark-to-market value a 4-year swap in which we pay fixed at 3.00% in these same market conditions. What explanation can you offer for this result?
$376,156. This can be calculated by changing the ‘Swap rate’ value in cell H22 to 3.00% and leaving the other input values in the workbook unchanged.
In this swap, we will be paying fixed at 3.00% in a market in which the current four-year swap rate is 4.00%. We are underpaying by 1.00% per year; on notional principal amount $10 million, this is $100,000 per year. Adding up over four years gives an undiscounted value of $400,000. Discounting these cash flows with the OIS discount factor gives a present value of $376,156, which is the MTM value of our position in the swap.
(c) Suppose that the swap in part (b) is our only derivative position with that counterparty. What is our credit risk exposure to the counterparty? What would our credit exposure be if we were the payer rather than receiver of fixed in the swap?
Year
Swap Rate
OIS Rate
1
1%
0.5%
2
2%
1.5%
3
3%
2.5%
4
4%
3.5%
Our credit risk exposure to a counterparty is the netted MTM value of our derivatives positions with the counterparty, conditional on that value being positive. For the swap in part (b), the swap has a positive MTM value of $376,156 from our point of view, so this is our credit risk exposure to the counterparty. If they were to default, we would have to go out an replace the swap. In current market conditions, however, a four-year swap to receive fixed has a MTM value of $376,156, so we would have to pay this amount upfront to our new counterparty to persuade them to enter into the swap.
If instead we were the payer of fixed in the swap in part (b), the MTM value of the position from our point of view would be negative $376,156. Our credit risk exposure to the counterparty would be zero, but they would have counterparty credit risk exposure of $376,156 to us. In derivatives markets, we are exposed to the credit risk of our counterparties when the derivatives positions into which we have entered with them have positive value.
In practice, of course, we may have multiple derivatives positions with the same counterparty that are part of the same netting set, in which case the netted MTM value of the positions that will determine our credit exposure. Central clearing or variation margin arrangements for non-centrally-cleared derivatives may also help limit counterparty risk.
(d) Use the Excel workbook to estimate the DV01 of the swap in part (b) for a one-basis-point shift in the swap curve.
[Hint: You can shift the swap curve by entering an appropriate value in cell B12.]
$2,766.
The MTM value of the swap before the shift is $376,156. After the shift, it is $378,922. The difference between these two values is $2,766. An increase in market swap rates increases the value of a swap in which we are the payer of the fixed.
(e) Restore the ‘Shift swap curve’ value in cell B12 to zero. What is the DV01 of the swap in part (b) for a one-basis-point in the OIS curve that leaves the market swap rates unchanged? Why is this different from the DV01 for a shift in the swap curve that you found in part (d)?
$91.
The MTM value of the swap before the shift is $376,156. After the shift, it is $376,065. The difference between these two values is $91.
In part (d), a shift in the swap curve increased the projected forward LIBORs (and hence the projected floating payments) but did not change the fixed
swap rate or the discount factors. The net effect was a significant increase in the MTM value of the swap. The impact of an increase in the OIS rates is more subtle. An increase in the OIS rates makes the OIS discount factors smaller, as you can confirm by looking at cells D7 to D10. This reduces the present value of the swap cash flows. It also, however, increases the projected forward LIBOR rates, as you can confirm by looking at the projected floating payments in cells J7 to J10. Under OIS discounting, the projected forward LIBORs must be consistent with both the OIS discount factors and the fixed-for-floating swap rates. Since we reduced the discount factors, the projected forward LIBORs had to increase to remain consistent with the unchanged swap rates. This offsets to some extent the impact of the increase in OIS rates. In practice, of course, the swap rates may well be changing along with the OIS rates, so separating out the two effects in this way is somewhat artificial.
Clearly, however, interest rate risk in swaps is more complicated under OIS discounting because we have to worry not just about changes in LIBOR-based swap rates but also about changes in OIS rates and changes in the spread between the two sets of rates.