The Swap Rate Curve

Fixed or floating? The type of interest rate received and paid makes a big difference to banks and other institutions. Most importantly, it should match (what's coming in and going out) for good risk management.

What characteristic would you assign to this swap market?

Absolutely! Highly liquid. Every market has frictions, and there is some counterparty risk involved, but the volume of this market makes it very liquid, and that leads to efficient swap rates. Two counterparties will agree to a rate for a given term based on some market reference rate (MRR).

No. This market has risk. There's always counterparty risk, since whoever should be doing the paying when rates change could suddenly walk away or default. It's rare, but possible.

Not really. These agreements have to be arranged, and there are contracts involved and eventually payment processing. So there are some frictions right from the start.

So swaps are arranged to trade one for the other. Not instruments, or even the notional amount of money the agreement is based on, but just any difference in the payments. That makes it easy. So USD 100,000,000 of notional value can be used in a swap to exchange fixed for floating, and a difference of a basis point means that one counterparty is paying the other counterparty $$\displaystyle 100,000,000 \times 0.0001 = 10,000 $$. A lot of institutions use this market, and total notional value is around USD 400,000,000,000,000. Wow.

This agreed upon rate is the __swap rate__, and it's set up so that the swap has zero value at the start. Think about that condition. Someone is agreeing to pay a fixed rate, like 3% for three years on USD 100,000,000 notional. The counterparty is agreeing to pay something like 12-month USD MRR plus 1.15% on the same notional amount for the same time period. If the swap has zero value to both parties, what is the implied expectation?

No. If that were the case, there would be no reason to set up the swap in the first place.

No, not necessarily. Consider the condition of an upward-sloping spot curve.

Exactly! Maybe the spot curve is upward sloping (as it usually is), and the spot curve suggests that the 12-month USD market reference rate (MRR) (it's just an example here, so call this "floating" from here on) is 1.72%. If the swap is for five annual payments at the end of the year, then the parties are really expecting a payment in Year 1. One counterparty party will pay the fixed rate 3%, and the floating party expects to pay the floating rate of $$\displaystyle 1.72 \% + 1.15 \% = 2.87 \% $$. That's a difference of 13 basis points on the notional amount, or a payment in USD of $$\displaystyle 100,000,000 \times 0.13 \% = 130,000 $$.

What must be the expectation of the floating rate?

No. That would leave the counterparty that is paying the fixed rate to pay the entire five years. If this was the expectation, then the swap definitely has a nonzero value at the start.

Yes! It has to. The fixed party is expecting to pay in the first year, so they must be expecting to receive later on. Remember that the swap needs to have zero value at the start.

No. That would eventually close the gap, but it would leave the counterparty paying the fixed rate at an expected loss. That firm wouldn't agree to such a swap.

Specifically, this equation must hold: $$\displaystyle \sum \limits_{t=1}^T \frac{s_T}{[1 + z_t]^t} + \frac{1}{[1 + z_T]^T} = 1 $$ The swap rate is $$s_T$$ here. This can be used with the government spot curve to derive the swap curve so that counterparties know what rate is fair for entering into these agreements. Assume the following discount factors and implied spot rates are given by the government spot curve. | Year | 1 | 2 | 3 | |---|---|---|---| | Discount Factor | 0.9709 | 0.9246 | 0.8638 | | Spot Rate | 3.00% | 4.00% | 5.00% |

The swap rates for each year can be found. $$\displaystyle \frac{s_1}{(1 + 0.03)^1} + \frac{1}{(1 + 0.03)^1} = 1$$ >$$ s_1 = 0.0300 = 3.00 \%$$. $$\displaystyle \frac{s_2}{(1 + 0.03)^1} + \frac{s_2}{(1 + 0.04)^2} + \frac{1}{(1 + 0.04)^2} = 1$$ >$$ s_2 = 0.0398 = 3.98 \% $$. $$\displaystyle \frac{s_3}{(1 + 0.03)^1} + \frac{s_3}{(1 + 0.04)^2} + \frac{s_3}{(1 + 0.05)^3} + \frac{1}{(1 + 0.05)^3} = 1$$ > $$s_3 = 0.0493 = 4.93 \%$$.

The government spot curve and the swap curve are connected by this math. You can also see that they are very close to each other. When a country has an active government debt market and swap market, institutions choose based on what market most closely matches what they are doing. What curve do you think is used more in countries where government debt issues are very few?

That's right.

No. Actually it's the swap curve.

When government debt is thinly traded, or even nonexistent beyond very short-term issues, the swap curve is a highly valuable tool for valuation.

To summarize: [[summary]]

Liquid

Riskless

Frictionless

All payments must be zero

The first payment must be zero

The summed present value of all payments must be zero

It is expected to fall

It is expected to rise beyond 1.85%

It is expected to rise to exactly 1.85%

The swap curve

The government spot curve

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