Could I get some help with this problem? SWAPS

Suppose that, in a two-year interest rate swap, the floating rate for each year is determined as the one-year rate at the beginning of the year, but payments are made at the end of the year. According to the relevant yield curve for the swap, the one year spot rate of interest is 5%, and two year spot rate of interest is 6%. Find the appropriate fixed-rate for the two-year interest swap.

My train of thought: I am looking at the problem and I am having trouble finding a similar problem to this to help me solve this particular problem. I see that the payments are made at the END of the year. That means that these payments do not accrue interest until one year after the interest rate at t= 1 and t =2. However, I am not sure what to do at this point. Does this mean I should calculate the future rate for [2,3]? Should I calculate the swap rate? or should I find the future rate and defer it one year? This particular section has been pretty rough in comparison to annuities/bonds/loans because it no longer relies on mathematical ability as much as more conceptual.

Another thought, could I possibly multiply both rates by one more factor of (1+i) and treat it as an ä?

Well I’m a bit rusty on this material, but let’s see if I can help.

Let’s just look at a single one of the swap payments, say in year T. At the beginning of year T (which is time t=T-1) we’ll know the 1-year rate i_1(T-1). Which means at the end of year T (which is time t=T) there will be a net payment of i_1(T-1) - r, where r is the fixed rate.

Now suppose we re-write that end-of-year-T payment as

i_1(T-1) - r = [1 + i_1(T-1)] - [1 + r]

Why do that? Well to get 1 + i_1(T-1) at time T we could invest $1 at time T-1 for 1 year (since we know at the beginning of year T that we’ll get a yield of i_1(T-1)). So that year-end 1 + i_1(T-1) payment is equivalent to a $1 payment at the beginning of the year.

But that means you could get the same net effect for the swap by saying that one side pays $1 at the beginning of each year, and the other pays (1+r) at the end of each year.

In that case if you put all the swap cash flows together then one side of the swap is paying an n-year annuity due with payments of $1, while the other side is paying an n-year annuity immediate with payments of (1+r). You just need to figure out the value for r that makes those two annuities worth the same amount.

Hopefully that helps (and I didn’t mess up any of the math).

Thank you so much! I was a bit confused by this one because it isn’t explicitly explained in the ASM manual/syllabus. I will let you know what I get for it in the next hour or so.

Just filling in the final details for posterity…

From the discussion above we want:
\ddot{a}_{\bar{2}|} = (1+r) a_{\bar{2}|}

\ddot{a}_{\bar{2}|} = 1 + 1 / 1.05 = 1.95238
a_{\bar{2}|} = 1 / 1.05 + 1 / 1.06^2 = 1.84238

r = [\ddot{a}_{\bar{2}|} / a_{\bar{2}|}] - 1 = 1.95238 / 1.84238 - 1 = 0.0597 = 5.97\%

Hey Jraven,

Thanks again for your help. I ended up getting the same answer. I also found another way to verify. I had to find the future rate and then doing a deferred swap

(1+.06)^2 = (1.05)(1+f[1,2])
f[1,2]= .07

Fixed interest rate swap R:





= 0.05966 = 6%

The formula I have seen for swaps is as follows:

R = (1 - P2)/(P1 + P2) where Pi = 1/(1 + i year spot rate)^i

P1 = 1/1.05 = 0.952381
P2 =1/(1.06^2) = 0.889996

R = (1-0.889996)/(0.952381+0.889996) = 0.059707

Yeah, that will give the same answer, though it does relate to a slightly different argument. That argument required a bit of a leap, though, so I avoided mentioning it initially.

Basically, if you take the floating leg of the swap and add a final maturity payment to it then you end up with the cash flows of a floating rate bond. But due to how the floating coupon rate is set and paid it will have market value equal to its par value. (That the market value equals the par value is the mental leap I was referring to. It’s not true of all floating rate bonds, but is true in this case.) So if we back out the value of the maturity payment then (assume for simplicity the face/par/nominal amount is $1)

value of floating leg = value of floating rate bond - value of maturity payment
value of floating leg = 1 - P2

On the other hand the fixed leg is just an annuity-immediate with payments equal to the fixed rate, so
value of fixed leg is

value of fixed leg = r x 2-yr annuity immediate

Since the two legs should have equal value at the start, that means that
r = (1 - P2) / (2-yr annuity immediate)