Interest rate swap contracts

[awilson]

This question is taken from ASM Manual, Volume 2, Section 10, #5

Two interest rate forward contracts are available for interest payments due 1 and 2 years from now. The forward interest rates in these contracts are based on a one-year spot rate of 5% and a 2-year spot rate of 5.5%. X is the level swap interest rate in a 2-year interest rate swap contract that is equivalent to the two forward contracts. Determine X.

Solution:
The forward interest rate under the two year contract is 1.055^2/1.05-1=6.0024%.

The PV of these rates is

PV=.05/1.05+.060024/1.055^2=.10155

If R is the level interest rate under the two-year interest rate swap, we have:

R/1.05 + R/1.055^2 = .10155

R = 5.49%

I DONT GET IT! :wacko:

I do not understand why you want to use the forward rates. I just don't get it. If someone could explain this to real simple like, I would appreciate it!

THANKS in advance.

[Junkmonkey]

What don't you get? Are you unsure why we are "discounting the forward rates" in the solution (because after all, how can you discount a rate?!)?

Imagine that you have to repay a $1 loan by making 2 annual interest payments and then repaying the balance at the end of the 2nd year. Each year, the interest rate on the loan varies (hence we have non-level interest payments). Say the first year's interest payment was 5% and the second years payment was, oh I don't know, 6.0024% . What does our payment schedule look like?

At time 0 we pay 0.
At time 1 we pay $1*0.05 = $0.05
At time 2 we pay $1*0.060024 = $0.060024
At time 2 we also repay the loan balance of $1

The motivation behind interest rate swaps is finding a level interest rate (who knows why! Maybe we don't want to keep track of things; maybe we want to make our calculator's life easier; maybe it makes for a good exam question; I'm sure there's good reason!). We go about finding this rate by setting the PV of the above non-level interest payment schedule equal to the same schedule WITH level interest payments! How? Easy: we just discount the amounts paid at each time back to time 0 and equate them (of course, we could pick ANY time to discount them back to but 0 is the easiest).

Now if you notice, I used the interest rates from the problem. But remember that these are forward rates and hence applicable over a one-year period (think of them as "effective" rates of interest!). So discounting from time 1 to time 0 is easy, we just use the effective interest rate over that period which is 5%. Discounting from time 2 to time 0 is trickier because we have to use the spot rate (if you don't recall why, then you'll have to review it!). But we know it's 5.5% (the problem says so but we could easily figure it out too). So we have:

PV = $0.05/1.05 + $0.060024/1.055^2 + $1/1.055^2 = 0.48884
(I'll explain why my equation looks a bit different in note 1 below)

Now we want to find the level interest rate that is equivalent to this. Start by calling the level interest rate r. Our schedule looks like this:

At time 0 we pay 0.
At time 1 we pay $1*r = $r
At time 2 we pay $1*r = $r
At time 2 we also repay the loan balance of $1

So our PV is given by PV = $r/1.05 + $r/1.55 + $1/1.55 and we want this to equal the PV from above. Thus we have,

$r/1.05 + $r/1.055^2 + $1/1.055^2 = 0.48884 = $0.05/1.05 + $0.060024/1.055^2 + 1/1.55^2

$r/1.05 + $r/1.055^2 = $0.05/1.05 + $0.060024/1.055^2 = 0.10155
$r/1.05 + $r/1.055^2 = 0.10155

Hence $r = $0.0549

But this is the level dollar amount and we want the level interest rate. Well, since the dollar amount was based on a loan value of $1, we see that r = 5.49%. And we're done.

Long winded, I know, but only for the sake of explanation. The actual solution takes a little under 2 minutes

---

Note 1: If you're wondering why my equation looked a little different (I had that extra $1/1.055^2 term and the manual's solution did not), it is because the solution in the manual 'knew' that the $1/1.055^2 term would cancel out of both present values. You should reread page 134 (just before Computing the Swap Rate in the General Case).

Note 2: Does my PV calculation look familiar? What type of payment plan makes interest payments for n years then a lump sum payment at time n equal to the value of a loan? Ans: bonds! How do we find the PV of a bond? Simple:

PV = Fr*v + Fr*v^2 + ... + Fr*v^n + C*v^n

Well in this case, F is $1 (because that is the amount by which we determine the 'coupons' each period); r is the level interest rate (so THAT'S why I was using r ); v is actually the discount factor given by the spot rates (that is, the yield curve isn't flat!); C is $1 (that's how much we pay at time 2); and n is 2. So we have:

PV = $r/1.05 + $r/1.055^2 + $1/1.055^2

Now the thing to remember is that our coupons are equal to r each year and our redemption value AND par value are both equal to 1, so what must our price be? Ans: $1. So we have,

PV = Price = $1 = $r/1.05 + $r/1.055^2 + $1/1.055^2

Do the math and you will get $r = $0.0549 so that r = 5.49%

---

So that is why they use the forward rates!

Does this help?

- junk

[EdKJ]

The part that confuses me is the 6.0024% at tyear two.

What don't you get? Are you unsure why we are "discounting the forward rates" in the solution (because after all, how can you discount a rate?!)?

Imagine that you have to repay a $1 loan by making 2 annual interest payments and then repaying the balance at the end of the 2nd year. Each year, the interest rate on the loan varies (hence we have non-level interest payments). Say the first year's interest payment was 5% and the second years payment was, oh I don't know, 6.0024% . What does our payment schedule look like?

At time 0 we pay 0.
At time 1 we pay $1*0.05 = $0.05
At time 2 we pay $1*0.060024 = $0.060024
At time 2 we also repay the loan balance of $1

The motivation behind interest rate swaps is finding a level interest rate (who knows why! Maybe we don't want to keep track of things; maybe we want to make our calculator's life easier; maybe it makes for a good exam question; I'm sure there's good reason!). We go about finding this rate by setting the PV of the above non-level interest payment schedule equal to the same schedule WITH level interest payments! How? Easy: we just discount the amounts paid at each time back to time 0 and equate them (of course, we could pick ANY time to discount them back to but 0 is the easiest).

Now if you notice, I used the interest rates from the problem. But remember that these are forward rates and hence applicable over a one-year period (think of them as "effective" rates of interest!). So discounting from time 1 to time 0 is easy, we just use the effective interest rate over that period which is 5%. Discounting from time 2 to time 0 is trickier because we have to use the spot rate (if you don't recall why, then you'll have to review it!). But we know it's 5.5% (the problem says so but we could easily figure it out too). So we have:

PV = $0.05/1.05 + $0.060024/1.055^2 + $1/1.055^2 = 0.48884
(I'll explain why my equation looks a bit different in note 1 below)

Now we want to find the level interest rate that is equivalent to this. Start by calling the level interest rate r. Our schedule looks like this:

At time 0 we pay 0.
At time 1 we pay $1*r = $r
At time 2 we pay $1*r = $r
At time 2 we also repay the loan balance of $1

So our PV is given by PV = $r/1.05 + $r/1.55 + $1/1.55 and we want this to equal the PV from above. Thus we have,

$r/1.05 + $r/1.055^2 + $1/1.055^2 = 0.48884 = $0.05/1.05 + $0.060024/1.055^2 + 1/1.55^2

$r/1.05 + $r/1.055^2 = $0.05/1.05 + $0.060024/1.055^2 = 0.10155
$r/1.05 + $r/1.055^2 = 0.10155

Hence $r = $0.0549

But this is the level dollar amount and we want the level interest rate. Well, since the dollar amount was based on a loan value of $1, we see that r = 5.49%. And we're done.

Long winded, I know, but only for the sake of explanation. The actual solution takes a little under 2 minutes

---

Note 1: If you're wondering why my equation looked a little different (I had that extra $1/1.055^2 term and the manual's solution did not), it is because the solution in the manual 'knew' that the $1/1.055^2 term would cancel out of both present values. You should reread page 134 (just before Computing the Swap Rate in the General Case).

Note 2: Does my PV calculation look familiar? What type of payment plan makes interest payments for n years then a lump sum payment at time n equal to the value of a loan? Ans: bonds! How do we find the PV of a bond? Simple:

PV = Fr*v + Fr*v^2 + ... + Fr*v^n + C*v^n

Well in this case, F is $1 (because that is the amount by which we determine the 'coupons' each period); r is the level interest rate (so THAT'S why I was using r ); v is actually the discount factor given by the spot rates (that is, the yield curve isn't flat!); C is $1 (that's how much we pay at time 2); and n is 2. So we have:

PV = $r/1.05 + $r/1.055^2 + $1/1.055^2

Now the thing to remember is that our coupons are equal to r each year and our redemption value AND par value are both equal to 1, so what must our price be? Ans: $1. So we have,

PV = Price = $1 = $r/1.05 + $r/1.055^2 + $1/1.055^2

Do the math and you will get $r = $0.0549 so that r = 5.49%

---

So that is why they use the forward rates!

Does this help?

- junk

[Junkmonkey]

The part that confuses me is the 6.0024% at tyear two.

Do you mean you are not sure where 6.0024% comes from? It is simply the forward rate applicable from year 1 to year 2. Note that there are many ways to denote a forward rate. All of the following are different ways of saying the same thing:

1. The 1-year forward rate

Since forward rates don't necessarily have to be over a one year period, saying "the t-year forward rate" can sometimes be ambiguous. Note, however, that it is usually understood that a forward rate applies for 1 year (or 1 period) only.

2. The forward rate applicable from time 1 to time 2

I like this one

3. The forward rate applicable to the 2nd year

I hate this one

4. The 1-year deferred one-year forward rate

I like this one

5. The 1-year forward rate for year 1

Seems like it should be "for year 2" since it is the rate applicable over the second year... but this is what is given on page 396 of the ASM Study Manual

6. The 1-year forward 1-year rate

This one sounds like a freaking riddle! Hate it!

As far as figuring out the forward rates, that is simple. Say you are given spot rates S_1, S_2, ... , S_t, ... , S_n and you want to find the forward rate applicable from time t to time (t+1). We have,

(1+S_t)^t * (1 + F_t) = (1 + S_(t+1))^(t+1)

where F_t is the one t-year deferred one-year forward rate. Just solve for F_t and we get,

F_t = [(1 + S_(t+1))^(t+1)] / [(1+S_t)^t] - 1

Does this help?

- junk

[EdKJ]

The spot rate S_n is the rate today at which you can yield until time n?

The forward rate is the future rate?

[url]http://en.wikipedia.org/wiki/Interest_rate_parity[/url]

[url]http://fx.sauder.ubc.ca/forward.html[/url]

[url]http://en.wikipedia.org/wiki/Spot_price[/url]

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