FRM Fun 25, interest rate swap comparative advantage

[David Harper CFA FRM]

Some Q&A on the tricky comparative advantage (of interest rate swap) concept (this thread here) got me thinking about how much we can infer from the visual.

Here is a interest rate swap, like Hull shows:

Clearly, Company A (on the left) has an advantage in fixed-rate markets, where it (Co A) can borrow at a fixed rate of 6.0% (the left arrow out into the void signifies borrowing from capital markets, at the going rate). And Company B (on the right) has an advantage in floating-rate markets, where it (Company B) can borrow at a floating rate of LIBOR + 2.0%.

Question: given only two further assumptions:

1. Company A can borrow (on its own) in floating-rate markets at LIBOR + 1.20%, and
2. The gains created by the swap are shared equally

What is Company B's borrowing rate (on its own) in fixed-rate markets?

 

[ABFRM]

Company A Company B
Fixed 6 8.2
Floating LIBOR+0.2 LIBOR+2

 

[David Harper CFA FRM]

Hi Abhiskek,

Indeed you have summarized the information already displayed, which includes:

• Company A borrows at a fixed rate of 6.0% in the capital markets, such that it must have comparative advantage in fixed-rate capital markets. Then, net of the swap, Company A effectively transforms this fixed-rate obligation into a floating-rate loan where it pays LIBOR + 0.2% (i.e., incoming 5.8% but outgoing LIBOR and outgoing 6.0% = pay LIBOR + 0.2%)
• Company B borrows at a floating rate of of LIBOR + 2.0%, such that it must have comparative advantage in floating-rate markets. Then, net of the swap, Company B effectively transforms this floating-rate obligation into a fixed-rate loan where it pays 8% (i.e., incoming LIBOR canceled by outgoing LIBOR, which leaves outgoing 2% + 6%).

But the question is a little tougher: given they shared the total gains from the swap (50/50), can we infer information that is not displayed:

• Company A's borrowing rate (on its own) in floating-rate markets (where it does not have C.A.)
• Company B's borrowing rate (on its own) in fixed-rate markets (where it does not have C.A.).

I only ask b/c i think (?) it's a good stress test of mastery ....

 

[David Harper CFA FRM]

Please note: the previous question was too difficult (perhaps unsolvable, I'm not quite decided). Apologies, but I just simplified the question by giving a further assumption, and i add the information to the image. Thanks,

 

[RiskNoob]

Let X be Company B's borrowing rate (on its own) in fixed-rate market.

Let Y be the basis for the comparative-advantage argument. (I borrowed the term from Hull Ch.7, Table 7-4) That is,

Y = (fixed rate for Company B - fixed rate for Company A) - (floating rate for Company B - floating rate for Company A)
= (X - 6%) - ( (LIBOR + 2%) - (LIBOR + 1.2%) )

On the other hand Y can be interpreted as

Y = Company A`s profit due to swap transaction using comparative advantage + Company B`s profit due to swap transaction using comparative advantage + FI`s profit.

David already clearly explained the first two (equal) terms, which is 1%. Also, as we can see from the diagram that FI profit is 0.2%.

So,

Y = 1% + 1% + 0.2% = 2.2%

Solving for X, we have X = 9%. - Company B's borrowing rate (on its own) in fixed-rate markets is 9%.

RiskNoob

 

[ShaktiRathore]

Let x be the fixed rate at which B can borrow,
B pays: 6% and L+2%
B receives: L%
so total net fixed rate to B= (6%+L+2%)-L%=8%
So total advantage to B in terms of fixed rate:x-8%[B wants to borrow at fixed rate but entered into swap to benefit from low fixed rate from A side who has comparative advantage in fixed rate]

A pays: 6% and L%
A receives: 5.8%
so total net floating rate to A= (6%+L)-5.8%=L+.2%
So total advantage to A in terms of floating rate:L+1.2%-(L+.2%)=1%[A wants to borrow at floating rate but entered into swap to benefit from low fixed rate from B side who has comparative advantage in floating rate]

total advantage to A=total advantage to B
=>x-8%=1%
=>x=8%+1%
=>x=9%

Ans2:
from diagram also we can infer directly that,
than total benefit to B=x-6%+L-(L+2%)=x-8%
total benefit to A=Libor+1.2%-Libor+5.8%-6%=1%
for benefit to B=benefit to A,
x-8%=1%
x=8%+1%
x=9%


thanks

 

[vt2012]

Agree, answer is quite obvious:
Company B's borrowing rate (on its own) in fixed-rate markets is 9%.
Gain =(9-6)-(L+2-L-1.2)=2.2% shared as 1%(CoA)+1%(CoB)+0.2%(FI).

Much more interesting the problem is WITHOUT asumption that CoA's borrowing rate (FL) is L+1.2
Strictly speaking, the problem HAS solution.
Let's denote CoB's borrowing rate (FX) as X; CoA's borrowing rate (FL) as L+sA.
Then gain (total) from swap =X-8+sA
After swap CoA's gain =L+s-6+5.8-L=sA-0.2; CoB's gain=X-(L+2)-L-6=X-8

Equations:
1. Gains are shared X-8=sA-0.2
2. Total gain minus FI is twice each gain X-8+sA-0.2=2(sA-0.2), or 2(X-8)
A bit algebra and we have solution X=sA+7.8

With your assumption sA=1.2 we have X=9

 

[vt2012]

Agree, answer is quite obvious:
Company B's borrowing rate (on its own) in fixed-rate markets is 9%.
Gain =(9-6)-(L+2-L-1.2)=2.2% shared as 1%(CoA)+1%(CoB)+0.2%(FI).

Much more interesting the problem is WITHOUT asumption that CoA's borrowing rate (FL) is L+1.2
Strictly speaking, the problem HAS solution.
Let's denote CoB's borrowing rate (FX) as X; CoA's borrowing rate (FL) as L+sA.
Then gain (total) from swap =X-8+sA
After swap CoA's gain =L+s-6+5.8-L=sA-0.2; CoB's gain=X-(L+2)-L-6=X-8

Equations:
1. Gains are shared X-8=sA-0.2
2. Total gain minus FI is twice each gain X-8+sA-0.2=2(sA-0.2), or 2(X-8)
A bit algebra and we have solution X=sA+7.8

With your assumption sA=1.2 we have X=9

P.S. Naturally, restriction is sA>0.2, otherwise we'll have losses, not gains

 

[David Harper CFA FRM]

vt2012 agreed, but you finish with one equation in two unknowns (X=sA+7.8), so you do require the s(A) assumption, yes? I could not improve on that .... I did originally expect it to be solvable w/o the sA assumption. Without knowing A's floating rate, there are three equations:

1. Fx(b) + Fl(a) - g = 8%; where g = gain before FI fee of 0.2%
2. 2*Fx(b) - g = 15.8%; i.e., B gets half of the post-fee gain such that Fx(b) - 8% = 0.5*(g-0.25%)
3. 2*Fl(a) -g = 0.2%; i.e., A gets half of the post-fee gain such that Fl(a) - 0.2% = 0.5*(g-0.2%) and omitting Libor throughout

But, unless i mistaken, that matrix is indeterminate, is why I added back an assumption. Although in Excel, I could not find other solutions, so I *feel* there is a solution in three equations but I could not find thanks for your help!

 

[ABFRM]

Hello David but there is an underlying assumption that benefits will be passed on two both equally.
Is this the case really? because in practice A will take more advantage becuse A has absolute advantage in bothe the markets so underlying assumption that both will share the advantage equally is doubtful.

 

[David Harper CFA FRM]

I agree the 50/50 share assumption is dubious (very interesting that the counterparty with absolute advantage ought to garner greater share of the gain!). The only reason I employed the assumption is that, without such an assumption (or sharing rule), I think the problem would have allow for an infinite number of solutions. So, as is often the case in writing questions, it is a way to "sneak in" an assumption that helps lead to only one answer. Thanks!

 

[vt2012]

vt2012 agreed, but you finish with one equation in two unknowns (X=sA+7.8), so you do require the s(A) assumption, yes? I could not improve on that .... I did originally expect it to be solvable w/o the sA assumption. Without knowing A's floating rate, there are three equations:

1. Fx(b) + Fl(a) - g = 8%; where g = gain before FI fee of 0.2%
2. 2*Fx(b) - g = 15.8%; i.e., B gets half of the post-fee gain such that Fx(b) - 8% = 0.5*(g-0.25%)
3. 2*Fl(a) -g = 0.2%; i.e., A gets half of the post-fee gain such that Fl(a) - 0.2% = 0.5*(g-0.2%) and omitting Libor throughout

But, unless i mistaken, that matrix is indeterminate, is why I added back an assumption. Although in Excel, I could not find other solutions, so I *feel* there is a solution in three equations but I could not find thanks for your help!

Hi David,
In your terms equations are:
1. Fx(b)+Fl(a)-L-g=8 -> g=Fx(b)+Fl(a)-L-8 this is g definition
2. Fx(b)-8=0.5(g-0.2)
3. Fl(a)-L-0.2=0.5(g-0.2)

Rank of the matrix is 2

Or, substituting g t0 2 and 3 we have

1. Fx(b)-8=Fl(a)-L-0.2
2. Fl(a)-L-0.2=Fx(b)-8

Of course, rank of the matrix =1; it means that there is no solution in terms Fx(b)=number and Fl(a)=number
However, remembering linear algebra course, solution of the equations is Fx(b)=Fl(a)-L+7.8
That is why I mentioned that "strictly speaking the problem Has solution"

 

[David Harper CFA FRM]

Hi vt2012, yes, well-played sir, "strictly speaking" the problem does have a solution indeed! I swear i must be have been lulled, from writing too many practice questions, into thinking that the solution must be a value. Your demonstration of the matrix rank is great, much more efficient than my indeterminate calculation. Terrific, thank you for "proof" of the solution, I am finally satisfied.

P.S. this is example of why i love finance, it's a onion you can just keep peeling, no concept is fully baked, there is always an n+1. I really think, too, that going an notch below the surface reinforces the surface itself. When I first read Hull's comparative advantage, it was a vague idea and an attempt to memorize the key formula, i think in this concept, that magically there is an apparent free lunch given by the total gain = difference of the rate differentials. Try to memorize the formula, and (for me) the formula is easily forgotten. But play with the (n+1), below the surface, and formula becomes almost an intuition. Or, maybe it's just practice, not sure