Chapter 8 - Swaps Flashcards
define a swap
A swap is an exchange of payments over time.
swaps are not related to a single transaction, but rather a stream of them.
name an example of a commodity swap
Say you need a number of barrels of oil each year. Since the oil price fluctuate, we can be interested in a swap to achieve a constant price.
what is a prepaid swap
Buying today and receiving multiple assets later on
what is typical in regards to how a swap is payed?
In equal installments every year.
In general, the only rule is that the present value of whatever we end up paying and when, must equal the prepaid swap price.
elaborate on the terminology of saying that “the 2-year swap price is 200 bucks”
it means that the price each year, or each installment, is 200 bucks, and since it is 2-year, it will pay 200 bucks each year.
consider a swap deal involving 2 payments.
The question is naturally what the price we pay each term should be.
The forward curve for the asset has priced it at 110 the first year and 111 the second year.
Elaborate on what we can do with this information?
interest rate 1 year is 6%, while 2 year is 6.5%
I mean, elaborate on what a swap actually entails
If we separate the two, we can treat them like individual forward contracts.
1)
The forward price is S_0e^(r-∂)
2)
The forward price is S_0e^(r-∂)2
we consider oil, so assume no dividends.
1)
S_0e^r
2)
S_0e^2r
Question is, what should we price the thing?
We know the values are 110 and 111. If we enter the swap now, this is what we lock in.
The present value of this stream of cash flows is:
110/1.06 + 111/1.065^2 = 201.638
The idea is that we KNOW we will pay 110 in 1 year and 111 in 2 years.
We discount and get this present value.
This amount is important, because it represent what we should be able to pay the supplier of the asset TODAY and receive the assets in the future.
However, it is more common for swaps to consider equal installments. Therefore, we need to solve the equation:
x/1.06 + x/1.065^2 = 201.638
As it turns out, x=110.483, which means that if we pay this amount each year, it would be the same as paying the present value at time=0.
It is worth noticing that 110.483 is CLOSE to the average, but still below it. It is interesting to see what would happen if it was equal to the average.
If x=average=110.5, at year 1 we pay 0.5 more than the forward price (from T=0) would indicate. At year 2, we pay 0.5 less than the forward price would indicate. So, in year 1, we’d lock in a higher price. This means that the counterparty receive more return that the risk free rate. However, he receive less the next period.
This means that at the year 1, while the counter party supply the asset, the oil barrels, we will pay him more than we “should”. Usually, this is balanced by paying him less then we “should” the next period. For the case of x=110.5, it would actually entail us lending the supplier a total of 0.5 USD the first period, and then we receive 0.5 USD back the next period.
For the balanced case, we overpay the first period with 0.483 USD. The second period, we underpay by 0.517 USD. This is basically a loan where we lend out 0.483 and receive a face value or whatever of 0.517. The loan has rate 7.04%.
The loan rate of 7% is called the 1-year implied forward yield from year 1 to year 2.
elaborate on the implied forward interest rate
We can think of it as the forward equivalent of what the interest rate should be, some time into the future.
what is back to back transaction
situation where a dealer matches a buyer and a seller. The dealer bear all the credit risk of both parties, but has no price risk.
as a dealer, consider the case where you want to serve as the counter party to someone who wants to buy an oil swap. What can you do?
We can hedge by going long the necessary forwards to oil. If the swap is for 2 years, we need to go long a 1 year and a 2 year contract.
When we compare the net payoff, it looks like this:
each year, we receive payment from the buyer. this payment is equal to the installment price less the spot price.
From our own long position, we pay the difference between spot price and our own forward price.
Now, recall how the swap introduced a loan feature. This comes into play here. For the first installment, we receive more than we pay. We net this amount.
for the second installment, we lose the new amount. however, the new amount should be equal to the first amount but with compounded value into the future value.
This has an important effect: Hedging a swap using only forwards will not completely hedge the position. we will still be exposed to the interest rate.
why do we say that we are exposed to the interest rate when hedging incompletely using forwards only?
if the interest rates fall, the dealer will not be able to invest the net profit from year 1 and make the implied forward rate, which means that he will not be able to repay the year 2 amount.
Therefore, he needs to hedge on the interest rate somehow.
how can the market value of a swap change over time
multiple factors. Firstly, underlying assets can change in value, interest rates can change in value.
Secondly, even if all else remain equal, the implicit lending will affect the value. For instance, when we make the first payment, we lend some amount to the seller of the swap. Therefore, if the seller of the swap wish to exit the agreement, he’d owe us this amount.
how can someone exit a swap
generally speaking there are 2 options:
1) Negotiate with the counterparty
2) Create an offsetting position. This would cancel the original obliugation except for the amount that the fixed payments now have changed.
say you buy a swap for oil 2 years. The forward price is 110 y1 and 111 y2.
After you buy it, the prices rise so that the forward prices are no 112 and 113.
How do we get the profit, and how much profit? Assume rates are 6% and 6.5%
The first deal goes like this:
110/rate + 111/rate^2 = something
x/rate + x/rate^2 = something
x will be x=110.483
For the second case, we get:
112/rate + 113/rate^2 = 205.2878
x=112.483 (not surprisingly).
This means that we will pay 110.483 to one party, while we receive the asset, but then we can enter a new agreement for selling the asset to get yearly cash of 112.483
We’d get 2 bucks more in each period than we pay. We take the present value of this to see how good of a deal this actually is:
2/1.06 + 2/1.065^2 = 3.65 USD
elaborate on the pricing of prepaid swaps
we need to account for the forward prices, and then discount these. to discount the,m, we use the zero coupon bond prices with face value 1USD.
prepaid swap price = ∑F_{0,i} P(0, i) [i=1, n]
so the prepaid swap price is basically the present value of the forward prices.
how do we find the swap price
We find the swap price by using the prepaid value of the swap, and forcing this amount to be equal to ∑RP(0,t_i) = R∑P(0, t_i)
So, we get:
R∑P(0, t_i) = ∑F_{0,t_i} P(0, t_i)
which we can solve for R:
R = ∑F_{0,t_i} P(0, t_i) / ∑P(0, t_i)
So what have we actually found here?
this is just the fixed quantity swap price installment price
In other words: R is the annuitized version of the swap stream. This means that we have converted the swpa stream into an annuitiy.
what is swap rate
swap rate is the same as swap price, but is the terminology we use when the swap is concerned with interest rates
if we dont want fixed prices, what options do we have?
variable prices, variable quantities.
Typically motivated by seasonalities
if we want to pay for variable quantities, how do we extend our model to add this? Consider prepaid swap
We introduce a parameter that holds the number of units we want to acquire at time t_i: Q_ti
The formula for the prepaid swap is as always the present value of the forward prices:
prepaid swap price = ∑Q_ti F_{0,t_i} P(0, t_i)
earlier, we only considered 1 unit. Now we can consider multiple.
With multiple and variable amount of units, we changed the prepaid swap price formula. How does this affect the swap prices?
We simply extend it to include quantity multiplied by the swap price for each period. We use the same logic of solving the equation to be equal to the present value of the swap.
∑Q_ti RP(0, t_i) = ∑Q_ti F_{0,t_i} P(0, t_i)
Solve for R:
R = ∑Q_ti F_{0,t_i} P(0, t_i) / ∑Q_ti P(0, t_i)
This is just a simple weighting. If we want to buy more units early on, we have to use more of these forward prices, rather than later ones.
what is LIBOR
London InterBank Offered Rate. Interst rate banks offer to other banks
Suppose a firm is exposed to floating rate debt. They want to change it to fixed rate debt. there are 3 ways to do this. Name and shortly introuce
1)
Retire the current floating rate debt, and issue fixed rate debt instead. however, this has transaction costs.
2)
enter into a strip of forward contract agreements. This will guarantee borrowing rate for the agreement duration. However, the company will lock in a different rate each year, which may not be ideal. It may not be ideal because the company’s cost of borrowing is different each year.
3)
Enter into a swap where they receive the floating rate debt and pay the fixed rate debt. Whether this turn out to be beneficial or not depends on what happens with the floating rate debt.
Recall that the firm is ALREADY paying floating rate debt. Therefore, they want an agreement where they receive floating rate, and pay fixed rate.
elaborate on how a swap on itnerest rate work to obtain fixed interest
Assume we have a floating rate debt that fluctuate according to LIBOR.
If we enter a swap, and obtain a fixed rate, we will end up paying the difference between LIBOR and the fixed rate to the swap counterparty. If the LIBOR is smaller than fixed rate, we pay this amount. If LIBOR is larger than fixed rate, the swap counterparty pay us.
The net payment therefore looks like this:
we are already obligatged to pay LIBOR.
From the swap, we receive (LIBOR - FIXED).
So: -LIBOR + (LIBOR - FIXED) = - FIXED
We always pay negative fixed, which means that we have converted a floating rate LIBOR debt into a fixed debt payment.
what is the role of the notional amount in swaps that regard interest rate?
The notional amount determine the cash flow. The interest rate in itself is multiplied by the notional amount to determine the actual payment that is either payed or received by a party.
two words used to describe the “life” of the swap?
term/tenor
does a bond or swaps (on interst) have more or less credit risk?
Swaps have less credit risk. this is because (for financially settled) the owed amount is at maximum (in th4e case of default) the present value of the net swap payments they are obligated to make at current market prices.
Bonds have principal involved, which can carry significant losses.
elaborate on the role of the market maker when regarding interest rate swaps
Assume a company use a swap to acquire a fixed rate debt. The market maker takes on the counter party in this transacation.
However, the market maker does not want the exposure to interest rate. They are paid by the spreads etc, and want to hedge this position.
The market maker will receive fixed payments, and will have to pay floating rate. Because of this, should the interest rate increase, they are at a loss. Should the interst rate decrease, they are at a profit. however, they dont want this exposure.
The market maker can hedge by entering into an FRA (Forward rate agreements).
elaborate on how the market maker reason to find out what he should do in order to hedge a swap on interest rates
The motivation is to hedge a swap.
he does this by entering into a series of forwards.
Each of the forwards, he will pay the differnece between the actual rate (underlying) and the forward price that was locked in. Recall that the forward price is now the interest rate that is locked in. It is equal to the implied forward rate that can be found using the swap.
so, each period, the market maker will PAY the difference between the underlying and the was-price for the locked in rate. The locked in rates are known, but the interest rate is obviously not.
however, the market maker will be payed the fixed rate each period as well. Or to be precise, he will be payed whatever is the difference between the fixed rate and the underlying. IF the underlying is large, we lose money, but if the underlying is small we earn a profit.
So, there are 2 things at work:
1)
The workings from the swap. We get (R - r_i). R is fixed rate, r_i is the underlying at that time.
2)
The workings from the forwards. Specific to each year. We have lcoek in the forward rate.
We pay: (r_i - ImpliedForwardRate_i)
the net effect is:
R - implied forward rate, except for the first year where the spot interst rate is known.
So, the question now is: what shuld we as a market maker accept as R?
Naturally, we want a R that makes the present value be 0.
We know all the rates, so this is trivial.
(R-spot year 1) / rate1 + (R - impliedForward 1)/rate2^2 + …. = 0
what can we say about the rate that the market maker accepts as R?
It is the same as the par coupon rate on a bond with the same maturty as the swap.
this is not surprising, as entering the fixed rate swap is actually the same as borrowing with a fixed yield.
what benefits does a swap entail compared to many forwards?
Convenience and transaction costs.
Ability to fix the price
what is a “strip”?
A set of forward prices for the same asset but with different maturities
What is a sophisticated way of looking at swaps?
A swap is a series of forward contracts coupled with implicit borrowing and lending
elaborate on a prepaid swap.
Why is this not commonly used?
1)
a prepaid swap would entail finding the present value of the price we’d pay for all the individual prepaid forwards in the strip.
The key is the present value. In theory, we could pay this amount to the supplier of some commodity etc today, and create a fair contract.
As a side note, the prepaid swap price also represent the amount that we should invest today (in risk free assets) in order to be able to pay for the forwards when their time comes.
2)
Credit risk. What happens if we pay a supplier, and he never deliver?
elaborate shortly on “ways to pay” for a swap
there are many different ways to do it. A very common one is to use fixed installments.
However, any payment that has a present value of the strip of forward prices is valid.
assume financial settlement,
what will be payed on expiration?
Then give the case of what happens if the long position actually want the asset, but hte shite is financially settled.
depends on the position.
For the dude who is long the swap, meaning the dude that will receive the assets, he will receive the difference (spot price at expiration - swap price). Since this is a swap, this will likely happen multiple times. Depending on how the spot price turns out, the swap can be profitable or not (not considering its risk management perspective).
this also means that the short swap position must be the differnece (spot - swap price). And paying a negative amount means getting payed isntead.
After the payment, the long position has either payed (swap - spot) or received (spot - swap). Then he needs to buy oil/whatever asset at the spot price.
net: spot-swap - spot = -swap.
This means, regardless of what the spot price is, he will pay the swap price in net effect.
I think this illustrate a very important point. When we use a swap to hedge prices, we are locking in the price. And the interesting result is that this would be teh same result AS IF IT WAS PHYSICALLY SETTLED.
how can a swap allow for pricing that is not the same as the forward prices?
Because of the lending that also takes place.
This is really done by the present value formulation.
Basically, some periods we underpay, some we overpay.
In general, we tend to overpay early, and underpay later on. This imply that we actually lend some money to the counter party and that we receive it back with interest later.
elaborate on hedign the short position as a dealer
So the dealer agree to take a short position in the swap. This means that he will be payed the swap price, while he will pay the spot price. Therefore, the dealer want to hedge situations where the spot price is greater than the swap price.
The short position in the swap is a natural short position in the udnerlying forwards. Therefore, we can (as the dealer) take long positions in the forward contracts. Taking long positions in the forwards will make us pay the forward price, but receive the spot price. So, for any given installment, we pay the spot price to the counter party of the swap, and then we get payed the spot price from the counterparty in the forward.
Now, we get payed the swap price each installment, and we need to pay the forward price for the specific period. These should cancel out in the duration of the swap, because the swap payments are constructed using the forward prices. If we take the present value of the swap payments and compare with the present value of the forward prices, we would end up at the same value. Therefore, while there is implicit lending and borrowing in the swap, the total value is the same for both.
The sophisticated way of saying that last part, is that the difference between the swap price and forward price is a loan. We make many loans throughout the swap period, but the key is that it will be payed back with interest.
however, this introduce a twist. the lending and borrowing depends on the spot interest rate at various times. when we find the swap price, we use the yield curve, but but this does not represent the interest we get on the loans later on. therefore, the short position is subject to interest rate risk.
can a swap change in value?
Yes. just the way as forwards do.
however, we also need to account for the loans.
