Swaps Note

8/11/2019 Swaps Note

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SAIS 380.760 Class Note on Valuing Swaps p. 1

Corporate FinanceProfessor Gordon Bodnar

Class Note on Valuing Swaps

A swap is a financial instrument that exchanges one set of cash flows for another set of cash flows of equal expected

value. Swaps allow parties to take speculative positions on certain financial prices or to alter the cash flows ofexisting assets or liabilities, most often to manage risk or to convert cash flows of one type of security into the cashflows of another type without physically have to sell the old one and buy the new one.

In all cases, when a swap is initially set up, the payment structures are set so that the PV of the expected amount aparty pays is equal to the expected amount that that party receives. Thus at issuance the swap is a zero NPV contract(ignoring transaction costs). This means the PV of the expected cash flows to one side of the swap equals theexpected cash flows to the other side of the swap at initiation.

However, if the financial prices or expected prices upon which the swap is based changes, the value of the swap willchange. When the value changes, one party to the swap will experience a gain equal to the increase in the value ofswap, while the other party to the swap will experience an equivalent loss (zero sum game). Below we will considerboth interest rate and currency swaps and consider how to measure their change in value in response to changes intheir underlying financial prices

Interest Rate Swaps

Lets consider an interest rate swap first. In an interest rate swap, parties are exchanging fixed interest ratepayments for floating interest rate payments on some notional value. To define an interest rate swap we start bydefining a notional value a principal amount upon which the interest payments are calculated. However, thisprincipal amount is not exchanged at the beginning or end of the contract, as it is not necessary (why give $100 justto receive $100?) As a result, interest rate swaps consist only of exchanges of periodic interest payments.

Consider the following situation. A firm enters into a two-year interest rate swap with a notional principal of$100M. The firm agrees to make four semi-annual payments at a fixed interest rate of 5.5% (APR) and receive foursemi-annual floating rate payments of LIBOR, denoted hereafter by L, plus 0.50% on the notional principal. Atinitiation of the swap, LIBOR is 4.75% (APR).

Below is a diagram of the cash inflows and outflows of this swap for the firm that entered into it. This firm would belong an interest rate swap as it is in a position to gain if interest rates rise. The counterparty to this swap has exactlythe opposite cash flow structure (they are short a swap). The cash flows are exchanged at the end of each semi-annual period.

While the fixed interest rate side payments each period are known with certainty, $100m x 5.5%/2 = $2.75M eachperiod, the floating rate side is known with certainty only for the first payment as it is based upon current LIBOR of4.75% plus a 0.50% spread. Thus the first floating rate payment is $100 x 5.25%/2 = $2.625m. Future floating ratepayments each period will depend on the future LIBOR rates. Their expectations are such that they have the samePV as the fixed side flows. Again notice that there are no notional principal exchanges with an interest rate swap.

Receive

Pay $100m x5.5%/2 $100m x5.5%/2 $100m x5.5%/2 $100m x5.5%/2

Year 0 0.5 1 1.5 2$100m x

LIBOR0/2

= $2.625M

$100m x

LIBOR0.5/2

$100m x

LIBOR1/2

$100m x

LIBOR1.5/2Receive

Pay $100m x5.5%/2 $100m x5.5%/2 $100m x5.5%/2 $100m x5.5%/2

$100m x

LIBOR0/2$100m x

LIBOR0.5/2

$100m x

LIBOR1/2

$100m x

LIBOR1.5/2


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We can also express the cash flows from this swap in table form:

Year 0 0.5 1 1.5 2

CFs to be received $2.625M $100M x $100M x $100M x(L0.5+ 0.5)/2 (L1+ 0.5)/2 (L1.5+ 0.5)/2

CFs to be paid $2.75M $2.75M $2.75M $2.75M

We could determine the expected floating rates (L0.5, L1, L1.5) using the Expectation Theory of the Term Structure,as this is how the market will form its expectations at the beginning of the swap. However, we know that the PV ofthe cash flows to the floating rate side of the swap must equal the PV of the cash flows to the fixed side of the swap,which are

PV ANNUITY (N = 4, I/Y = 5.5%/2, PMT = $2.75m) = $10.2834m

This comes from realizing that the swap is a zero NPV security at initiation.

Valuing an Existing Interest Rate Swap

Now lets consider how to determine the value of an interest swap at some point in the future when economicconditions have changed relative to the origination of the swap. Suppose 6 months (0.5 year) into the swap, at thedate of the first interest payments, interest rates are now lower than originally expected. This means both the currentfixed rate and the current and expected future LIBOR rates are lower than they were expected to be at the beginningof the swap. The firm will suffer a loss on the swap as a result of this drop in rates as it is stuck paying the old(higher) fixed interest rates, and receiving the (now lower) set of LIBOR rates, both currently and forecasted into thefuture. The question is how much does the firm lose on this swap? This depends on how much interest rateschanges and the maturity of the swap.

Assume that rates change such that the fixed interest rate on a new swap with a maturity of only 1.5 years is 4.5%(APR), and current LIBOR is 3.5% (APR). There are two ways we can determine the value of the original swap.

Method 1: Discount Remaining Fixed Cash Flows and Phantom Principal Repayment

Determine the PV of the remaining fixed interest rate payments including the phantom repayment of the notionalvalue at the maturity on the original swap at the new fixed rate of interest for a swap with that remaining maturityand compare this to the PV of the floating rate side of the original swap, which is by definition equal to the notionalprincipal of the original swap.

Below is a table with the remaining fixed rate cash flows on our interest rate swap (defined above) as of time 0.5,just after the first semiannual interest payments are made, plus the phantom notional principal repayment atmaturity. (Note repayment of notional principal at maturity does not really occur but we must include thisphantom cash flow to get the value correct).

Year 0.5 1 1.5 2Remaining Fixed Rate CFs $2.75M $2.75M $2.75Mon original swap + $100M

Take the PV of these CFs at the new fixed interest rate for a 2-year swap of 4.5% (APR) or 2.25% per period.

PV of Fixed interest payments = -$2.75 /(1.0225) + -$2.75/(1.0225)2+ -$102.75/(1.0225)3 = -$101.43495M

Thus the current value of the fixed payments to the swap is $101.43495M. This is the PV of the payment the firmmust make on the swap. The market value of the floating rate side of the swap will, by definition, be $100M (thePV of floating rate payments on $100M where the rates adjust for interest rate movements). So, with the PV of whatthe firm must pay at $101.43495M and the PV of what the firm will receive at $100M, the firm has lost $1.43495Mof value on the swap as a result of the interest rate changes.


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Method 2: The Offsetting Swap Approach

This method involves imagining that the firm enters into a new swap at current market prices that offsets one side ofthe remaining cash flows on the existing swap. We take the present value of the net cash flows of the two swapstogether, which is typically a net fixed rate payment at the current fixed interest rate on the new swap. Thisapproach takes advantage of the fact that the new swap has zero NPV, so when we combine its cash flows with theexisting swaps cash flows the PV of these net cash flows will, be definition, produce the value of the original swap

at the new market conditions.

Consider the original swap above at time 0.5, with the same set of mew market conditions (the fixed interest rate ona new swap with a maturity of 1.5 years is 4.5% (APR). and current LIBOR is 3.5% (APR)). To implement thisapproach on our swap above at time 0.5, we would need to enter into a new swap with a $100M notional value onwhich we pay the floating LIBOR rates plus 0.50% spread ($100M x (L + 0.5)/2) and receive fixed payments at thenew lower fixed interest rates 4.5% (APR) ($100M x 4.5%/2 = $2.25M). Since no principal amounts are exchangedin an interest rate swap and we assume that the interest payments on the original swap at time 0.5 have just beenmade, there are no net cash flows at the current moment. These cash flows on the original and new swap are asfollows:

Year 0.5 1 1.5 2

CFs to be received $100M x $100M x $100M x(L0.5+ 0.5)/2 (L1+ 0.5)/2 (L1.5+ 0.5)/2

CFs to be paid $2.75M $2.75M $2.75M

CFs to be received $2.25M $2.25M $2.25M

CFs to be paid $100M x $100M x $100M x(L0.5+ 0.5)/2 (L1+ 0.5)/2 (L1.5+ 0.5)/2

You can immediately see that the new swap is set up so that netting the cash flows from these two swaps results in aperfect offset of the floating rate payments and a small net payment (outflow) on the floating rate side. Because theunknown floating rate payments cancel out, the net cash flows each period are the differences between the paymentsmade on the old swap, $100M x 5.5%/2 = $2.75M and the payments received on the new swap, $10M x 4.5%/2 =2.25M, resulting in a net cash outflow of $0.5M Thus the combined net cash flows of these two swaps are:

Year 0.5 1 1.5 2

Net CFs -$0.5M -$0.5M -$0.5M(+ inflow, - outflow)

Taking the PV of these net cash flows as of time 0.5 at the current fixed rate of 4.5% (APR) yields us

-$0.5M / 1.0225 + -$0.5M / (1.0225)2 + -$0.5M / (1.0225)3= -$1.43495M

Thus with this technique we get that the combined swaps have a PV of $1.43495M. Since the new swap has, bydefinition, an NPV of 0, this means that the value of the original swap to the firm must be -$1.43495M, exactly aswith the previous method.

This is the method for valuing interest rate swaps where the cash flows re all in the same currency and the firms issimply swapping fixed interest rate payment for floating rate payments (or vice verse). Now we turn to currencyswaps where the flows will be in different currencies.

Original Swap

Plus Imaginary

New Market-

Rate Swap

Combination of

Original and

New Swa s Above


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Currency Swaps

A currency swap works much the same way as an interest rate swap. The primary difference is that we account forthe initial and final principal exchanges as they involve exchanges of different currencies, which can change inrelative value over time. In addition, currency swaps can involve swapping currencies at fixed rates, or floatingrates, or one at fixed and one at floating rates. Finally, unexpected changes in exchange rates as well as interest

rates can lead to changes in value.

Lets consider a 3-year fixed-to-fixed rate US$ - Australian$ currency swap with a principal value of US$50M. Atinitiation, the exchange rate is US$0.625 per A$. Currency swaps are constructed so that at initiation the principalamounts exchanged are equal at the currency exchange rate. This implies that the A$ principal of this swap will be$50M x (A$/$0.625) = A$80M. Assume that the firm enters into the swap agreeing to pay US$50M to thecounterparty in exchange for A$80M today. This is an even-up exchange today at the current exchange rate. Thefirm then makes semiannual payments to the counterparty in A$ (the borrowed currency) at the fixed rate of 5.0%APR (A$80M x 2.5% = A$2M) and receives semiannual payments from the counterparty in US$ (the lent currency)at a fixed rate of 4.0% APR (US$50M x 2.0% = US$1M). At maturity, the firms re-exchange the initial principals.The cash flows on the swap to from the perspective of the firm look as follows:

Note the fixed currency swap is like a spot exchange rate transaction, US$50M for A$80M, and a series of forwardtransactions, US$1M for A$2M, over the next 6 semi-annual periods, and a final forward transaction, A$80M forUS$50M, at maturity. Thus a fixed interest rate currency swap can be thought of as a bunch of forward contracts(all mostly all at the same ratewhich is not the actual forward rate at any time).

Valuing an Existing Currency Swap : Exchange Rate Change Only

Suppose that one year into this swap, the interest rates remained the same but the exchange rate had changed relativeto expectations. In particular, the spot exchange rate at time 1 is at US$0.75/A$. The impact of this exchange ratechange on the value of the swap can be examined using either the close out method of the discounting methoddiscussed above.

Method 1: Using the Close out Swap Technique

If we are interested in the US$ value of the original swap to the firm under the new exchange rates, we need toconsider a new 2-year swap for A$80M notional principal to eliminate the A$ side of the existing swap (note: weneed not actually enter into this swap we just use it to help us value the original swap). At the same interest rates asoriginally, this swap will have the firm paying out today a principal of A$80M, and receiving semi-annual interestpayments at the fixed rate of A$80M x 2.5% (A$2M), and then receiving back the A$80M principal at maturity.The other side of this swap would involve receiving a US$ principal equivalent to the A$80M at the currentexchange rate of $0.75/A$, which is A$80M x $0.75/A$ = US$60M. The firms would then make semi-annual US$interest payments of US$60M x 2% ($1.2M) and pay back the US$60M principal at maturity (time 3).

receive

pay

US$50

A$80 US$50

US$CF1 US$CF5US$CF4US$CF3US$CF2

A$CF1

US$50

A$80 US$50

US$1m1 US$1mUS$1m

A$2m

US$1m US$1m US$1m

A$2m A$2m A$2m A$2m A$2m

A$80

+

+

Time 0 0.5 1 1.5 2 2.5 3

receive

pay

US$50

A$80 US$50


A$CF1

US$50

A$80 US$50

US$1m1 US$1mUS$1m

A$2m

US$1m US$1m US$1m


A$80

+

+

receive

pay

US$50

A$80 US$50


A$CF1

US$50

A$80 US$50

US$1m1 US$1mUS$1m

A$2m

US$1m US$1m US$1m


A$80

+

+

Time 0 0.5 1 1.5 2 2.5 3


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Looking at the future cash flows to the old and new swap (no diagram this time) we have:

Year 1 1.5 2 2.5 3

CFs to be received US$1M US$1M US$1M US$1M+US$50M

CFs to be paid A$2M A$2M A$2M A$2M + A$80M

New Swap at time 0.5 for on A$80M

CFs to be received US$60M A$2M A$2M A$2M A$2M + A$80M

CFs to be paid A$80 US$1.2M US$1.2M US$1.2M US$1.2M+US$60M

Net Flows on both swaps: 0 -US$0.2M -US$0.2M -US$0.2M -US$10.2M(+ receive - pay )

Again, the A$ cash flows drop out, including the current principal payment which cancels out the equivalent valueUS$60M received by the firm at time 0.5 (at current XR, A$80m = US$60m). Taking the present value of the netcash flows on the two swaps combined we obtain:

PV1of Net CF at 4% (APR) = -$0.2/1.02 + -$0.2/1.022+ -$0.2/1.023+ -$10.2/1.024

PV1= -US$10m

The exchange rate change (US$ depreciation) has resulted in a loss on the swap to the firm of US$10M. Thereasoning behind this is that in the original swap the firm was receiving UD$ and paying A$. When the value of A$went up (exchange rate moving from US$0.625/A$ to US$0.75/A$), the value of paying A$ and receiving US$became less valuable. The market value of this loss at time 1 is US$10M

Alternative Approach: Taking PV of Both Sets of Cash Flows at Currency Rates

Another way to do this is to apply the same concept as Method 1 above for the interest rate swap. One discounts theremaining cash flows on the original swap in each currency at the current interest rate for that currency. The PV ofthe A$ cash flows is converted into US$ using the current exchange rate and this value is compared to the PV of theUS$ cash flows to determine the gain or the loss. The difference between the value US$ value of the cash flows tobe received and the value of the US$ cash flows to be paid out is the gain on the swap (its value compared toorigination).

In this example, with no change in interest rates, the PV of the cash flows remains the same, and only the exchangerate used to compare the PV at time 1 is different.PV ($ side at 2% per period) = US$1M/1.02 + US$1M/1.022+ US$1M/1.023+ US$51M/1.024= US$50M

PV(A$ side at 2.5% per period) = A$2M/1.025 + A$2M/1.0252+ A$2M/1.0253+ $82M/1.0254 = A$80M

With the XR at US$0.75/A$, the US$ value of the PV A$ side of the swap at time 1 = A$80M x US$0.75 = $60M

Net difference in PV of cash flows in US$ = $50M - $60M = -$10M

So with the present value of paying the A$ side measured at US$60M, and the present value of receiving the US$payments measured at US$50M, the firm clearly loses US$10M in value on the swap. If it wanted to get out of thisswap today, it would cost the firm US$10M, either to pay someone to take over their position or to close it out withthe counterparty.

Original Swap

Plus New Market Rate Swap


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Valuing Currency Swap when Interest Rates and Exchange Rates Change

When interest rates as well as exchange rates change, the approach to determining the value of the swap is similar.If only one interest rate and or the exchange rate changes, we can use the offsetting swap approach (method 1outlined above for the currency swap). In such a case, we would want to eliminate the cash flows in the currencywhose interest rate remains the same. (If this is the US$, then we would value the swap in FC by discounting net

cash flows using the FC interest rate and convert the PV of net FC cash flows into US$ at the current exchange rate.)

Close out Swap Method

Consider the situation with our currency swap from above. Suppose that at time 1, (just after the interest payments)the A$ interest rate for a 2 year swap has fallen to 4.0% (APR) and the exchange rate has risen to $0.80/A$. TheUS$ interest rate for a 2 year swap remains at 4.0% (APR)

At time 1, we can value the original swap by considering a new 2-year swap constructed with a notional value of$50M designed to eliminate the US$ cash flows on the old swap. This new swap (which we need not enter, butconstruct only to help us get the net cash flows and value the old swap) will involve receiving US$50M at time 1,and making interest payments of US$50M x 2% = US$1M in each of the remaining 4 semi-annual periods of theoriginal swap and then repaying the $50M principal at time 3. The other side of this new swap will involve payingA$62.5M (the US$ notional value of $50M converted into A$ at the current exchange rate =>US$50M/(US$0.8/A$) = A$62.5M) and receiving interest payments of A$1.25M (A$62.5M x 2%) in each of

remaining 4 semi-annual periods of the original swap and then receiving the A$62.5M principal at time 3.

Thus at time 1 (after the interest payments) the remaining 4 semi-annual cash flows (plus principal payments) to theoriginal swap and the cash flows to the new market swap would be as follows:

Year 1 1.5 2 2.5 3

CFs to be received US$1M US$1M US$1M US$51M

CFs to be paid A$2M A$2M A$2M A$82M

CFs to be received US$50M A$1.25M A$1.25M A$1.25M A$63.75M

CFs to be paid A$62.5 US$1M US$1M US$1M US$51M

Net Flows: 0* -A$0.75M -A$0.75M -A$0.75M -A$18.25M(+ receive - pay )

Taking the PV of each stream of CF at the appropriate interest rate for that currency as of time 1 we obtain:

PV1of Net CF at 4% (A$ APR) = -A$0.75/1.02 - $0.75/1.022- $0.75/1.023- A$18.25/1.024

PV1= -A$19.02M

This PV is in A$ and needs to be multiplied by the currency spot price ($/A$) in order to get the market value of the

swap in US$. => A$19.02M x US$0.80/A$ = -US$15.22MThus the swap would have a negative value of -US$15.22M to the firm.

Alternative Method: Taking PV of Both Sets of Cash Flows at Currency Rates

We could also have determined the value of the swap by using the multiple currency version of method 1 from theinterest rate swaps where we take the PV of the remaining cash flows in each currency at the current market interestrate, convert the PVs into a common using the spot rate and compared the PVs of the amount to be received withthat to be paid. In this case the remaining cash flows are given by

Original Swap

Plus New Market Rate Swap


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Year 1 1.5 2 2.5 3



The rate to discount the US$ flows is 2% per period (iUS$= 4% (APR)) and the rate to discount the A$ flows is 2%per period (iA$= 4% (APR)). With the spot rate of US$0.80/A$ this technique would yield:

PV ($ side at 2% per period) = US$1M/1.02 + US$1M/1.022+ US$1M/1.023+ US$51M/1.024 = US$50M

PV(A$ side at 2% per period) = A$2M/1.02 + A$2M/1.022+ A$2M/1.023+ $82M/1.024 = A$81.52M

With the exchange rate at US$0.80/A$, the US$ value of the PV of the A$ side of the swap equals= A$81.52 x US$0.80/A$ = US$65.22M

Thus under the new conditions, the swap is equivalent to receiving CF with a PV = US$50M and paying CF with aPV = US$65.22M. Thus the swap has a negative value to the firm of US$65.22M - US$50M = US$15.22M (sameas other method)

General Case:In the case when the interest rates of both currencies (and possibly the exchange rate) change, we must use thediscounting method rather than the offsetting swap method. When both interest rates change, it will not be possibleto devise a new swap that will completely eliminate the cash flows in one of the two currencies, so we must take thePV of the stream of CF in each currency and convert the present values into a common currency at the current spotrate and compare the present value of the CFs being received with those being paid out to determine whether theswap owner gains or loses.

Suppose for our currency swap above, at time 1 the new market conditions are that US$ interest rate for a 2-yearswap has fallen to 2% (APR), the A$ interest rate for a 2-year swap has fallen to 4% (APR) and the exchange ratehas risen to US$0.75/A$.

What would be the market value of the swap at time 1? Here are the remaining cash flows to the original swap:

Year 1 1.5 2 2.5 3



PV ($ side at 1% per semi-annual period) = $1M/1.01 + $1M/1.012+ $1M/1.013+ ($51M)/1.014 = US$53.05M

PV(A$ side at 2% per period) = A$2M/1.02 + A$2M/1.022+ A$2M/1.023+ $82M/1.024 = A$81.52M

With the exchange rate at US$0.75/A$, the US$ value of the PV of the A$ side of the swap equals= A$81.52 x US$0.75/A$ = US$61.14M

Thus under these new conditions where everything has changed, US$ interest rate, A$ interest rates, and theexchange rate, the original swap from the perspective of the firm is now equivalent to receiving CF with a PV =US$53.05M and paying CF with a PV = US$61.14M. Thus the swap has a negative value to the firm of

$61.14M - $53.05M = US$8.09M

Original Swap

Original Swap

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