# Class Note on Valuing Swaps

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1 Corporate Finance Professor 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 of existing assets or liabilities, most often to manage risk or to convert cash flows of one type of security into the cash flows 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 a party 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 the expected 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 will change. When the value changes, one party to the swap will experience a gain equal to the increase in the value of swap, while the other party to the swap will experience an equivalent loss (zero sum game). Below we will consider both interest rate and currency swaps and consider how to measure their change in value in response to changes in their underlying financial prices Interest Rate Swaps Let s consider an interest rate swap first. In an interest rate swap, parties are exchanging fixed interest rate payments for floating interest rate payments on some notional value. To define an interest rate swap we start by defining a notional value a principal amount upon which the interest payments are calculated. However, this principal amount is not exchanged at the beginning or end of the contract, as it is not necessary (why give \$100 just to 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 four semi-annual floating rate payments of LIBOR, denoted hereafter by L, plus 0.50% on the notional principal. At initiation 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 be long an interest rate swap as it is in a position to gain if interest rates rise. The counterparty to this swap has exactly the opposite cash flow structure (they are short a swap). The cash flows are exchanged at the end of each semiannual period. Year Receive LIBOR 0 /2 = \$2.625M LIBOR 0.5 /2 LIBOR 1 /2 LIBOR 1.5 /2 Pay While the fixed interest rate side payments each period are known with certainty, = \$2.75M each period, the floating rate side is known with certainty only for the first payment as it is based upon current LIBOR of 4.75% plus a 0.50% spread. Thus the first floating rate payment is \$100 x 5.25%/2 = \$2.625m. Future floating rate payments each period will depend on the future LIBOR rates. Their expectations are such that they have the same PV as the fixed side flows. Again notice that there are no notional principal exchanges with an interest rate swap. SAIS Class Note on Valuing Swaps p. 1

2 We can also express the cash flows from this swap in table form: Year CFs to be received \$2.625M \$100M x \$100M x \$100M x (L )/2 (L )/2 (L )/2 CFs to be paid \$2.75M \$2.75M \$2.75M \$2.75M We could determine the expected floating rates (L 0.5, L 1, L 1.5 ) 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 of the 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 =, PMT = \$2.75m) = \$ m This comes from realizing that the swap is a zero NPV security at initiation. Valuing an Existing Interest Rate Swap Now let s consider how to determine the value of an interest swap at some point in the future when economic conditions have changed relative to the origination of the swap. Suppose 6 months (0.5 year) into the swap, at the date of the first interest payments, interest rates are now lower than originally expected. This means both the current fixed rate and the current and expected future LIBOR rates are lower than they were expected to be at the beginning of 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 the future. The question is how much does the firm lose on this swap? This depends on how much interest rates changes 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 notional value at the maturity on the original swap at the new fixed rate of interest for a swap with that remaining maturity and compare this to the PV of the floating rate side of the original swap, which is by definition equal to the notional principal 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 at maturity. (Note repayment of notional principal at maturity does not really occur but we must include this phantom cash flow to get the value correct). Year Remaining Fixed Rate CFs \$2.75M \$2.75M \$2.75M on 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 = -\$ M Thus the current value of the fixed payments to the swap is \$ M. This is the PV of the payment the firm must make on the swap. The market value of the floating rate side of the swap will, by definition, be \$100M (the PV of floating rate payments on \$100M where the rates adjust for interest rate movements). So, with the PV of what the firm must pay at \$ M and the PV of what the firm will receive at \$100M, the firm has lost \$ M of value on the swap as a result of the interest rate changes. SAIS Class Note on Valuing Swaps p. 2

3 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 of the remaining cash flows on the existing swap. We take the present value of the net cash flows of the two swaps together, which is typically a net fixed rate payment at the current fixed interest rate on the new swap. This approach takes advantage of the fact that the new swap has zero NPV, so when we combine its cash flows with the existing swap s 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 on a new swap with a maturity of 1.5 years is 4.5% (APR). and current LIBOR is 3.5% (APR)). To implement this approach on our swap above at time 0.5, we would need to enter into a new swap with a \$100M notional value on which we pay the floating LIBOR rates plus 0.50% spread (\$100M x (L + 0.5)/2) and receive fixed payments at the new lower fixed interest rates 4.5% (APR) (\$100M x 4.5%/2 = \$2.25M). Since no principal amounts are exchanged in an interest rate swap and we assume that the interest payments on the original swap at time 0.5 have just been made, there are no net cash flows at the current moment. These cash flows on the original and new swap are as follows: Year CFs to be received \$100M x \$100M x \$100M x (L )/2 (L )/2 (L )/2 CFs to be paid \$2.75M \$2.75M \$2.75M Plus Imaginary New Market- Rate Swap CFs to be received \$2.25M \$2.25M \$2.25M CFs to be paid \$100M x \$100M x \$100M x (L )/2 (L )/2 (L )/2 You can immediately see that the new swap is set up so that netting the cash flows from these two swaps results in a perfect offset of the floating rate payments and a small net payment (outflow) on the floating rate side. Because the unknown floating rate payments cancel out, the net cash flows each period are the differences between the payments made on the old swap, \$100M x = \$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: Combination of Original and New Swaps Above Year 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 / \$0.5M / (1.0225) 2 + -\$0.5M / (1.0225) 3 = -\$ M Thus with this technique we get that the combined swaps have a PV of \$ M. Since the new swap has, by definition, an NPV of 0, this means that the value of the original swap to the firm must be -\$ M, exactly as with 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 is simply swapping fixed interest rate payment for floating rate payments (or vice verse). Now we turn to currency swaps where the flows will be in different currencies. SAIS Class Note on Valuing Swaps p. 3

4 Currency Swaps A currency swap works much the same way as an interest rate swap. The primary difference is that we account for the initial and final principal exchanges as they involve exchanges of different currencies, which can change in relative value over time. In addition, currency swaps can involve swapping currencies at fixed rates, or floating rates, 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. At initiation, the exchange rate is US\$0.625 per A\$. Currency swaps are constructed so that at initiation the principal amounts 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 the counterparty in exchange for A\$80M today. This is an even-up exchange today at the current exchange rate. The firm 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: Time A\$80 US\$50 receive pay US\$CF US\$1m 1 US\$CF US\$1m 2 US\$CF US\$1m 3 US\$1m US\$CF 4 US\$1m US\$CF 5 US\$1m + + A\$CF A\$2m 1 A\$2m A\$2m A\$2m A\$2m A\$2m US\$50 A\$80 Note the fixed currency swap is like a spot exchange rate transaction, US\$50M for A\$80M, and a series of forward transactions, US\$1M for A\$2M, over the next 6 semi-annual periods, and a final forward transaction, A\$80M for US\$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 rate which 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 relative to expectations. In particular, the spot exchange rate at time 1 is at US\$0.75/A\$. The impact of this exchange rate change on the value of the swap can be examined using either the close out method of the discounting method discussed 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 to consider a new 2-year swap for A\$80M notional principal to eliminate the A\$ side of the existing swap (note: we need not actually enter into this swap we just use it to help us value the original swap). At the same interest rates as originally, this swap will have the firm paying out today a principal of A\$80M, and receiving semi-annual interest payments 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 current exchange 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). SAIS Class Note on Valuing Swaps p. 4

5 Looking at the future cash flows to the old and new swap (no diagram this time) we have: Year 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 Plus New Market Rate Swap 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 value US\$60M received by the firm at time 0.5 (at current XR, A\$80m = US\$60m). Taking the present value of the net cash flows on the two swaps combined we obtain: PV 1 of Net CF at 4% (APR) = -\$0.2/ \$0.2/ \$0.2/ \$10.2/ PV 1 = -US\$10m The exchange rate change (US\$ depreciation) has resulted in a loss on the swap to the firm of US\$10M. The reasoning 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 the remaining cash flows on the original swap in each currency at the current interest rate for that currency. The PV of the A\$ cash flows is converted into US\$ using the current exchange rate and this value is compared to the PV of the US\$ cash flows to determine the gain or the loss. The difference between the value US\$ value of the cash flows to be received and the value of the US\$ cash flows to be paid out is the gain on the swap (its value compared to origination). In this example, with no change in interest rates, the PV of the cash flows remains the same, and only the exchange rate used to compare the PV at time 1 is different. PV (\$ side at 2% per period) = US\$1M/ US\$1M/ US\$1M/ US\$51M/ = US\$50M PV(A\$ side at 2.5% per period) = A\$2M/ A\$2M/ A\$2M/ \$82M/ = 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 this swap today, it would cost the firm US\$10M, either to pay someone to take over their position or to close it out with the counterparty. SAIS Class Note on Valuing Swaps p. 5

6 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 1 outlined above for the currency swap). In such a case, we would want to eliminate the cash flows in the currency whose 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\$. The US\$ 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, but construct 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 the original swap and then repaying the \$50M principal at time 3. The other side of this new swap will involve paying A\$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 the original swap and the cash flows to the new market swap would be as follows: Year 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 Plus New Market Rate Swap 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: PV 1 of Net CF at 4% (A\$ APR) = -A\$0.75/ \$0.75/ \$0.75/ A\$18.25/ PV 1 = -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\$15.22M Thus 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 the interest rate swaps where we take the PV of the remaining cash flows in each currency at the current market interest rate, convert the PVs into a common using the spot rate and compared the PVs of the amount to be received with that to be paid. In this case the remaining cash flows are given by SAIS Class Note on Valuing Swaps p. 6

7 Year CFs to be received US\$1M US\$1M US\$1M US\$51M CFs to be paid A\$2M A\$2M A\$2M A\$82M The rate to discount the US\$ flows is 2% per period (i US\$ = 4% (APR)) and the rate to discount the A\$ flows is 2% per period (i A\$ = 4% (APR)). With the spot rate of US\$0.80/A\$ this technique would yield: PV (\$ side at 2% per period) = US\$1M/ US\$1M/ US\$1M/ US\$51M/ PV(A\$ side at 2% per period) = A\$2M/ A\$2M/ A\$2M/ \$82M/ 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 = US\$50M = A\$81.52M Thus under the new conditions, the swap is equivalent to receiving CF with a PV = US\$50M and paying CF with a PV = US\$65.22M. Thus the swap has a negative value to the firm of US\$65.22M - US\$50M = US\$15.22M (same as other method) General Case: In the case when the interest rates of both currencies (and possibly the exchange rate) change, we must use the discounting method rather than the offsetting swap method. When both interest rates change, it will not be possible to devise a new swap that will completely eliminate the cash flows in one of the two currencies, so we must take the PV of the stream of CF in each currency and convert the present values into a common currency at the current spot rate and compare the present value of the CFs being received with those being paid out to determine whether the swap 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-year swap has fallen to 2% (APR), the A\$ interest rate for a 2-year swap has fallen to 4% (APR) and the exchange rate has 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 CFs to be received US\$1M US\$1M US\$1M US\$51M CFs to be paid A\$2M A\$2M A\$2M A\$82M PV (\$ side at 1% per semi-annual period) = \$1M/ \$1M/ \$1M/ (\$51M)/ PV(A\$ side at 2% per period) = A\$2M/ A\$2M/ A\$2M/ \$82M/ = 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 = US\$53.05M Thus under these new conditions where everything has changed, US\$ interest rate, A\$ interest rates, and the exchange 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 SAIS Class Note on Valuing Swaps p. 7

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