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In this section, we explore how to price and value three types o f equity swaps: ( I ) a swap to pay a fixed rate and receive the return on the equity, (2) a swap to pay a floating rate and receive the return on the equity, and ( 3 )a swap to pay the return on one equity and receive the return on another.
To price or value an equity swap, we must determine a combination of stock and bonds that replicates the cash flows on the swap. As we saw with interest rate and currency swaps, such a replication is not difficult to create. W e issue a bond and sell a bond, with one being a fixed-rate bond and the other being a floating-rate bond. I f we are dealing with a currency swap, we require that one o f the bonds be denominated in one currency and the other be denominated in the other currency. With an equity swap, it would appear that a replicating strategy would involve issuing a bond and buying the stock or vice versa, but this is not exactly how to replicate an equity swap. Remember that in an equity swap, we receive cash payments representing the return on the stock, and that is somewhat different from payments based on the price.
Pricing a Swap to Pay a Fixed Rate and Receive the Return on the Equity: By example, we will demonstrate how to price an n-payment m-day rate swap to pay a fixed rate and receive the return on equity. Suppose the notional principal is $1, the swap involves annual settlements and lasts for two years (n = 2),and the returns on the stock for each o f the two years are 10 percent for the first year and 15 percent for the second year. The equity payment on the swap would be $0.10 the first year and $0.15 the second. I f ,    however, we purchased the stock instead o f doing the equity swap, we would have to sell the stock at the end o f the first year or we would not generate any cash. Suppose at the end o f the first year, the stock is at $1.10. W e sell the stock, withdraw $0.10, and reinvest $1.OO in the stock. At the end o f the second year the stock would be at $1.15. W e then \ell the stock, taking cash of $0.15. But we have $1.OO left over. To get rid of,or offset,this cash flow, suppose that when we purchased the stock we borrowed the present value of $1.00 for two years. Then two years later, we would pay back $1.00 on that loan. This procedure would offset the $1.00 in cash we have from the stock. The fixed payments on the swap can be easily replicated. I f the fixed payment is denoted as FS(O,n,m),we simply borrow the present value o f FS(O,n,m) for one year and also borrow the present value of FS(O,n,m)for two years. When we pay those loans back, we will have replicated the fixed payments on the swap.Equity

Recall the four types of currency swaps: (1) pay one currency fixed, receive the other fixed, (2) pay one currency fixed, receive the other floating, (3) pay one currency floating, receive the other fixed, and (4) pay one currency floating, receive the other floating. In determining the fixed rate on a swap, we must keep in mind one major point: The fixed rate is the rate that makes the present value of the payments made equal the present value of the payments received. In the fourth type of currency swap mentioned here, both sides pay floating so there is no need to find a fixed rate. But all currency swaps have two notional principals, one in each currency. We can arbitrarily set the notional principal in the domestic currency at one unit. We then must determine the equivalent notional principal in the other currency. This task is straightforward: We simply convert the one unit of domestic currency to the equivalent amount of foreign currency, dividing 1.0 by the exchange rate.
Consider the first type of currency swap, in which we pay the foreign currency at a fixed rate and receive the domestic currency at a fixed rate. What are the two fixed rates? We will see that they are the fixed rates on plain vanilla interest rate swaps in the respective countries.
Because we know that the value of a floating-rate security with $1 face value is $1, we know that the fixed rate on a plain vanilla interest rate swap is the rate on a $1 par bond in the domestic currency. That rate results in the present value of the interest payments and the hypothetical notional principal being equal to 1.0 unit of the domestic currency. Moreover, for a currency swap, the notional principal is typically paid, so we do not even have to call it hypothetical. We know that the fixed rate on the domestic leg of an interest rate swap is the appropriate domestic fixed rate for a currency swap in which the domestic notional principal is 1.0 unit of the domestic currency.
What about the fixed rate for the foreign payments on the currency swap? To answer that question, let us assume the point of view of a resident of the foreign country. Given the term structure in the foreign country, we might be interested in first pricing plain vanilla interest rate swaps in that country. So, we know that the fixed rate on interest rate swaps in that country would make the present value of the interest and principal payments equal 1.0 unit of that currency.
Now let us return to our domestic setting. We know that the fixed rate on interest rate swaps in the foreign currency makes the present value of the foreign interest and principal payments equal to 1.0 unit of the foreign currency. We multiply by the spot rate, So, to obtain the value of those payments in our domestic currency: 1.0 times So equals So, which is now in terms of the domestic currency. This amount does not equal the present value of the domestic payments, but if we set the notional principal on the foreign side of the swap equal to l/So, then the present value of the foreign payments will be So(l/So) = 1.0 unit of our domestic currency, which is what we want.

Pricing an interest rate swap means finding the fixed rate that equates the present value of the fixed payments to the present value of the floating payments, a process that sets the market value of the swap to zero at the start. Using the time line illustrated earlier, the swap cash flows will occur on days hl, h2,…,hn-l, and h,, so there are n cash flows in the swap. Day h, is the expiration date of the swap. The time interval between payments is m days. We can thus think of the swap as being on an m-day interest rate, which will be LIBOR in our examples.
As previously mentioned, the payments in an interest rate swap are a series of fixed and floating interest payments. They do not include an initial and final exchange of notional principals. As we already observed, such payments would be only an exchange of the same money. But if we introduce the notional principal payments as though they were actually made, we have not done any harm. The cash flows on the swap are still the same. The advantage of introducing the notional principal payments is that we can now treat the fixed and floating sides of the swap as though they were fixed- and floating-rate bonds.
So we introduce a hypothetical final notional principal payment of $1 on a swap starting at day 0 and ending on day h,, in which the underlying is an m-day rate. The fixed swap interest payment rate, FS(O,n,m), gives the fixed payment amount corresponding to the $1 notional principal.