Equity Swap Pricing
Back in 2008 and 2009, the global financial crisis was in full force, and officials were dumbfounded by the consistent selling in the marketplace, regardless of the numerous attempts at liquidity injections.
And as you know now, equity swaps were a part of the problem, especially equity swaps with a floating or fixed payment attached.
Why would that be the case?
No, actually.
Bingo!
Equity swaps with a pay fixed or pay floating can really increase risk through increased leverage.
That's because a significant decrease in the price of the underlying equity reference can easily lead to large payments, versus the smaller change in the referenced interest rate.
For example, start with the equity leg of an equity swap. It can be shown as
$$\displaystyle S_i = NA_E R_E$$.
_R__E_ is the periodic return of the equity either with or without dividends, and _NA__E_ is the notional amount.
Now, compare that with a fixed interest rate leg of the equity swap.
$$\displaystyle FS = NA_E \times AP_{FIX} \times r_{FIX}$$
_AP__FIX_ indicates the constant accrual period for the fixed leg, and _r__FIX_ notes the fixed rate on the equity swap.
If you were going to price out an equity swap with a fixed interest rate, you'd start by analyzing the setup from an arbitrageur's perspective.
For the initial simultaneous transactions, the arbitrageur would start by entering into an equity swap, and then buy the equity with the notional amount _NA__E_. There would also need to be a transaction with a fixed-rate bond.
How would that be structured?
No.
The arbitrageur needs to offset the cost of buying the equity.
You got it!
The short-sale bond proceeds would offset the cost of the equity.
At this point, you've got an equity swap, a long equity position, and a short fixed-rate bond. Now think about how the cash flows would be structured. The fixed interest payments would be offset by the bond coupons, so the cash flow is zero. For the equity, any increase or decrease above or below the initial notional amount would need to be sold (above the notional) or bought (below the notional). But again, the difference in the interest payments versus the equity price performance would make these cash flows equal. Note that all dividends are considered reinvested.
So if you assume that the notional amount invested in the equity equals the notional amount of par value in the bond, how would you price the swap?
No.
The notional value of the bond should equal the equity amount so that cash flow offsets.
Not quite.
The notional value of the equity should equal the bond amount so that cash flow offsets.
Now, think about how this applies to a receive-floating, pay-equity swap. Since the floating rate isn't fixed, but it changes over the period between each reset, an interest rate swap is essentially priced off the fixed rate.
So with two fluctuating payments over the contract period and two notional amounts that are equal, what's the price of a receive-floating, pay-equity swap?
No.
The notional values are equal, so the price of the stock is offset by the notional amount in the floating-rate bond.
That's not it.
The notional value of the floating-rate bond is offset by the notional amount in the equity.
Bingo!
It's zero. The price at initiation is zero, because all the cash flows will settle the value over time.
And the same thought process applies to pay-equity, receive-equity. The notional value at initiation will cancel, and the cash flows are indeterminable, so the price is zero.
To sum it up:
[[summary]]
You got it!
Pricing the receive-fixed, pay-equity swap is the same as pricing a fixed-rate swap. Just remember that the notional amount is equal to 1, which is also the par value.
$$\displaystyle r_{FIX} = \frac{1 - PV_n(1)}{\sum^{n}_{i=1} PV_i(1)}$$
Fixed or floating swaps decrease risk
Fixed or floating equity swaps increase leverage
Buy the bond long
Sell the bond short
The price of the bond
The price of the stock
The price of the fixed-rate interest swap
Zero
Price of equity stock
Price of floating-rate bond
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