Equity Swaps

"Equity swap is an exchange of cash flows between two parties that allows each party to diversify its income, while still holding its original assets. The two sets of nominally equal cash flows are exchanged as per the terms of the swap, which may involve an equity-based cash flow (such as from a stock asset) that is traded for a fixed-income cash flow (such as a benchmark rate), but this is not necessarily the case. Besides diversification and tax benefits, equity swaps also allow large institutions to hedge specific assets or positions in their portfolios" (Investopeida, 2008).

Equity swaps can exist where one counter party agres to make to the other counterparty a series of payments that are determined by the return on a stock or stock index and the other party ameks a series of payments to the first party based on a floating money market index or a fixed rate or return from another stock or index. Basically, the exchange of cash flows can be three different types as follows:

1. Cash flow from stock or stock index in exchange for cash flow from a floating money market index or a fixed rate

2. Cash flow from a floating money market index or fixed rate and cash flow based on percentage of the stock index movement. In this case, a notional/principal amount of money is agreed

upon between the two counterparties

3. Cash flow from stock or stock index in exchange for cash flow from another stock or stock index

"Equity swaps differ from interest rate swaps in several ways. Since a stock return can be negative, the party receiving the retutn on the stock will have to pay the retutn on the stock when the stock goes down. Thus, in constrast to an interest rate swap where party A makes an itnerest payment to party B and party B makes an interest payment to party A, an equity swap can have one party amking and one party receiving both payments" (Rich, 2008). There is however a similarity between equity swaps and interest rate swaps in that the exchange of cash is usually done by netting the amonts owed, so there is only one cash amount transferred from one party to the other.

"Most equity swaps today are conducted between large financing firms such as auto financiers, investment banks and capital lending institutions. LIBOR rates are a common benchmark for the fixed income portion of equity swaps, which also tend to be held at intervals of one year or less, much like commercial paper" (Investopedia, 2008).

Source Website[http://www.investopedia.com/terms/e/equityswap.asp]

1. Total Return Equity Swaps and Equity Forwards

Overview

"A total return equity swap or a total return swap (TRS) on an equity is similar to a total return swap on a bond. It is a bilateral financial contract in that one counterparty pays out the total return of the equity, including its dividends and capital appreciation or depreciation, and in return, receives a regular fixed or floating cash flow. For convenience we call the asset’s total return a TR-leg and the fixed or floating cash flow a non-TR leg. A total return swap can be settled at the terminating date only or periodically, e.g., quarterly" (FinCAD, 2008).

"The underlying equity can be a local equity, i.e., an equity denominated in the local currency, or a currency translated equity, i.e., an equity that is denominated in a foreign currency and is translated into a local currency; or a quanto equity, a foreign equity that is treated as a local equity without a currency translation although a constant is multiplied oftentimes. To make clear the differences between a currency-translated equity and a quanto equity, suppose an equity X is traded in UK with a spot price of £20 and the spot FX rate of USD/Sterling is 1.65. Then the spot dollar price of the currency-translated equity X is $33, but the spot dollar price of the quanto equity X is $20, although it can be multiplied by a constant FX rate that does not change during the life of a swap contract" (FinCAD, 2008).

"The equity used in a total return swap contract can be a single publicly traded stock or a private stock, a portfolio of stocks, a stock index, or even any market index. The buyer of a total return equity swap can gain the economic exposure to certain equity or index market without physically owning such assets while the seller of a total return equity swap can reduce or eliminate the market risk of his/her stock portfolio without selling the assets and gain stable returns. Speculators can use equity TRS to trade a market index which may not be traded physically" (FinCAD, 2008).

Formulas & Technical Details

Valuation

The payoff of the TR leg of a TRS on equity is:

N * (S - S_{0} + D ) / S_{0}

where

N is the swap’s notional,

S_{0} is the stock’s initial price,

S is the stock’s price at the maturity date of the swap and

D is the stock’s dividends paid during the swap’s life.

Note that the payoff of the TR leg is independent of the dividends, if the swap does not reset. Set ^{k = N / S0}. Then is ^{k} is the number of shares of the portfolio of the reference equity. If the swap resets, then the return of the TR leg on a reset period is the price change of the portfolio in this period:

k * (S_{i} - S_{i-1} + D_{i})

where

S _{i} denotes the stock price at the *i*th reset date and

D_{i} the dividends paid in this period. The notional of the non-TR leg grows or shrinks with the stock price. In period the notional of the non-TR leg is:

k * S_{i-1}

"To make this clearer, let’s look at an example. Suppose a bank holds a large portion of the stock of a company. The bank wants to reduce the risk of holding the stock without selling it. It then enters into a total return equity swap with an investor. The swap has an initial notional of $1,000,000 and the initial price of the stock is $20. The stock does not pay dividends. On any reset date the investor will receive from the bank any price appreciation of the stock of 50,000 (=$1,000,000/$20) shares and pay any depreciation and a coupon of 4% on the current notional semi-annually. The TRS resets semi-annually" (FinCAD, 2008).

"To get a picture of the cash flows of the swap we check the first two reset periods. Suppose at the end of the first half year the stock price goes up to $22. Then at the first cash flow date the investor will receive $100,000 (=($22-$20)*50,000) from the bank and pay $20,000 (=0.04*$1,000,000/2). Suppose at the end of one year, i.e., the second cash flow date, the stock price goes down to $16. Then the investor will pay the bank $250,000 (= ($22-$16)*50,000) for the stock price depreciation and $22,000 (=0.04*22*50,000/2) for the interest payment" (FinCAD, 2008).

"To calculate the value of a TRS on an equity, we only need to discount all the future cash flows of the swap to the valuation date. The value of the non-TR leg is simply the value of a fixed or floating leg of an interest rate swap with variable notional amount. For the TR leg, from the above example we see that the most important part is to forecast the price of the underlying equity, in other words, to calculate the forward prices of the equity at reset dates. The formulas are given in the following, assuming the current time is 0 " (FinCAD, 2008).

For a local equity the forward price at time T is:

S(T) + S exp ((r - q)T)

where

S is the equity spot price and

r and q are the risk-free rate and dividend yield respcetively. If the dividends are discrete, the forward price is:

S(T) = (S - D) exp (rT)

where D is the sum of the present values of the dividends paid up to time T.

For a currency-translated equity we need a model for the spot stock price and the spot FX rate. We assume that the stock price follows the Black-Scholes lognormal model and the FX rate the Garman-Kohlhagen lognormal model. With these models in hand we can calculate the forward price of a currency-translated equity with a dividend yield as follows:

S(T) = S*X exp ((r - q + ρσσ_{X})T)

where

σ and σ_{x} are the volatilities of the equity and the FX rate, respectively, and

X is the FX rate, in units of domestic currency per one unit of foreign currency.

The factor

ρ is the correlation between these volatilities.

For a quanto equity with dividend yield q :

S(T) = S * X_{0} exp ((r - q - ρσσ_{X})T)

where X_{0} is the constant FX rate specified in a quanto contract. If a foreign stock pays discrete dividend, simply replace S in the above formulas with S - D and set q = 0 as is done for a local equity, but D is the present value of dividends discounted with a foreign risk-free rate.

Source: [http://www.fincad.com/support/developerFunc/mathref/equity_trs.htm]

2. Equity Swap : Equity index and Money Market Index

"The exchange of cash flows in an equity swap can exist between cash flow calculated as a percentage of the increase in one of the stock market equity indices and cash flow calculated from a floating money market index such as LIBOR or a fixed rate. Again," two cash flows are usually referred to as 'legs'" (Wikipedia, 2008). The cash flow stream that is pegged to floating rate of interest or pays a fixed rate is named "equity leg". The cash flow that is based "on the performance of a share of stock or stock market index is named 'floating leg'" (Wikipedia, 2008).

Like any other basic swap, the basic equity swap involves a notional principal, a specified tenor and prespecified payment intervals.

Examples

Parties may agree to make periodic payments or a single payment at the maturity of the swap ("bullet" swap), the simplest case.

Take a simple index swap where Party A swaps £5,000,000 at LIBOR + 0.03% (also called LIBOR + 3 basis points) against £5,000,000 (FTSE to the £5,000,000 notional). In this case Party A will pay (to Party B) a floating interest rate (LIBOR +0.03%) on the £5,000,000 notional and would receive from Party B any percentage increase in the FTSE equity index applied to the £5,000,000 notional.

In this example, assuming a LIBOR rate of 5.97% p.a. and a swap tenor of precisely 180 days, the floating leg payer/equity receiver (Party A) would owe (5.97%+0.03%)*£5,000,000*180/360 = £150,000 to the equity payer/floating leg receiver (Party B).

At the same date (after 180 days) if the FTSE had appreciated by 10% from its level at trade commencement, Party B would owe 10%*£5,000,000 = £500,000 to Party A. If, on the other hand, the FTSE at the six-month mark had fallen by 10% from its level at trade commencement, Party A would owe an additional 10%*£5,000,000 = £500,000 to Party B, since the flow is negative.

As a mitigant to credit exposure, the trade can be reset or "marked-to-market" during its life. In that case, appreciation or depreciation since the last reset is paid and the notional is increased by any payment to the floating rate payer or decreased by any payment from the floating leg payer.

Applications

"Typically Equity Swaps are entered into in order to avoid transaction costs (including Tax), to avoid locally based dividend taxes, limitations on leverage (notably the US margin regime) or to get around rules governing the particular type of investment that an institution can hold.

Investment banks that offer this product usually take a riskless position by hedging the client's position with the underlying asset. For example, the client may trade a UK cash equity swap - say Vodafone. Bank credits the client with 1000 Vodafone at GBP1.45. The bank pays the return on this investment to the client, but also buys the stock in the same quantity for its own trading book (1000 Vodafone at GBP1.45). Any equity-leg return paid to or due from the client is offset against realized profit or loss on its own investment in the underlying asset. The bank makes its money through commissions, interest spreads and dividend rake-off (paying the client less of the dividend than it receives itself). It may also use the hedge position stock (1000 Vodafone in the previous example) as part of a funding transaction such as stock lending, repo or as collateral for a loan" (Wikipedia, 2008).

Source: [http://en.wikipedia.org/wiki/Equity_swap]

3. Equity Swap: Cash Flow From Stock or Stock Index in Exchange for Cash Flow From Another Stock or Stock Index

Let us suppose that there is an agreement between two counterparties for one-year equity swap with quarterly settlements to pay the return on NASDAQ and receive the return on the S&P 500 on Notional Principal of $20 million. Suppose that nn December 15 of a given year Dynamic Money Management enters into a swap to pay the return on the NASDAQ Composite index and receive the return on the S&P 500 with payments to occur on March 15, June 15, September 15, and December 15 for one year. Payments will be calculated on a notional principal of $20 million. The counterparty is the swaps dealer Total Swaps, Inc. The S&P 500 is at 1105.15 and NASDAQ is at 1705.51.

The table below shows a set of hypothetical payments based on an assumed set of values for the S&P 500 and NASDAQ. Note in this case that since Dynamic Money Management is paying the NASDAQ return, it can actually end up receiving the NASDAQ return. This occurs on June 15. The NASDAQ index lost 3.5144% from March 15 to June 15, resulting in a negative payment of $702,880. Since Dynamic is obligated pay the NASDAQ return, which is negative, Dynamic would receive the NASDAQ payment. Dynamic receives the S&P 500 payment, which is a negative $800,020, so the net effect is that Dynamic pays $97,140. On September 15, however, the NASDAQ payment is more negative than the S&P 500 payment, so Dynamic ends up receiving a net of $407,100.

Date | S&P 500 Index | Periodic Return on S&P 500 | S&P 500 Cash Flow | NASDAQ Index | Periodic Return on NASDAQ | NASDAQ Cash Flow | Net Cash Flow |

December 15 | 1105.15 | 1705.51 | |||||

March 15 | 1129.48 | 2.2015% | $440,300 | 1750.78 | 2.6543% | -$530,860 | -$90,560 |

June 15 | 1084.30 | -4.0001% | -800,020 | 1689.25 | -3.5144% | +702,880 | -97,140 |

September 15 | 1055.29 | -2.6755% | -535,100 | 1609.67 | -4.7110% | +942,200 | 407,100 |

December 15 | 1099.52 | 4.1913% | 838,260 | 1678.51 | 4.2767% | -855,340 | -17,080 |

Source: [http://www.bus.lsu.edu/academics/finance/research/working%20papers/EquitySwapsandEquityInvesting.pdf]