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# Pricing and Valuing Equity Swap Contracts

## Equity Swaps

An equity swap is an OTC derivative contract in which two parties agree to exchange a series of cash flows. In this arrangement, one party pays a variable series determined by equity. The other party, on the other hand, pays a variable series determined by different equity or rate or a fixed series.

### Types of Equity Swaps

• Pay a fixed rate and receive equity return
• Pay floating rate and receive equity return
• Pay one equity return and receive another equity return

We can look at an equity swap as a portfolio of an equity position and a bond.

The equity swap cashflows are expressed as :

• NA(Equity return – Fixed rate) (for pay fixed, receive equity party)
• NA(Equity return – Floating rate) (for pay floating, receive equity)
• NA(Equity returnX – Equity returnY) (for pay equity, receive equity) where X and Y denote different equities.

### Pricing Equity Swaps

An equity swap is priced at the same rate as a comparable interest rate swap. Note, however, that the cashflows involved are very different.

The fixed swap rate is:

$$r_{FIX}=\frac{1-{PV}_{0,t_n}\left(1\right)}{\sum_{i=1}^{n}{{PV}_{0,t_i}\left(1\right)}}$$

#### Example: Calculating the price of an Equity Swap

Consider a four-year annual reset Libor floating-rate bond trading at par. A comparable interest rate swap has a fixed rate of 1.117%. The information used to price the interest rate swap is given in the following table:

$$\begin{array}{c|c} \textbf{Year} & \textbf{Discount factor} \\ \hline 1 & 0.9723 \\ \hline 2 & 0.9667 \\ \hline 3 & 0.9625 \\ \hline 4 & 0.9569 \end{array}$$

Using the same data, the fixed interest rate for a 4-year pay fixed rate and receive equity return equity swap is closest to:

#### Solution

The fixed-rate on an equity swap is identical to the fixed rate on a comparable interest rate swap. This means that the fixed rate on the equity swap will be 1.117%, which is similar to the fixed rate on a comparable interest rate swap.

### Valuing an Equity Swap

Valuing an equity swap after it is initiated is comparable to valuing an interest rate swap. However, instead of adjusting the floating-rate bond for the last floating rate observed (advanced set), the value of the notional amount of equity is adjusted.

Therefore, the value of an equity swap is expressed as:

$$V_t = FB_t\left(C_0\right)- \frac {S_t}{S_{t-}}NA_E – PV(Par – NA_E)$$

Where

$$FB_t(C_0)$$ = time t value of a fixed-rate bond initiated with coupon C0 at time 0

$$S_t$$ = the current equity price,

$$S_{t–}$$ = the equity price observed at the last reset date

## Question

An equity swap has an annual swap rate of 4% and a notional principal of $2 million. The underlying index is currently trading at 2,000. After 30 days, the index trades at 2,200, and the LIBOR spot rates are as given in the following table: $$\begin{array}{c|c} \textbf{Year} & \textbf{Spot rates} \\ \hline 60 -\text{day Libor} & 3.90\% \\ \hline 150-\text{day Libor} & 4.55\% \\ \hline 240-\text{day Libor} & 5.20\% \\ \hline 330-\text{day Libor} & 5.85\% \end{array}$$ The value of the equity swap to the fixed-rate payer is closest to: 1.$ 301,800.
2. $23,980. 3.$ 223,980.

#### Solution

The correct answer is C:

The first step is to calculate the discount factors:

\begin{align*} D_{60} &=\frac{1}{1+\left(0.0390\times\frac{60}{360}\right)}=0.9935 \\ D_{150} &=\frac{1}{1+\left(0.0455\times\frac{150}{360}\right)}=0.9814 \\ D_{240} &=\frac{1}{1+\left(0.0520\times\frac{240}{360}\right)}=0.9665 \\ D_{330} &=\frac{1}{1+\left(0.0585\times\frac{330}{360}\right)}=0.9491 \end{align*}

The value of the fixed-rate bond is then calculated as:

\begin{align*} P(\text {fixed}) & =\frac{\left(4\%\right)}{4} \times(0.9935+0.9814+0.9665+0.9491)+1\times0.9491 \\ & = 0.98801 \end{align*}

The value of the index investment :

$$P(\text{Index}) =\frac {2200}{2000} = 1.1$$

The swap value to the fixed-rate payer is, therefore:

\begin{align*} V & = [P(\text{index}) -P(\text{fixed})]\times \text{notional principal} \\ & = (1.1-0.98801)\times 2 \text{ million} \\ & =223,980 \end{align*}

Reading 33: Pricing and Valuation of Forward Commitments

LOS (g) Describe how equity swaps are priced, and calculate and interpret their no-arbitrage value.

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