###### Unit Roots for Time-Series Analysis

The Unit Root Problem An AR(1) series is said to be covariance stationary... **Read More**

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 pays a variable series determined by different equity or rate or a fixed series.

- 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 return X – Equity return Y) (for pay equity, receive equity) where X and Y denote different equities.

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:

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

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:

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 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.

\(S_t\) = Current equity price.

\(S_{t–}\) = 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:

- $301,800.
- $23,980.
- $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 33 (g) Describe how equity swaps are priced, and calculate and interpret their no-arbitrage value.*