Pricing and Valuing Equity Swap Contracts

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 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 return X – Equity return Y) (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:


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:

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


\(FB_t(C_0)\) = 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.


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.


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.

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep

    Daniel Glyn
    Daniel Glyn
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.
    Nyka Smith
    Nyka Smith
    Every concept is very well explained by Nilay Arun. kudos to you man!
    Badr Moubile
    Badr Moubile
    Very helpfull!
    Agustin Olcese
    Agustin Olcese
    Excellent explantions, very clear!
    Jaak Jay
    Jaak Jay
    Awesome content, kudos to Prof.James Frojan
    sindhushree reddy
    sindhushree reddy
    Crisp and short ppt of Frm chapters and great explanation with examples.