Pricing and Valuation of Interest Rate Swaps

Swaps are typically derivative contracts in which two parties exchange (swap) cash flows or other financial instruments over multiple periods for a give-and-take benefit, usually to manage risk.

Both swap contract parties have future obligations. Thus, similar to forwards and futures, swaps are forward commitments as both parties are committed in the future. The net initial value of a swap to each party should be zero, and as one side of the swap contract gains, the other side loses by the same amount.

Interest Rate Swaps

An interest rate swap allows the parties involved to exchange their interest rate obligations (usually a fixed rate for a floating rate) to manage interest rate risk or to lower their borrowing costs, among other reasons.

Interest rate swaps have two legs, a floating leg (FLT) and a fixed leg (FIX). The floating rate cash flows are expressed in the following equation:

$$S_{i}=\bigg(\frac{NAD_{FLT,i}}{NTD_{FLT,i}}\bigg)r_{FLT,i}$$

On the other hand, the fixed-rate cash flows are given by: 

$$FS=\bigg(\frac{NAD_{FIX,i}}{NTD_{FIX,i}}\bigg)r_{FIX}$$

Where: 

\(r_{FLT}=\) Observed floating rate appropriate for the time i

\(r_{FIX}=\) Fixed swap rate

\(NAD_{i}=\) Number of accrued days during the payment period 

\(NTD_{i}=\) Total number of days during the year applicable to cash flow \(i\)

Suppose that the accrual periods are constant. Then the receive-fixed, pay-floating net cash flow can be determined as:

$$FS-S_{i}=AP(r_{FIX}-r_{FLT,i})$$

On the other hand, the receive-floating, pay-fixed net cash flow can be expressed as:

$$S_{i}-FS=AP(r_{FLT,i}-r_{FIX})$$

Example: Interest Rate Swaps Cashflows

Suppose that the fixed rate is 5%, and the floating rate is 4.25%. Given that the accrual period is 60 days based on a 360-day year, the payment of a receive-fixed, pay-floating swap is closest to:

$$\begin{align*}FS-S_{i}&=AP(r_{FIX}-r_{FLT,i})\\&=\bigg(\frac{60}{360}\bigg)(5\%-4.25\%)\\&=\text{0.00125 per notional of 1}\end{align*}$$

Pricing of Interest Rate Swaps

The value of a swap to the receiver of a fixed rate and payer of a floating rate is given by:

$$\begin{align*}\text{V}&=\text{Value of fixed bond}-\text{Value of floating bond}\\&=\text{FB}-\text{VB}\end{align*}$$

Where: 

Value of fixed bond (FB) \(=FB=C\sum_{i=1}^{n}PV_{0,t_{i}}(1)+PV_{0,t_{n}}(1)\)

\(C=\) Coupon payment for the fixed-rate bond

\(PV_{0,t_{i}}\)= Appropriate present value factor for the i^th fixed cash flow.

The value of a floating rate bond is par, assumed to be I. The assumption is that we are on a reset date, and the interest payment matches the discount rate.

At the contract inception, the fixed rate is determined such that the present value of the floating rate payments equates to the present value of the fixed-rate payments. The fixed-rate is known as the swap rate. Determining the fixed (swap) rate is similar to pricing the swap.

Thus, 

$$r_{FIX}=\frac{1-PV_{0,t_{n}}(1)}{\sum_{i=1}^{n}PV_{0,t_{i}}(1)}$$

In other words, the fixed swap rate is simply one minus the final present value term divided by the sum of present values.

Example: Calculating the Price of an Interest Rate Swap

Consider a one-year LIBOR based interest rate swap with quarterly resets. The annualized Libor spot rates are given below:

$$\begin{array}{|c|c|}\hline\textbf{Year} & \textbf{Spot rates} \\ \hline\text{90-day Libor} & 1.90\% \\ \hline\text{180-day Libor} & 2.30\% \\ \hline\text{270-day Libor} & 2.60\% \\ \hline\text{360-day Libor} & 3.00\%\\ \hline\end{array}$$

The swap rate is closest to:

Solution

Recall that the swap rate is equivalent to the fixed rate.

Formula:

$$r_{FIX}=\frac{1-PV_{0, t_{n}}(1)}{\sum_{i=1}^{n}PV_{0,t_{i}}(1)}$$

We first need to calculate the discount factor:

$$D_{90}=\frac{1}{1+(0.019\times\frac{90}{360})}=0.9953$$

$$D_{180}=\frac{1}{1+(0.023\times\frac{180}{360})}=0.9886$$

$$D_{270}=\frac{1}{1+(0.026\times\frac{270}{360})}=0.9809$$

$$D_{360}=\frac{1}{1+(0.03\times\frac{360}{360})}=0.9709$$

The quarterly swap rate is then calculated as:

$$r_{FIX}=\frac{1-0.9709}{(0.9953+0.9886+0.9809+0.9709)}=0.0074=0.74\%$$

We the calculate the annualize fixed rate as follows:

$$\text{Annualize fixed rate}=0.74\%\times\frac{360}{90}=2.96\%$$

Notice that the swap rate (fixed rate) is very close to the last spot rate. You can use this tip to check whether your resulting swap rate is close to the last spot rate. Additionally, the swap rate should lie within the spot rates range as it is seen as the average of spot rates.

Valuation of an Interest Rate Swap

The value of a fixed-rate swap at some future point in time t is determined as the sum of the present value of the difference in fixed swap rates times the notional amount.

The swap value to the receive fixed party is:

$$V=NA(FS_{0}-FS_{t})\sum_{i=1}^{n’}PV_{t,t_{i}}$$

Note that the above equation provides the value to the party receiving fixed.

Example: Calculating the Value of an Interest Rate Swap

A bank entered a $500,000, five-year receive-fixed LIBOR-based interest rate swap, which is reset annually one year ago. Suppose that the fixed rate in the swap contract entered one year ago was 1.5%. The estimated discount factors are given in the following table;

$$\begin{array}{|c|c|}\hline\textbf{Year} & \textbf{Discount factor}\\ \hline1 & 0.9723 \\ \hline2 & 0.9667\\ \hline3 & 0.9625 \\ \hline4 & 0.9569\\ \hline\end{array}$$

a) The fixed rate of the swap is closed to:

$$r_{FIX}=\frac{1-PV_{0, t_{n}}(1)}{\sum_{i=1}^{n}PV_{0,t_{i}}(1)}$$

$$r_{FIX}=\frac{1-0.9569}{0.9723+0.9667+0.9625+0.9569}=1.117\%$$

b) The value for the party receiving the floating rate will be closest to:

The equivalent receive-floating swap value is simply the negative of the receive-fixed swap value.

The swap value to the receive fixed party is:

$$V=NA(FS_{0}-FS_{t})\sum_{i=1}^{n’}PV_{t,t_{i}}$$

$$V=$500,000(1.5\%-1.117\%)\times(0.9723+0.9667+0.9625+0.9569)=$7,389$$

Therefore, the swap value to the receive floating party is -$7,389.

Since the fixed rate exceeds the floating rate, the party that receives fixed (and pays floating) would receive this amount from the party that pays fixed (and receives floating).

Currency Swaps

A currency swap is an agreement between two counterparties to exchange future interest payments in different currencies. The payments can be based on either a fixed interest rate or a floating interest rate. By swapping future interest obligations, the two parties can manage currency risk.

Currency swaps may also involve exchanging notional amounts at both the starting of the contract and the contract expiration. The counterparties can exchange payments denominated in one currency to equivalent payments denominated in another currency.

Pricing Currency Swaps

Pricing a currency swap involves solving the appropriate notional amount in one currency, given the notional amount in the other currency, and determining the two fixed interest rates such that the currency swap value is zero at the initiation.

Similar to interest rate swaps, currency swaps are priced by determining the fixed swap rate. The equilibrium fixed swap rate equation for a currency X is given as:

$$r_{FIX,X}=\frac{1-PV_{0,t_{n},X}(1)}{\sum_{i-1}^{n}PV_{0,t_{i},X}(1)}$$

Example: Calculating the Price of a Currency Swap

A France company needs to borrow 500 million dollars ($) for one year for one of its American Subsidiaries. The company decides to issue Euro-denominated bonds in an amount equivalent to $500 million. The company enters into a one-year currency swap that reset quarterly and agrees to exchange notional amounts at the contract inception and maturity. The following spot rates and present values are observed at time 0.

Given that the spot exchange rate of EUR/USD is 0.8163;

$$\small{\begin{array}{|c|c|c|}\hline\textbf{Days to Maturity} & \textbf{EUR Spot Interest Rates} & \textbf{US\$ Spot Interest Rates (%)} \\ \hline90 & 2.13\% & 0.09\% \\ \hline180 & 2.21\% & 0.13\% \\ \hline270 & 2.30\% & 0.17\% \\ \hline360 & 2.38\% & 0.21\% \\ \hline\end{array}}$$
The annual fixed swap rates for EUR and USD are closest to:

The present values for each reset date are calculated as follows:

EUR present values:

$$D_{90}=\frac{1}{1+\bigg(0.0213\times\frac{90}{360}\bigg)}=0.9947$$

$$D_{180}=\frac{1}{1+\bigg(0.0221\times\frac{180}{360}\bigg)}=0.9891$$

$$D_{270}=\frac{1}{1+\bigg(0.0230\times\frac{270}{360}\bigg)}=0.9831$$

$$D_{360}=\frac{1}{1+\bigg(0.0238\times\frac{360}{360}\bigg)}=0.9768$$

Annual Fixed rate for EUR:

$$r_{FIX,EUR}=\frac{1-PV_{0,t_{n},EUR}(1)}{\sum_{i=1}^{n}PV_{0,t_{i}, EUR}(1)}$$

$$r_{FIX,EUR}=\frac{1-0.9768}{0.9947+0.9891+0.9831+0.9768}=0.5895\%$$

$$\text{Annual Rate} =0.5895\%\times\frac{360}{90}=2.358\%$$

USD present values: 

$$D_{90}=\frac{1}{1+\bigg(0.0009\times\frac{90}{360}\bigg)}=0.9998$$

$$D_{180}=\frac{1}{1+\bigg(0.0013\times\frac{180}{360}\bigg)}=0.9994$$

$$D_{270}=\frac{1}{1+\bigg(0.0017\times\frac{270}{360}\bigg)}=0.9987$$

$$D_{360}=\frac{1}{1+\bigg(0.0021\times\frac{360}{360}\bigg)}=0.9979$$

Annual Fixed rate for USD: 

$$r_{FIX, USD}=\frac{1-PV_{0,t_{n},USD}(1)}{\sum_{i=1}^{n}PV_{0,t_{i}, USD}(1)}$$

$$r_{FIX, USD}=\frac{1-0.9979}{0.9998+0.9994+0.9987+0.9979}=0.0531\%$$

$$\text{Annual rate}=0.0531\%\times\frac{360}{90}=0.212\%$$ 

The notional amount in EUR is calculated as:

The EUR notional amount is calculated as USD 500 million multiplied by the current spot exchange rate at which US$1-dollar trades for EUR 0.8163.

$$\text{EUR Notional}=\text{500 million}\times0.8163=\text{EUR 408.15 Million}$$

The fixed swap payments in currency units equal the periodic swap rate times the appropriate notional amounts:

$$\begin{align*}FS_{EUR}&=NA_{EUR}(AP)r_{FIX}\\&=EUR408.15M\times\bigg(\frac{90}{360}\bigg)(2.358\%)\\&=\text{EUR 2.406 Million}\end{align*}$$

$$\begin{align*}FS_{US$}&=NA_{US$}(AP)r_{FIX},US$\\&=$500m\bigg(\frac{90}{360}\bigg)(0.212\%)\\&=\text{\$0.265 Million}\end{align*}$$

In summary, currency swap pricing has three key variables: two fixed interest rates and one notional amount.

Valuing Currency Swaps

The value of a currency swap is 0 at the contract inception.

The value of a fixed-to-fixed currency swap at some future point in time t is determined as the difference in a pair of fixed-rate bonds, one expressed in currency a and one expressed in currency b.

$$V_{a}=NA_{a,0}\Bigg(r_{FIX,a,0}\sum_{i=1}^{n’}PV_{t},t_{i},a+PV_{t,t_{n}{‘a}}\Bigg)$$

$$-S_{t}NA_{b,0}\Bigg(r_{FIX, b, 0}\sum_{i=1}^{n’}PV_{t},t_{i},b+PV_{t, t_{n}{‘b}}\Bigg)$$

Example: Calculating a Currency Swap Value

Bright Investment firm has entered into a one-year currency swap with quarterly reset (30/360-day count). The exchange of notional amounts is done at the initiation and maturity of the swap. The annualized fixed rates are 1% (0.25%/quarter) for GBP and 0.50% (0.125%/quarter) for AUD. The notional amounts were AUD 500,000 and GBP 200,000.

After one month, the GBP/AUD spot exchange rate changes to 0.60. Consider the following market information:

$$\small{\begin{array}{|c|c|c|c|c|}\hline\textbf{Days to Maturity} & \textbf{£ Spot Interest Rates} & \textbf{A\$ Spot Interest Rates} & \textbf{PV (£1)} & \textbf{PV (A\$1)} \\ \hline60 & 6.000\% & 2.000\% & 0.9901 & 0.9967 \\ \hline150 & 7.000\% & 3.000\% & 0.9717 & 0.9877 \\ \hline240 & 8.000\% & 4.000\% & 0.9494 & 0.9740 \\ \hline 330 & 9.000\% & 5.000\% & 0.9238 & 0.9562 \\ \hline\textbf{Sum} & & & \textbf{3.8349} & \textbf{3.9145}\\ \hline\end{array}}$$

The value of the swap \((V_{a})\) entered into 60 days ago is closest to:

$$V_{a}=NA_{a,0}\Bigg(r_{FIX,a,0}\sum_{i=1}^{n’}PV_{t},t_{i},a+PV_{t,t_{n}{‘a}}\Bigg)$$

$$-S_{t}NA_{b,0}\Bigg(r_{FIX, b, 0}\sum_{i=1}^{n’}PV_{t},t_{i},b+PV_{t, t_{n}{‘b}}\Bigg)$$

$$\begin{align*}V_{a}&=200,000[(0.0025\times3.8349)+0.9238]-0.60\times500,000(0.00125\times3.9145+0.9562)\\&=186,677.45-288,327.94\\&=-£101,650.49\end{align*}$$

Equity Swaps

An equity swap is an OTC derivative contract in which two parties agree to exchange a series of cash flows. One party pays a variable series determined by equity, and the other party pays either 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 similarly to a comparable interest rate swap, although the cashflows involved are very different.

The fixed swap rate is:

$$r_{FIX}=\frac{1-PV_{0,t_{n}}(1)}{\sum_{i=1}^{n}PV_{0,t_{i}}(1)}$$

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 . The information used to price the interest rate swap is given in the following table:

$$\small{\begin{array}{|c|c|}\hline\textbf{Year} & \textbf{Discount factor} \\ \hline1 & 0.9723 \\ \hline2 & 0.9667 \\ \hline3 & 0.9625 \\ \hline4 & 0.9569\\ \hline\end{array}}$$
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 identical 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.

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

$$V_{t}=FB_{t}(C_{0})-\bigg(\frac{S_{t}}{S_{t-}}\bigg)NA_{E}-PV(Par-NA_{E})$$

Where: 

\(FB_{t}(C_{0})=\) time t value of a fixed-rate bond initiated with coupon \(C_0\) at time 0

\(s_{t}=\)the current equity price,

\(S_{t}-=\)the equity price observed at the last reset date 

Example: Calculating the Value of an Equity Swap

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:

$$\small{\begin{array}{|c|c|}\hline\textbf{Year} & \textbf{Spot rates} \\ \hline\text{60 -day Libor} & 3.90\% \\ \hline\text{150-day Libor} & 4.55\% \\ \hline \text{240-day Libor} & 5.20\% \\ \hline\text{330-day Libor} & 5.85\%\\ \hline\end{array}}$$

The value of the equity swap to the fixed-rate payer is closest to:

The first step is to calculate the discount factors:

$$D_{60}=\frac{1}{1+\bigg(0.0390\times\frac{60}{360}\bigg)}=0.9935$$

$$D_{150}=\frac{1}{1+\bigg(0.0455\times\frac{150}{360}\bigg)}=0.9814$$

$$D_{240}=\frac{1}{1+\bigg(0.0520\times\frac{240}{360}\bigg)}=0.9665$$

$$D_{330}=\frac{1}{1+\bigg(0.0585\times\frac{330}{360}\bigg)}=0.9491$$

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

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

The value of the index investment:

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

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

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

Question 1

Arth Shah, CFA, is pricing a four-year LIBOR based interest rate swap with annual resets assuming the 30/360-day count. Shah has determined the following discount factors.

$$\small{\begin{array}{|c|c|}\hline\textbf{Maturity (years)} & \textbf{Present value factors} \\ \hline1 & 0.9604 \\ \hline2 & 0.9485 \\ \hline3 & 0.9364 \\ \hline4 & 0.9103\\ \hline\end{array}}$$

The swap rate is closest to:

1. 0.0239
2. 0.2423
3. 0.0315

Solution

The correct answer is A:

The swap rate is equivalent to the fixed-rate calculated as follows:

$$r_{FIX}=\frac{1-PV_{0,t_{n}}(1)}{\sum_{i=1}^{n}PV_{0,t_{i}}(1)}$$

$$r_{FIX}=\frac{1-0.9103}{0.9604+0.9485+0.9364+0.9103}=0.02388\approx0.0239$$

Question 2

ABC Investment Co. has entered into a one-year currency swap with quarterly reset (30/360-day count). The exchange of notional amounts is done at initiation and at maturity of the swap. The annualized fixed rates were 0.78% (0.1938%/quarter) for AUD and 0.25% (0.0625%/quarter) for US dollars. The notional amounts were US$300,000 and AUD 200,000.

After one month, the AUD/US$ spot exchange rate changes to 0.76. Consider the following market information:

$$\small{\begin{array}{|c|c|c|c|c|}\hline\textbf{Days to Maturity} & \textbf{€ Spot Interest Rates} & \textbf{US\$ Spot Interest Rates} & \textbf{PV (€1)} & \textbf{PV (US\$1)}\\ \hline30 & 0.788\% & 0.315\% & 0.9993 & 0.9997 \\ \hline120 & 0.893\% & 0.368\% & 0.9970 & 0.9988 \\ \hline210 & 0.998\% & 0.420\% & 0.9942 & 0.9976 \\ \hline300 & 1.103\% & 0.473\% & 0.9909 & 0.9961 \\ \hline\textbf{Sum} & & & \textbf{3.9815} & \textbf{3.9921}\\ \hline\end{array}}$$

The value of the swap \((V_{a})\) entered into 30 days is closed to:

1. -A$27,956.44
2. -A$99,849.04
3. A$76,000.00

Solution

The correct answer is A: 

$$V_{a}=NA_{a,0}\Bigg(r_{FIX,a,0}\sum_{i=1}^{n’}PV_{t},t_{i},a+PV_{t,t_{n}{‘a}}\Bigg)$$

$$-S_{t}NA_{b,0}\Bigg(r_{FIX, b, 0}\sum_{i=1}^{n’}PV_{t},t_{i},b+PV_{t, t_{n}{‘b}}\Bigg)$$

$$\begin{align*}V_{a}&=€200,000[0.001938(3.9815) +0.9909]-0.76($300,000)[0.000625(3.9921)+0.9961]\\&=199,723.23-227,679.67\\&=-27,956.44\end{align*}$$

Reading 37: Pricing and Valuation of Forward Commitments

LOS 37 (c) describe and compare how the interest rate, currency, and equity swaps are priced and valued;

LOS 37(d) Calculate and interpret the no-arbitrage value of interest rate, currency, and equity swaps.