###### The Scope of Private Company Valuation

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A * currency swap *is an agreement between two counterparties to exchange future interest payments in different currencies. The payments can be based either on 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 both at the beginning of the contract and the contract expiration. The counterparties can exchange payments denominated in one currency to equivalent payments denominated in another currency.

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. The currency swap value is zero at the time of 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:

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

The equilibrium fixed swap rate equation for currency Y is given as:

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

A French 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 a one-year currency swap agreement that resets 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.

$$ \begin{array}{c|c|c} \textbf{Days to} & \textbf{EUR Spot} & \bf{\text{US}$ \text{ Spot} (\%)} \\ \textbf{Maturity} & \textbf{Interest Rates} & \textbf{Interest Rates} \\ \hline 90 & 2.13\% & 0.09\% \\ \hline 180 & 2.21\% & 0.13\% \\ \hline 270 & 2.30\% & 0.17\% \\ \hline 360 & 2.38\% & 0.21\% \end{array} $$

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

$$ \begin{align*} D_{90} &=\frac{1}{1+\left(0.0213\times\frac{90}{360}\right)}=0.9947 \\ D_{180} & =\frac{1}{1+\left(0.0221\times\frac{180}{360}\right)}=0.9891 \\ D_{270} &=\frac{1}{1+\left(0.0230\times\frac{270}{360}\right)}=0.9831 \\ D_{360} &=\frac{1}{1+\left(0.0238\times\frac{360}{360}\right)}=0.9768 \end{align*} $$

Annual fixed rate for EUR:

$$ \begin{align*} r_{FIX,EUR} &=\frac{1-{PV}_{0,t_n,EUR}\left(1\right)}{\sum_{i=1}^{n}{{PV}_{0,t_i,EUR}\left(1\right)}} \\ r_{FIX,EUR} &=\frac{1-0.9768}{0.9947+0.9891+0.9831+0.9768}=0.5895\% \\ \text{Annual rate} & = 0.5895\% \times\frac{360}{90}=2.358\% \end{align*} $$

USD present values:

$$ \begin{align*} D_{90} &=\frac{1}{1+\left(0.0009\times\frac{90}{360}\right)}=0.9998 \\ D_{180} &=\frac{1}{1+\left(0.0013\times\frac{180}{360}\right)}=0.9994 \\ D_{270} &=\frac{1}{1+\left(0.0017\times\frac{270}{360}\right)}=0.9987 \\ D_{360} &=\frac{1}{1+\left(0.0021\times\frac{360}{360}\right)}=0.9979 \end{align*} $$

Annual fixed rate for USD:

$$ \begin{align*} r_{FIX,USD} & =\frac{1-{PV}_{0,t_n,USD}\left(1\right)}{\sum_{i=1}^{n}{{PV}_{0,t_i,USD}\left(1\right)}} \\ r_{FIX,USD}&=\frac{1-0.9979}{\ 0.9998+0.9994+0.9987+0.9979}=0.0531\% \\ \text{Annual rate} & =0.0531\%\times\frac{360}{90}=0.212\% \end{align*} $$

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

$$ \text{EUR Notional} = 500\text{ million} \times 0.8163=\text{EUR } 408.15 \text{ million} $$

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

$$ \begin{align*} FS_{EUR} &=NA_{EUR}\left(AP\right)r_{FIX} =EUR 408.15M\times\left(\frac{90}{360}\right)\left( 2.358\%\right) \\& =EUR 2.406 \text{ million} \\ FS_{US$}& = NA_{US$}\left(AP\right)r_{FIX,US$} =$500m\left(\frac{90}{360}\right)\left(0.212\%\right) \\ & = $0.265 \text{ million} \end{align*} $$

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

The value of a currency swap is 0 at the time of 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 \(b\).

$$ \begin{align*} V_a &=NA_{a,0}\left(r_{FIX,a,0}\sum_{i=1}^{n^{`}}{PV_t,t_i,a}+PV_{t,t_n^{`}a}\right) \\ & -S_tNA_{b,0}\left(r_{FIX,b,0}\sum_{i=1}^{n^{`}}{PV_t,t_i,b}+PV_t,t_{n^{`}},b\right) \end{align*} $$

## Question

Bright Investment firm has entered a one-year currency swap agreement 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:

$$ \begin{array}{c|c|c|c|c} \textbf{Days to} & \bf{£ \text{ Spot Interest}} & \bf{\text A$ \text{Spot Interest}} & \textbf{PV} & \textbf{PV} \\ \textbf{Maturity} & \textbf{Rates} & \textbf{Rates} & \bf{(£ 1)} & \bf{(\text A$1)} \\ \hline 60 & 6.000\% & 2.000\% & 0.9901 & 0.9967 \\ \hline 150 & 7.000\% & 3.000\% & 0.9717 & 0.9877 \\ \hline 240 & 8.000\% & 4.000\% & 0.9494 & 0.9740 \\ \hline 330 & 9.000\% & 5.000\% & 0.9238 & 0.9562 \\ \hline \textbf{Sum} & & & 3.8349 & 3.9145 \end{array} $$

The value of the swap entered 60 days ago is closest to:

- £186,677.45.
- £288,327.94.
- −£101,650.49.
## Solution

The correct answer is C.$$ \begin{align*} V_a &=NA_{a,0}\left(r_{FIX,a,0}\sum_{i=1}^{n^{`}}{PV_t,t_i,a}+PV_{t,t_n^{`}a}\right) \\ & -S_tNA_{b,0}\left(r_{FIX,b,0}\sum_{i=1}^{n^{`}}{PV_t,t_i,b}+PV_t,t_{n^{`}},b\right) \\ V_a & =200,000 \left[(0.0025\times3.8349)+0.9238 \right] – 0.60 \\ & ×500,000 (0.00125× 3.9145)+ 0.9562] \\ & =186,677.45-288,327.94=- £ 101,650.49 \end{align*} $$

*Reading 33: Pricing and Valuation of Forward Commitments*

*LOS 33 (f) Describe how currency swaps are priced, and calculate and interpret their no-arbitrage value.*