SWAPS - solutions (2024)

RS/GZ, contact: [email protected], [email protected]

8 April 2020

Exercise 1

Interest rate swap

Assume that company A has agreed to pay a 6-month Libor and receive a fixed interest rate of 8% per annum (with interest payable every six months) from the face value of $ 100 million. Swap is 1.25 years to expire. The interest rates for 3, 9 and 15 months are: 10%, 10.5% and 11% respectively. Assume that interest rates are continously compounded. The 6-month Libor is currently 10.2%. Calculate the value of this swap for company A.

Solution
In our case:
$Q = \$100\ mln$ - the principal of the swap (in bond notation it is face value FV)
$\text{LIBOR} = 0.102$ - 6 months LIBOR
$T = 1.25$ - the maturity of the bond as a fraction of the year (15 months is 1.25 of year)
$r_{3m} = 0.10$ - 3 months interest rate
$r_{9m} = 0.105$ - 9 months interest rate
$r_{15m} = 0.11$ - 15 months interest rate
$r_{fix} = 0.08$ - fixed interest rate
$2$ - number of payments in a year

General concept of valuation:

We use the formula (1) from materials since the company A pays floating interest and receives fixed one.

$V_I = B_{fix} - B_{fl}$

$B_{fix}$ - the value of a fixed rate bond being part of the swap,
$B_{fl}$ - the value of bond with floating interest as part of the swap,

$B_{fix} = \sum_{i=1}^n\frac k {e^{(r_i*t_i)}} + \frac Q {e ^{(r_n*t_n)}}$

Fixed interest rate cash flows:
$k = 0.08 * 0.5 * \$100 mln = \$4 mln$, we multiply by $0.5$ because the interests are paid every 6 months and the interest rate is expressed annually. $t_1=0.25$, $t_2=0.75$, $t_3=1.25$.

Therefore we have:
$B_{fix} = 4 * e^{-0.25 *0.1} + 4 * e^{-0.75 *0.105} + 104 * e^{-1.25 *0.11} = \$98.24 mln$

For floating-rate bond the valuation is expressed by formula (4):
$B_{fl} = \frac {k^*} {e^{(r_1*t_1)}} + \frac Q {e ^{(r_1*t_1)}}$
The floating payment is based on LIBOR. The nearest (first) payment is: $k^* = 0.102 * 0.5 * \$100 mln = \$5.1 mln$

Therefore we have:
$B_{fl} = 5.1 * e^{-0.25 *0.1} + 100 * e^{-0.25 *0.1} = \$102.51 mln$

And the final calcualtion:

$V_I = \$98.24 mln - \$102.51 mln = -\$4.27 mln$

Therefore the swap value for company A is $-\$4.27 mln$. If the company A pays fixed interest and receives floating interests then the walue would be $+\$4.27 mln$.

Exercise 2

Interest rate swap
Determine the value of the swap from Exercise 1 in the way as described in the Relationship between interest rate swap and FRA part.

Solution

General concept of valuation:
Step a)

The cash flows that will be exchanged after three months can be calculated at the beginning. The interest rate of 8% will be exchanged for a rate of 10.2%. Let us call it $FRA_{1}$:

$FRA_{1} = 0.5 * 100 * (0.08 – 0.102) * e^{-0.25*0.1} = -1.07$

Step b)
Determination of flows after 9th and 15th month requires two more formulas.

First we have to calculate forward interest rate for the half-year period for 3 months from now. This is required to calculate the cash flow after 9 months. We will use the following formula: $e^{r_{9m}*0.75} = e^{r_{3m}*0.25} * e^{F_{0.25,\ 0.5}^{'}*0.5}$. Solving the formula for $F_{0.25,\ 0.5}^{'}$ we have:
$F_{0.25,\ 0.5}^{'} = \frac{r_{9m}*0.75 - r_{3m}*0.25}{0.5} = 0.10750$

The rate is continously compounded therefore we should convert it into semiamnual compounding with the formula $R_m = m(e^{\frac{R_c}{m}}-1)$, where $r_c$ is continously compounded and $r_m$ is discretely compounded Note that $m$ in this formula is the number of payments within a year (not the number of months).
$F_{0.25,\ 0.5} = 2*(e^{\frac{F_{0.25,\ 0.5}^{'}}{2}} - 1) = 2*(e^{\frac{0.1075}{2}} - 1)=0.11044$

The present value of the cash flow exchanged in 9 months is:

$FRA_{2} = 0.5 * 100 * (0.08 – 0.11044) * e^{-0.75*0.105} = -1.41$

Note that the above formula is the formula for FRA contract valuation:
$FRA = (T_2 - T_1)*Q * (r_{fix} – r_{fl}) * e^{-R*T_2}$ where:
$T_1$ - the begining of the period of FRA contract
$T_2$ - the end of the period of FRA contract
$Q$ - the principal of the FRA contract
$r_{fix}$ - the fixed interest rate
$r_{fl}$ - the floating interest rate
$R$ - risk free interest rate

step c)

Exercise 3

Currency swap

Assume that yield curves in Japan and in the US are flat. The interest rate in Japan is equal to 4% per annum, and in the US to 9% per annum (with continuous compounding). The financial institution takes position in the swap contract, under which it receives 5% on an annual basis of the amount denominated in yen and pays 8% per annum of the amount denominated in dollars. These amounts are respectively 10 million USD and 1200 million yen. The contract is valid for 3 years and the current exchange rate is 110 USDJPY. What is the value of this currency swap?

Solution

In our case:
$r_{f} = 0.04$ - interest rate for JPY (foreign currency)
$r_{d} = 0.09$ - interest rate for USD (domestic currency)
$r_{f,\ fix} = 0.05$ - fixed interest rate for JPY
$r_{d,\ fix} = 0.08$ - fixed interest rate for USD
$10 \text{ mln USD}$ - the principal amount in USD
$1200 \text{ mln JPY}$ - the principal amount in JPY
$S = 110$ - USDJPY foreign exchange rate (110 yen per 1 dollar)

We should use the formula (5) for swap valuation (): $V = \frac{1}{S} * B_f - B_d$ Note that $S$ should be appropriately entered to the formula to calculate V in USD.

The cash flows in domestic (USD) and foreign (JPY) currency:
$k_d = r_{d,\ fix} * 10 \text{ mln USD} = 0.08 * 10 \text{ mln USD} = 0.8 \text{ mln USD}$
$k_f = r_{f,\ fix} * 1200 \text{ mln JPY} = 0.05 * 1200 \text{ mln JPY} = 60 \text{ mln JPY}$

Now we can calculate bond values by discounting the cash flows with respective interest rates ($r_{f}$ and $r_{d}$):
$B_d = 0.8 * e^{-0.09 * 1} + 0.8 * e^{-0.09 * 2} + 10.8 * e^{-0.09 * 3} = 9.64 \text{ mln USD}$

$B_f = 60 * e^{-0.04 * 1} + 60 * e^{-0.04 * 2} + 1260 * e^{-0.04 * 3} = 1230.55 \text{ mln JPY}$

The formula for swap value:
$V = S * B_f - B_d = \frac{1230.55 \text{ mln JPY}}{110 \text{ USDJPY}} - 9.64 \text{ mln USD} = 1.55 \text{ mln USD}$

The swap value for a financial institution is 1.55 million USD. If the institution paid in yen and received cash flows in dollars, the value of the swap would be -1.55 million USD.

Exercise 4

Currency swap
Determine the value of the swap from Exercise 3 as the sum of forward contracts.

Solution

We will calculate all the values in USD. The exchange rate is 110 USDJPY and says how many yens one should pay for 1 USD. We need the ratio in USD therefore we can write it as a rate of the form of JPYUSD and then ($S' = 0.009091 = \frac{1}{110}$) JPYUSD (0.009091 USD per 1 yen). At the beginning, we will calculate forward exchange rates for one, two and three years:
$F = S'*e^{(r_d-r_f)T}$:

$F_1 = 0.009091 * e^{0.05 * 1} = 0.0096$
$F_2 = 0.009091 * e^{0.05 * 2} = 0.0101$
$F_3 = 0.009091 * e^{0.05 * 3} = 0.0106$

Using the formulas given in the theoretical part, we can calculate the values of cash flows that will occur at the end of the first, second and third year:

$FC_1 = (60 * 0.0096 – 0.8) * e^{-0.09 * 1} = -0.21$
$FC_2 = (60 * 0.0101 – 0.8) * e^{-0.09 * 2} = -0.16$
$FC_3 = (60 * 0.0106 – 0.8) * e^{-0.09 * 3} = -0.13$

In addition, the swap notional will also be exchanged at the end of the third year. Let’s calculate the value of this flow:

$FC_n = (1200 * 0.0106 – 10) * e^{-0.09 * 3} = 2.05$

The final value of a currency swap is the sum of the above four values:

$V = -0.21 – 0.16 – 0.13 + 2.05 = 1.55 \text{ mln USD}$

Exercise 5

Currency swap
Company X intends to borrow US dollars based on a fixed interest rate, while company Y intends to borrow in Japanese yens also based on a fixed interest rate. The loan amounts are approximately the same, taking into account the current exchange rate. The following loan rates have been proposed to companies to reflect the risks associated with their current situation:

Company X gets offers: JPY 5.0%, USD 9.6%
Company Y gets offers: JPY 6.5%, USD 10.0%

Design a swap that will allow the bank as an intermediary institution to achieve a profit of 50 bp per year (assuming that the entire currency risk is taken over by the bank) and at the same time will bring the same profit to X and Y, compared to the initial situation, without a swap contract.
What will the swap contract look like if the currency risk is taken over by company X?
What will change in the contract if the risk is taken over by Y? Calculate the value using forward contracts approach.

Solution

a)
X wants to borrow dollars and Y wants to borrow yen. X has a comparative advantage in yen market and Y has a comparative advantage in dollars market. The differential in yen and in dollar are respectively:
$YEN_{diff} = 6.5\%-5.0\% = 1.5\%$ per annum
$USD_{diff} = 10.0\% - 9.6\% = 0.4\%$ per annum

Therefore the total gain from the transaction is:

$YEN_{diff} - USD_{diff} = 1.5\% - 0.4\% = 1.1\%$ per annum

The bank requires $0.5\%$ per annum, hence $0.3\%$ is left for X and Y ($0.3\% = 1.1\% - 0.5\%$).

The swap should allow the situation when:
X borrows dollars at $9.6\% - 0.3\% = 9.3\%$
Y borrows yen at $6.5\% - 0.3\% = 6.2\%$
Note that both cash flows (in an out) in the same currency are equal in the case of the company X. The same works for the company Y. Thus the currency risk is taken by the bank. The design of the contract is presented below.

The direction of arrows show flow of payments

Exercise 6

Currency swap

The currency swap has 3 years to expire. Agreement assumes the exchange of the following payments: A pays to B 14% from GBP 20 million, while B pays to A 10% from USD 30 million. The term structure of interest rates is flat in both the United Kingdom and the United States. At present, US rates are 8%, and Great Britain rates are 11% (continous compounding). Interest is paid once a year. The current exchange rate is 1.65 GBPUSD.

What is the value of the swap for the company A and for the company B?
How will the answer to the above question change if we assume a decreasing structure of interest rates, both in GBP and USD?
USD: 8%, 7%, 6% for Y1, Y2, Y3 respectively
GBP: 11%, 10%, 9% for Y1, Y2, Y3 respectively
Will the value of the swap change if we assume that the parties pay floating payments based on market rates? Use the forward contracts to calculate swap value.

Solution

Company A receives payments in USD and pays in GBP. Therefore we should use the formula for swap valuation using bonds portfolio ($B_{USD}$ - the present value of USD denominated bond, $B_{GBP}$ - present value of GBP denominated bond):
$V = B_{USD} - FX * B_{GBP}$

The value of GBP cash flow and the bond underlying the swap is:

$CF_{GBP} = 20\text{ mln} * 14\% = 2.8 \text{ mln GBP}$

$B_{GBP} = \frac{2.8}{e^{0.11*1}}+\frac{2.8}{e^{0.11*2}} +\frac{20 + 2.8}{e^{0.11*3}} = 21.15\text{ mln GBP}$

The value of USD cash flow bond underlying the swap is:

$CF_{USD} = 30\text{ mln} * 10\% = 3 \text{ mln USD}$

$B_{USD} = \frac{3}{e^{0.08*1}}+\frac{3}{e^{0.08*2}} +\frac{30 + 3}{e^{0.08*3}} = 31.28\text{ mln USD}$

The value of the contract to the A (paying GBP) is

$V_A = B_{USD} - FX * B_{GBP} = 31.28 - 1.65 * 21.15 = -3.62\text{ mln USD}$
And for company B:
$V_B = 3.62\text{ mln USD}$

In this case we should use appropriate interest rate term structure to calculate bond values.

The value of GBP bond underlying the swap is:

$CF_{GBP} = 20\text{ mln} * 14\% = 2.8 \text{ mln GBP}$

$B_{GBP} = \frac{2.8}{e^{0.11*1}}+\frac{2.8}{e^{0.10*2}} +\frac{20 + 2.8}{e^{0.09*3}} = 22.21\text{ mln GBP}$

The value of USD bond underlying the swap is:

$CF_{USD} = 30\text{ mln} * 10\% = 3 \text{ mln GBP}$

$B_{USD} = \frac{3}{e^{0.08*1}}+\frac{3}{e^{0.07*2}} +\frac{30 + 3}{e^{0.06*3}} = 32.94\text{ mln USD}$

The value of the contract to the A (paying GBP) is

$V_A = B_{USD} - FX * B_{GBP} = 32.94 - 1.65 * 22.21 = -3.71 \text{ mln USD}$
And for company B:
$V_B = 3.71 \text{ mln USD}$

At the beginning we have to calculate market rates i.e.forward rates for GBP and USD. Because rates are continously compounded we will use:

$e^{r_{t+n}*(t+n)} = e^{r_{t}*t} * e^{F_{t,\ n}*n}$.

Solving the formula for $F_{t,\ n}$ we have:

$F_{t,\ n} = \frac{r_{t+n}*(t+n) - r_{t}*t}{n}$

In our case:
$r_{1, GBP} = 11\%$
$r_{2, GBP} = 10\%$
$r_{3, GBP} = 9\%$

$r_{1, USD} = 8\%$
$r_{2, USD} = 7\%$
$r_{3, USD} = 6\%$

Therefore:
$F_{0, 1, GBP}^{'} = 0.11$
$F_{1, 1, GBP}^{'} = \frac{0.1*2 - 0.11*1}{1} =0.09$
$F_{2, 1, GBP}^{'} = \frac{0.9*3-0.1*2}{1} =0.07$

$F_{0, 1, USD}^{'} = 0.08$
$F_{1, 1, USD}^{'} = \frac{0.7*2 - 0.8*1}{1} =0.06$
$F_{2, 1, USD}^{'} = \frac{0.6*3-0.7*2}{1} =0.04$

The rates are continously compounded therefore we have to convert them into annual compounding (for details see exercise 2):
$F_{0, 1, GBP} =e^{F_{0, 1, GBP}^{'}}-1 = 0.1163$
$F_{1, 1, GBP} =e^{F_{1, 1, GBP}^{'}}-1 = 0.0942$
$F_{2, 1, GBP} =e^{F_{2, 1, GBP}^{'}}-1 = 0.0725$

$F_{0, 1, USD} =e^{F_{0, 1, USD}^{'}}-1 = 0.0833$
$F_{1, 1, USD} =e^{F_{1, 1, USD}^{'}}-1 = 0.0618$
$F_{2, 1, USD} =e^{F_{2, 1, USD}^{'}}-1 = 0.0408$

Now have to calculate also forward exchange rate ($F = Se^{(r_d-r_f)T}$):

$F_1 = FX*e^{(r_{1, USD} - r_{1, GBP})T} =1.65*e^{-0.03*1} = 1.6012$
$F_2 = FX*e^{(r_{1, USD} - r_{1, GBP})T} =1.65*e^{-0.03*2} = 1.5539$
$F_3 = FX*e^{(r_{1, USD} - r_{1, GBP})T} =1.65*e^{-0.03*3} = 1.5050$

Now we can calculate the cash flows:

$FC_1 = (F_{0, 1, USD}*30 - F_{0, 1, GBP}*20*F_1)e^{-r_{1, USD}*1}$
$FC_1 = (0.0833*30 - 0.1163*20*1.6012)*e^{0.08*1} = -1.1310$

$FC_2 = (F_{1, 1, USD}*30 - F_{1, 1, GBP}*20*F_2)e^{-r_{2, USD}*2}$
$FC_2 = (0.0618*30 - 0.0942*20*1.5539)*e^{0.07*2} = -0.9317$

$FC_3 = (F_{2, 1, USD}*30 - F_{2, 1, GBP}*20*F_2)e^{-r_{3, USD}*3}$
$FC_3 = (0.0408*30 - 0.0725*20*1.5080)*e^{0.06*3} = -0.8040$

And the principal cash flow:
$FC_n = (30 - 20*F_2)e^{-r_{3, USD}*3}$
$FC_n = (30 - 20*1.5080)*e^{0.06*3} = -0.1334$

The value of the swap is:

$V = FC_1 + FC_2 + FC_3 + FC_n = -1.1310 -0.9317 -0.8040 -0.1334 = -3.00\ mln USD$

Note that the solution of this point would be a one line if we use the bond valuation based approach (since both bonds are valued at beginning of its life so the values are equal to face value ):
$(30 – 20*1.65) = -3$
If time at which we value a swap contract is later than moment of bond issue then one should use the full formula for bond valuations.