Forward contracts on fixed-income securities are priced and valued in a virtually identical manner to their equity counterparts.

• To price a fixed-income forward contract, take the bond price, subtract the present value of the coupons over the life of the contract, and compound this amount at the risk-free rate to the expiration date of the contract.
  Forward price = (Bond price - Present value of coupons over life of contract) (1 + r)^T
  = (Bond price) (1+r)^T - Future value of coupons over life of contract
  Alternatively, the price of forward contract on bond with coupons CI is:
  F(0, T) = [B₀^c(T+Y) - PV(CI, 0, T)] (1 + r)^T
  = [B₀^c(T+Y)](1 + r)^T - FV(CI, 0, T)
  Where:
  • B₀^c(T+Y) is the bond price at time 0.
  • T is the expiration date of the forward contract.
  • Y is the remaining maturity of the bond on the forward contract expiration.
  • T + Y is the time to maturity of the bond at the time the forward contract is initiated.
  • PV(CI, 0, T) is the present value of the coupon interest over the life of the forward contract.
  • FV(CI, 0, T) is the future value of the coupon interest over the life of the forward contract.

• The value of a fixed income forward contract is the bond price minus the present value of the coupons minus the present value of the forward price that will be paid at expiration.
  Vt(0, T) = B_t^c(T + Y) - PV(CI, t, T) - F(0, T)/(1 + r)^{(T - t)}
  Where:
  • B_t^c(T + Y) is the bond price at time t.

Example 1

Consider a bond with semi-annual coupons.

• The par value is $1,000.
• The coupon rate is 6%. This implies that each coupon is $30.
• It has a current maturity of 593 days.
• It pays four coupons: the next occurs in 47 days, followed by coupons in 229 days, 411 days, and 593 days.
• The bond price is $994.45. This includes accrued interest.
• The risk-free interest rate is 4.75%.
• The forward contract expires in 305 days.

Analysis:

• T = 305.
• Y = 288, which means the bond has 288 days remaining after the forward contract expires.
• Only the first two coupons occur during the life of the forward contract. The present value of the coupons is: $30/(1.0475)^47/365 + $30/(1.0475)^229/365 = $58.96.
• The forward price if the contract is initiated now is F(0, T) = ($994.45 - $58.96)(1.0475)^305/365 = $972.48. This means that we shall be able to enter into this contract to buy the bond in 310 days at the price of $972.48.
• Suppose it is 30 days later and the new bond price is $985.14. The risk-free rate is now 5.75%.
  • The present value of the remaining coupons is $30/(1.0575)^17/365 + $30/(1.0575)^199/365 = $59.02.
  • The value of the forward contract is thus $985.14 - $59.02 - $972.48/(1.0575)^275/365 = -$6.25.
  • The contract has gone from a zero value at the start to a negative value, primarily because of the decrease in the price of the underlying bond.

Now let's price and value a FRA.

In the FRA market, contracts are created with specific day counts. Consider the time line shown below.

• On day 0 we shall initiate an FRA.
• The FRA expires on day h.
• The rate underlying the FRA is the rate on an m-day Eurodollar deposit.
• The date indicated by g will simply be a date during the life of the FRA at which we want to determine a value for the FRA.

A LIBOR deposit is a loan from one bank to another. Let L_i(j) represent the rate on a j-day LIBOR deposit on an arbitrary day i, which falls somewhere in the above period from 0 to h. Consider a bank that borrows $1 on day I for j days. It will pay back $1[1 + L_i(j) (j/360)] in j days.

Let's denote the fixed rate on the FRA as FRA(0, h, m), which stands for the rate on an FRA established on day 0, expiring on day h, and based on m-day LIBOR. If we use a $1 notional principal for the FRA, at expiration its payoff is

The numerator is the difference between the underlying LIBOR on the expiration day and the rate agreed on when the contract was initiated, multiplied by the adjustment factor m/360. The adjustment is necessary as both of these rates are annual rates applied to a Eurodollar deposit of m days. These rates apply to Eurodollar deposits created on day h and paying off m days later.

The following is the formula for pricing an FRA:

This is actually the formula for a LIBOR forward rate, given the interest payment conventions in the FRA market.

• The numerator is the future value of a short-term Eurodollar deposit of h + m days.
• The denominator is the future value of a shorter-term Eurodollar deposit of h days.
• The ratio is 1 plus a rate: subtracting 1 and multiplying by 360/m annualizes the rate.

The value of FRA on day g is:

The idea behind this formula is quite simple:

• The first term on the right-hand side is the present value of $1 received at day h.
• The second term is the present value of 1 plus the FRA rate to be received on day h + m, the maturity date of the underlying Eurodollar time deposit.

Example 2

Consider a 3 x 6 FRA, which expires in 90 days and is based on 90-day LIBOR. (3 x 6 refers to the fact that the contract expires in 3 months and 3 months later, or 6 months from the contract initiation date, the interest is paid on the underlying Eurodollar time deposit on whose rate the contract is based). As we are on day 0, h = 90, m = 90.

Suppose the current rates are:

• L₀(h) = L₀(90) = 0.054
• L₀(h + m) = L₀(180) = 0.059

Using the formula: F(0, h, m) = F(0, 90, 90) = [(1 + 0.059 x 180/360)/(1 + 0.054 x 90/360) - 1] x (360/90) = 0.0631.

Therefore, to enter into an FRA on day 0, the rate would be 6.31%.

Then assume that we go long the FRA, and it is 20 days later. As we are on day 20, g = 20, h - g = 90 - 20 = 70, and h + m - g = 90 + 90 - 20 = 160. We also need information about the new term structure:

• L_g(h - g) = L₂₀(70) = 0.058
• L_g(h + m - g) = L₂₀(160) = 0.064

Using the formula, the value of the FRA is V_g(0, h, m) = V₂₀(0, 90, 90) = 1/(1 + 0.058 x 70/360) - (1 + 0.0631 x 90/360)/(1 + 0.064 x 160/360) = 0.001167.

Therefore, the FRA now has a value of $0.001167 per $1 notional principal. If the notional principal is any amount other than $1, we multiply the notional principal by $0.001167 to obtain the full market value of the FRA.

The pricing and valuation of currency forwards is remarkably similar to that of equity forwards.

F(0, T) = [S₀/(1 + r(f))^T] (1 + r)^T

Where:

• S₀ is the current spot exchange rate (direct quote: # of domestic currency/ one unit of foreign currency)
• r(f) is the foreign interest rate.
• r is the domestic interest rate.

The term in brackets is the spot exchange rate discounted by the foreign interest rate. This term is then compounded at the domestic interest rate to the expiration day.

This formula is called interest rate parity. It states that

• The forward rate premium (or discount) of a currency should reflect the differential in interest rates between the two countries.
• The discounted interest rates differential equals the percentage forward discount.

It is equivalent to the statement that one unit of foreign currency, deliverable on a particular future date, must cost the same amount independently of whether it is obtained through the forward market or by means of the spot market. The difference in the spot and forward rates for currencies are due solely to differentials in interest rates.

For example, if one-year interest rates are 5% in the U.S. and 10% in the U.K. and the current spot exchange rate in dollars per foreign currency is 2, the one-year forward rate will be 1.91. The 5% interest rate advantage that could be obtained in the U. K. will be offset by a 5% depreciation in the value of the pound. As pounds are bought spot and sold forward, the forward discount will widen. Simultaneously, as money flows from the U. S., interest rates here will tend to rise while the inflow to funds to the U.K. will tend to depress interest rates there.

One should not, on the basis of this information, conclude that a currency selling at a premium is expected to increase or one selling at a discount is expected to decrease.

Covered interest arbitrage, a technique resulting in riskless profits, involves the short-term investment in a foreign currency which is covered by a forward contract to sell that currency when the investment matures. Covered interest arbitrage is plausible when the forward premium does NOT reflect the interest rate differential between the two countries specified by the interest rate parity formula.

The two concepts, interest rate parity and covered interest arbitrage, are discussed in Level I already.

The value of a currency forward contract is simply the spot rate discounted at the foreign interest rate over the life of the contract, minus the present value of the forward rate at expiration.

V_t(0, T) = S_t/(1 + r(f))^(T-t) - F(0, T)/(1 + r)^(T-t)

If we assume that interest is compounded continuously:

F(0, T) = S₀ e ^-r(f)T e^rT

and

V_t(0, T) = [S_t e ^{- r(f)(T-t)}] - F(0, T) e^{-r(T - t)}

Where:

• r(f) is the continuously compounded foreign interest rate, defined as r(f) = ln(1 + r(f)).
• r is the continuously compounded foreign interest rate, defined as r = ln(1 + r).

See basic questions for examples.

Practice Question 1

An investor bought a bond when it was originally issued with a maturity of 30 years. The bond pays semi-annual coupons of $60. The first coupon occurs 181 days after issue, the second 365 days, the third 547 days, and the fourth 730 days. It is now 120 days into the life of the bond and the price of the bond is $1,012.85 (includes accrued interest). The investor wants to sell the bond two days after its fourth coupon. The risk-free rate is currently 7 percent. At what price could the investor enter into a forward contract to sell the bond two days after its fourth coupon?

Correct Answer: Only the first four coupons occur during the life of the forward contract. At this time point the next occurs in 61 days, followed by coupons in 245 days, 427 days, and 610 days. The present value of the coupons is: $60/(1.07)^61/365 + $60/(1.07)^245/365 + $60/(1.07)^427/365 + $60/(1.07)^610/365 = $225.68.

The forward contract expires in 612 days (732 - 120). F(0, T) = F(0, 612/365) = ($1,012.85 - $225.68) (1.07)^612/365 = $881.73.

Practice Question 2

Kevin, a corporate treasurer, wants to hedge against an increase in future borrowing costs. He plans to enter into a long 3 x 9 FRA. The current term structure for LIBOR is 30 day - 4.89%; 90 day - 5.10%; 180 day - 5.27%; 270 day - 5.52%; 360 day - 5.65%. What is the rate Kevin would receive on a 3 x 9 FRA?

Correct Answer: Here the notation would be: h = 90, m = 180, and h + m = 270.
L₀(h) = L₀(90) = 5.10%, and L₀(h + m) = L₀(270) = 5.52%.
FRA(0, h, m) = FRA(0, 90, 180) = [(1 + 0.0552 x 270/360)/(1 + 0.051 x 90/360) - 1] x (360/180) = 5.6579%.

Practice Question 3

You have the following information: - One year Swiss interest rate is 5% (compounded quarterly) - One year Venezuelan interest rate is 25% (compounded quarterly) - The spot rate is 425.00 Bolivar per Swiss Franc What is forward, half-year exchange rate that creates interest rate parity?

A. 389.52
B. 468.01
C. 385.94

Correct Answer: B

(1 + .25/4)^(4x1/2) = 1/425 x (1 + .05/4)^(4x1/2) x F
F = 468.01

Practice Question 4

The spot rate on the pound is $1.5005/£, the 360-day forward is $1.5020/£, the 360-day interest rate in the U.S. is 2.25%, and in the U.K. it is 1.9%. Is covered interest arbitrage possible and if so, what are your profits assuming $1,000,000 or its equivalent in the other currency?

A. covered interest arbitrage is not possible.
B. covered interest arbitrage is possible with a profit of $2,481.
C. covered interest arbitrage is possible with a profit of $1,652.

Correct Answer: B

The interest rate differential and forward premium/discount are such that you want to invest in the U.S. money market. Borrow £666,444.52, then convert it in the spot market to $1,000,000 (666,444.52 x 1.5005). Invest in U.S. money market at 2.25%. Enter forward contract to deliver $1,022,500 in 360 days for £680,758.76 ($1,022,500/1.5020). In 360 days, exercise forward and repay loan of £679,106.96 (£666,444.52 x 1.019), leaving a profit of £1652 or $2,481. In practice, you would only enter a forward contract for $1,020,019, the amount necessary to repay the loan, leaving you with the $2,481.

Practice Question 5

Suppose the domestic currency is the U.S. dollar and the foreign currency is the Canadian dollar.

• The spot exchange rate is $0.7321.
• The U.S. interest rate is 3.5%.
• The Canada interest rate is 4.25%.
• Assume these interest rates are fixed and don't change over the life of the forward contract. They are based on annual compounding and are not quoted as LIBOR-type rates.
• Assume a currency forward contract has a maturity of 90 days.

What should be the forward price if you want to enter into a forward contract to long Canadian dollars in 90 days? What if interests are continuously compounded?

Correct Answer: With discrete compounding:
F(0, T) = F(0, 90/365) = [0.7321/(1.0425)^90/365] (1.035)^90/365 = 0.7308.

With continuously compounding:
r = ln(1.035) = 3.44%, and r(f) = ln(1.0425) = 4.16%.
F(0, T) = (0.7321 e ^{-0.0416 (90/365)}) e ^{0.0344 (90/365)} = 0.7308.

Note that the two rates are equal.

Practice Question 6

The current spot rate between the Japanese yen and the U.S. dollar is ¥124.56/$. If the risk-free rate in Japan is 2.6% and in the U.S. its 6.4%, should be the 6-month forward rate?

A. ¥120.11/$.
B. ¥126.85/$.
C. ¥122.31/$.

Correct Answer: C

F = [¥124.56/$ / (1 + 0.064)^180/360] * (1 + 0.026^)180/360 = ¥122.31/$.

Practice Question 7

An investor bought a bond (par value: $1,000) when it was originally issued with a maturity of 30 years. The bond pays semi-annual coupons of $60. The first coupon occurs 181 days after issue, the second 365 days, the third 547 days, and the fourth 730 days, and so. 120 days later he entered into a forward contract which would allow him to sell the bond in 612 days (from the contract initiation date) at a then no-arbitrage price of $875.44. Now, 365 days since he entered into the forward contract, the risk-free rate is 7% and the new price of the bond is $1,034.75. What's the value of the forward contract for the investor?

A. $81.84.
B. $74.26.
C. $93.79.

Correct Answer: A

Now it is the 485th day of the bond's life (120 + 365). There are two coupons to go, once occurring in 547 - 485 = 62 days, and the other in 730 - 485 = 245 days. The present value of the coupons is now: $60/1.07^62/365 + $60/1.07^245/365 = $116.65.

There are 612 - 365 = 247 days now remain until the contract's expiration. The value of the forward contract is then $1,034.75 - $116.65 - $875.44/1.07^247/365 = $81.84.

This positive value represents a loss to the investor as his position is "short".

Practice Question 8

Kevin, a corporate treasurer, wants to hedge against an increase in future borrowing costs. He plans to enter into a long 3 x 6 FRA. The current term structure for LIBOR is 30 day - 4.89%; 90 day - 5.10%; 180 day - 5.27%; 270 day - 5.52%; 360 day - 5.65%. What is the rate Kevin would receive on a 3 x 6 FRA?

A. 4.93%.
B. 5.37%.
C. 5.15%.

Correct Answer: B

Here the notation would be: h = 90, m = 90, and h + m = 180.
L₀(h) = L₀(90) = 5.10%, and L₀(h + m) = L₀(180) = 5.27%.
FRA(0, h, m) = FRA(0, 90, 90) = [(1 + 0.0527 x 180/360)/(1 + 0.051 x 90/360) - 1] x (360/90) = 5.37%.

Practice Question 9

Ten days ago a portfolio manager went short an FRA which would expire in 30 days and was based on 210-day LIBOR. The no-arbitrage rate he received was 6.16%. Now the current LIBOR term structure is: 20 day - 5.45%; 170 day - 5.75%; 200 day - 5.95%. What's the market value of the FRA for a $10 million notional principal?

A. -$8352.55
B. $9278.31
C. $7894.52

Correct Answer: A

Here the notation would be: g = 10, h - g = 20, h + m - g = 200.
L_g(h - g) = L₁₀(20) = 5.45%, and L_g(h + m - g) = L₁₀(200) = 5.95%.
V_g(0, h, m) = V₁₀(0, 30, 180) = 1/(1 + 0.0545 x 20/360) - (1 + 0.0616 x 180/360)/(1 + 0.0595 x 200/360) = -0.00084.
Thus, for a notional principal of $10 million, the value would be 10,000,000 x (-0.00084) = -$8352.55.

Practice Question 10

To hedge against a possible decrease in short-term interest rates, a financial manager of a company went short on a 1 x 2 FRA 30 days ago. The rate she received based on the LIBOR term structure of that time was 6.32%. Today is the contract expiration day. The 30-day LIBOR is 6.10% and the 60-day LIBOR is 6.52%. For a notional principal of $5 million, what is her payoff on the FRA?

A. The company should pay $828.83.
B. The company should pay $912.03.
C. The company should receive $912.03.

Correct Answer: C

Here the notation would be: g = 30, h - g = 0, h + m - g = 30.
L_g(h - g) = L₃₀(0) = 0%, L_g(h + m - g) = L₃₀(30) = 6.10%.
V_g(0, h, m) = V₃₀(0, 30, 30) = 1/(1 + 0) - (1 + 0.0632 x 30/360)/(1 + 0.0610 x 30/360) = -0.00018.
Thus, for a notional principal of $5 million, the value would be 5,000,000 x (-0.00018) = -$912.03. Thus, the $912.03 would be received by the company from the counterparty because it is short and the rate on expiration is lower than the FRA rate.
Note that the 60-day LIBOR rate is not relevant here.

Practice Question 11

Given U.S. interest rates at 2.00% and Japanese interest rates at 1.00% for the same period, where would the Interest Rate Parity Theory expect the JPY/USD exchange rate mostly likely to trend toward given a current spot exchange rate of 134.00.

A. 135.00.
B. 132.00.
C. 136.00.

Correct Answer: B

The Interest Rate Parity Theory says that given disequilibrium in interest rates, the market foreign exchange rate will compensate owners of the lower yielding currency, by increasing the value of that currency vis-a-vis the higher yielding currency.

Since, U.S. interest rates are higher than Japanese interest rates for the same period, Interest Rate Parity should cause the value of the dollar to Decline and the value of the Yen increase.

Practice Question 12

The spot lira/$ exchange rate is 833 lira/$ and the one year forward rate is 863 lira/$. If the annual interest rate on dollar CDs is 9.5%, what would you expect the annual interest rate to be on lira CDs?

A. 5.3%.
B. 13.4%.
C. 25.6%.

Correct Answer: B

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Practice Question 13

Which is the following statement is not true?

A. According to the interest rate parity theory, the return on a hedged foreign investment will not equal the domestic interest rate on investments of identical risk.
B. According to the interest rate parity theory, the currency of the country with a lower interest rate should be at a forward premium in terms of the currency of the country with the higher rate. In an efficient market with no transaction costs, the interest differential should be about equal to the forward differential.
C. If the covered interest differential between two money markets is nonzero, there is an arbitrage incentive to move money from one market to the other.

Correct Answer: A

Interest parity ensures that the return on a hedged foreign investment will equal the domestic interest rate on investments of identical risk.

Practice Question 14

The current exchange rate between Thai bahts and US$ is 35 baht/$1. The one year interest rate available on U.S. treasury securities is 4.5%, and the equivalent rate on Thai debt instruments is 13.5%. According to interest rate parity, what should the one year forward Baht/$ exchange rate be?

A. 38.014.
B. 36.575.
C. 32.225.

Correct Answer: A

F/35 = (1 + 0.135)/(1 + 0.045); F = 38.014, remember the higher interest rate currency is expected to depreciate.

Practice Question 15

How much will a trader make through covered interest arbitrage under the following exchange and interest rate conditions (assume the investment horizon is 1 year). Borrow $100 at an interest rate of 4%. Convert the dollars to CNY at the spot rate of $0.1920. Invest the CNY in China at 8% interest. Sell the CNY at the forward rate of 0.1879. What is the profit on the transaction?

A. $2.50.
B. $1.69.
C. $5.69.

Correct Answer: B

You borrow $100 and owe $104 at the end of a year. Convert the dollars to CNY, at $100/.192 = CNY 520.83. Invest the CNY 520.83 at 8% interest, yielding CNY 562.50 at the end of a year. Sell CNY 562.5 forward for on year at the one year forward rate of 0.1879, for 562.5 x .1879 = $105.69. Repay the loan for $104, and earn a riskless profit of $1.69.

Practice Question 16

Suppose the domestic currency is the US$ and the foreign currency is the British pound.

• The spot exchange rate is $1.4718.
• The U.S. interest rate is 4.5%.
• The British interest rate is 2.25%.
• Assume these interest rates are fixed and don't change over the life of the forward contract. They are based on annual compounding and are not quoted as LIBOR-type rates.
• Assume a currency forward contract has a maturity of 120 days.

With discrete compounding, what should be the forward price if you want to enter into a forward contract to long British pounds in 120 days?

A. 1.4824.
B. 1.4792.
C. 1.4932.

Correct Answer: A

F(0, T) = F(0, 120/365) = [1.4718/(1.0225)^120/365] (1.045)^120/365 = 1.4824.

Practice Question 17

Suppose the domestic currency is the Japanese Yen and the foreign currency is the US$. Thirty days ago Takashi, an investor in Tokyo, entered into a forward contract to buy US$ in 180 days. The forward contract price was Y107.42 with a notional principal of $3million. Today's exchange rate is Y102.13 per dollar. The annualized U.S. interest rate is 3.25%, and the annualized Japanese interest rate is 0.75%. With continuous compounding, what is the value of this contract (ignore sign)?

A. 5.42 million yen.
B. 3.79 million yen.
C. 18.88 million yen.

Correct Answer: C

With continuously compounding:

r = ln(1.0075) = 0.7472%, and r(f) = ln(1.0325) = 3.1983%.
V30(0, 180) = (102.13 e ^{-0.031983 (150/365)} ) - 107.42 e ^{-0.007472 (150/365)} = -6.29424.

The value is -6.29424 x 3,000,000 = -18,882,707.

Practice Question 18

Which of the following combinations of interest rate options would replicate a 6-month by 9-month forward rate agreement to pay 5% fixed? The LIBOR below refers to the 3-month LIBOR.

A. Long a 6-month call on LIBOR and short a 6-month put on LIBOR.
B. Long a 9-month call on LIBOR and short a 9-month put on LIBOR.
C. Long a 6-month call on LIBOR and long a 6-month put on LIBOR.

Correct Answer: A

If LIBOR exceeds 5%, this combination will result in an inflow of funds just like an FRA. If LIBOR drops below 5%, this combination will result in an outflow of funds just like an FRA.