Alternatively, the price of forward contract on bond with coupons CI is:
Consider a bond with semi-annual coupons.
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.
A LIBOR deposit is a loan from one bank to another. Let Li(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 + Li(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 value of FRA on day g is:
The idea behind this formula is quite simple:
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:
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:
Using the formula, the value of the FRA is Vg(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.
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
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.
If we assume that interest is compounded continuously:
See basic questions for examples.
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.
(1 + .25/4)(4x1/2) = 1/425 x (1 + .05/4)(4x1/2) x F
A. covered interest arbitrage is not possible.
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 $2,481.
With continuously compounding:
Note that the two rates are equal.
F = [¥124.56/$ / (1 + 0.064)180/360] * (1 + 0.026)180/360 = ¥122.31/$.
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: $60/1.0762/365 + $60/1.07245/365 = $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 - $875.44/1.07247/365 = $81.84.
This positive value represents a loss to the investor as his position is "short".
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%.
Here the notation would be: g = 10, h - g = 20, h + m - g = 200.
Lg(h - g) = L10(20) = 5.45%, and Lg(h + m - g) = L10(200) = 5.95%.
Vg(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.
A. The company should pay $828.83.
Here the notation would be: g = 30, h - g = 0, h + m - g = 30.
Lg(h - g) = L30(0) = 0%, Lg(h + m - g) = L30(30) = 6.10%.
Vg(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.
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.
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.
Interest parity ensures that the return on a hedged foreign investment will equal the domestic interest rate on investments of identical risk.
F/35 = (1 + 0.135)/(1 + 0.045); F = 38.014, remember the higher interest rate currency is expected to depreciate.
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.
F(0, T) = F(0, 120/365) = [1.4718/(1.0225)120/365] (1.045)120/365 = 1.4824.
A. 5.42 million yen.
With continuously compounding:
r = ln(1.0075) = 0.7472%, and r(f) = ln(1.0325) = 3.1983%.
The value is -6.29424 x 3,000,000 = -18,882,707.
A. Long a 6-month call on LIBOR and short a 6-month put on LIBOR.
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.