Question

Suppose that the 2-year and 2.5-year zero rates with continuous compounding are 2.6% and 3.0%, respectively. (a) What is the forward rate for the six-month period beginning in 2 years (2R2.5) (from Year 2 to Year 2.5) with continuous compounding? (b) What is the forward rate for the six-month period beginning in 2 years (2R2.5) (from Year 2 to Year 2.5) with semiannual compounding? (c) What is the (Year 0) value of an FRA that promises to pay the lender 4.5% (compounded semiannually) on a principal of $2 million for the six-month period starting in 2 years (from Year 2 to Year 2.5)?

Please give me the process, thank you!

Answer 1

(a) With continuous compounding

e: natural exponent

e^(R2.5*2.5) = e^(R2*2)*e^({2R2.5}*0.5)

e^(R2.5*2.5) = e^(R2*2+{2R2.5}*0.5)

Since e is common, cancelling it out

R2.5*2.5 = R2*2+{2R2.5}*0.5

3.0%*2.5 = 2.6%*2 +{2R2.5}*0.5

2R2.5 = 4.60% pa (compounded continuously)

(b) With semi-annual compounding

(1+R2.5/2)^(2*2.5) = (1+R2/2)^(2*2)x(1+(2R2.5)/2}^(2*0.5)

(1+3%/2)^5 = (1+2.6%/2)^4 x (1+(2R2.5)/2}^1

2R2.5 = 4.61% pa (compounded semi-annually)

(c) 2 years from now:

Six month theoretical interest rate = 4.61% pa (compounded semi-annually)

FRA rate = 4.5% pa (compounded semi-annually)

Interest lost (as bank will receive only 4.5% while actual rate is 4.61%) = 4.61%-4.5% = 0.11%

Interest lost in 6-months = $2million * 0.11%/2 = $1100

Value of $1100 at (t=2) ie discounting by 4.5% = 1100/(1+4.5%/2)^1 = $1075.78

Value at (t=0) discounting by 2.6% = 1075.78/(1+2.6%/2)^(2*2) = $1021.61

Bank will have to pay $1021.61 for the FRA