Question

The six-month LIBOR rate observed three months ago was 4.85% with semi-annual compounding. Today's three- and nine-month LIBOR rates are 5.3% and 5.8% (continuously compounded), respectively.

(a) calculated that the forward LIBOR rate for the period between three and nine months with semi-annul compounding.

(b) A semi-annual pay interest rate swap where the fixed rate is 5% (with semi-annual compounding) has a remaining life of nine months. If the swap has a principal value of $15,000,000, what is the value of the swap to the party receiving a fixed rate of interest?

Answer 1

Current Year = 3 omnths or 0.25 year

And the swap would last 9 more months or 0.75 years. therefore Life of Swap = 0.25 + 0.75

= 1 year

Loan amount	15	million
Fixed	5%	with semi annual compounding
Life	1	Years

LIBOR rate are:-

Time Period	LIBOR Rate
0.25	5.30%
0.75	5.80%

6-month LIBOR = 4.85%

(a)

Therefore Forward Rate = (LIBOR Rate_{year 0.75}*Year_0.75 + LIBOR Rate_{year 0.25}*Year_0.25)/(Year_0.75 - Year_0.25)

= (5.8%*0.75 + 5.2%*0.25)/(0.75-0.25)

= 6.05% semiannualy

(b)

Using the following formula to convert the annual rates to continuous rate:-

Rm = m*(e^(R_c/m) - 1) {Where m= number of compounding periods in a year ; R_m = Annual rate of the the period ; Rc = continuous compounding rate}

using the Above formula we find the below continuous compounding rate:-

Time Period	Forward Rate (S.D.)
0.25	4.85%
0.75	6.14%

Fixed Rate Cash Flows = semi annual rate*Loan amount

= (5%/2)*15 million

= 0.375 million at year- 0.5 and year-1

Cash Flows for the floating rate coupons = Forward continous compounding rate*Loan amount

= 4.85%*15 at year-0.5 & 6.14%*15 at year-1

= 0.36375 at year-0.5 & 0.46068 at year-1

Now find the Net Cash Flow and discountign it with countinous compounding method:-

Time	Fixed Cash Flow	Floating Cash Flow	Net cash Flow	Discount Factor	PV of Net Cash Flow
0.25	0.375	0.36375	0.01125	0.986837	0.011102
0.75	0.375	0.46068	-0.08568	0.957433	-0.082035
					-0.070933	Million

Dscount factor is calculated as = e^(-r*t)

Time Period	LIBOR Rate	Discount rate(e^-(LIBOR rate*Times Period))
0.25	5.30%	0.986837
0.75	5.80%	0.957433

Therefore the value of the Swap at Year-0.25 or 3 months into the swap = -$70,933.50

So the fixed rate receiver would be in loss of 70,933.5 USD on the swap.