Question

The 1-year, 2-year, 3-year and 4-year zero rates are 2%, 3%, 4% and 5% per annum (APR) with quarterly compounding/payment.

a) What are the corresponding per annum zero rates with continuous compounding?

b) What is today’s forward rate for an investment initiated one year from today and maturing 3 years from today?  (Give your answer per annum with continuous compounding)?

c) What is today’s forward rate for a one-year investment initiated three years from today?  (Give your answer per annum with continuous compounding)?

Answer 1

a)

Rate with continuous compounding

Formula

Continuous Compounding rate = Times Compounding Takes place  * Log( 1 + rate / Times Compounding Takes place )

1 year rate = 2% per annum quarterly compounding

Here

Times Compounding Takes place = 12/ 3 => 4

Continuous Compounding rate = 4 * Log( 1 + 2%/ 4)

= 4 * Log( 1 + 0.005)

use excel or scientific calculator to calculate Log(1.005)

= 4 * 0.00498 => 0.01995 or 1.995%

Similarly for all years

Continuous compounding Rate = Compounding Factor * LN( 1 + Zero Rate/ Compounding Factor)

Compounding Factor	Period	Zero Rates	Continuous compounding Rate	or %
4	1	2%	0.019950	1.995
4	2	3%	0.029888	2.988
4	3	4%	0.039801	3.98
4	4	5%	0.049690	4.969

B)

Forward rate =[ (1 + spot rate_b )^tb / (1 + spot rate_a)^ta ] -1

Forward rate of investment initiated one year from today and maturing 3 years from today (at continuous compounding rate)

Three year spot rate = 3.98%

One year spot rate = 1.995

Spot Rate_b = 3.98%

Spot Rate_a = 1.995%

tb = 3 years

ta = 1 year

putting these values in formula

= [ (1 + 3.98%)³/ ( 1 + 1.995%)¹ ] - 1

=[ 1.0398³ / 1.01995 ] -1

Use Excel

=( 1.0398^3 / 1.01995 ) - 1

=1.124 / 1.01995 - 1

=0.1022 or 10.22%

C)

Forward rate for a one-year investment initiated three years from today

Investment is done at 3 year for one year i.e. maturing in 4 the year

Applying same formula

Forward rate =[ (1 + spot rate_b )^tb / (1 + spot rate_a)^ta ] -1

Spot Rate_b = 4.969%

Spot Rate_a = 3.98%

tb = 4 years

ta = 3 year

putting these values in formula

= [ (1 + 4.969%)⁴/ ( 1 + 3.98%)³ ] - 1

=[ 1.04969³ / 1.0398³ ] -1

Use Excel

=( 1.049696^3 / 1.0398³ ) - 1

=1.214 /1.124 - 1

1.0799 -1 => 7.99 %