Q1. The 6-month, 12-month, 18-month, and 24-month interest rates are 1.50%, 1.75

Question

Q1. The 6-month, 12-month, 18-month, and 24-month interest rates are 1.50%, 1.75%, 2.00%, and 2.25% with continuous compounding.

a. Calculate the present value of $100 in 1.5 years (=18 months).

b. Calculate are the equivalent 6-month, 12-month, 18-month, 24-month interest rates with quarterly compounding.

c. Calculate the 6-month forward rate between 1.5 years (=18 months) and 2 years (=24 months). Answer the forward rate with continuous compounding.

Q2. A short 18-month forward contract on a non-dividend-paying stock was entered into a year ago. The delivery price of the contract is $48.

Today, this contract has 6 months to maturity. The risk-free rate with continuous compounding is 2% per annum, the current stock price is $51. Calculate the value of this short forward contract.

Hint: value of short forward contract = value of long forward contract * -1

Explanation / Answer

Q1.

(a)

With continuous compounding,

Future Value = Present Value x (er x N). where

r: interest rate and N: number of years

Future Value of $100 after 1.5 years = $100 x (e0.02 x 1.5) = $103.045

(b)

With quarterly compounding, frequency of compounding per year = 4

If annual interest rate is 'r', & frequency of compounding is 'm',

Equivalent interest rate with 'm' compoundings per year is

= [1 + (r /m)]m - 1  

Here m = 4

So:

(i) 6- month rate with quarterly compounding = [1 + (0.015 / 4)]4 - 1 = 0.0151 = 1.51%

(ii) 12- month rate with quarterly compounding = [1 + (0.0175 / 4)]4 - 1 = 0.0176 = 1.76%

(iii) 18- month rate with quarterly compounding = [1 + (0.02 / 4)]4 -1 = 0.0202 = 2.02%

(iv) 24- month rate with quarterly compounding = [1 + (0.0225 / 4)]4 - 1 = 0.0227 = 2.27%

(c) Spot rate at 1.5 years = 2%

  Spot rate at 2 years = 2.25%

The 6-month forward rate between these two periods is calculated by following formula:

f = [(1.0225)2 / (1.02)] - 1 = 2.5%

Q2.

Value of a long forward contract at time t

= ST - [F / (1 + r)(T - t)] where

ST: Asset value (Here, S: Stock Price = $51)

F: Forward price = $48

T = 18 months = 1.5 years

t: time to maturity = 6 months = 0.5 year

So, T - t = 1

r: Risk free interest rate = 2%

So, Value of forward contract = $51 - [$48 / (1.02)1] = $3.94

This amount is value to the Long. Since this is a Short forward, its value is a ngative $3.94.

So value of short forward = - $3.94

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