Question

Betty and Bob a 2-year coupon bond with a face and maturity value of $1,000 and a coupon rate of 8% per annum payable semiannually and a yield to maturity of 10% per annum compounded semiannually.

A. Algebraically find the price of the bond. Your final answer should be correct to 2 places after the decimal point.
The price of the portfolio is __________________.

B. Algebraically find the exact Macaulay Duration of the portfolio. Your final answer should be correct to 2 places after the decimal point and expressed in years.
The Macaulay Duration is ______________ years.

Answer 1

a) Bond valuation using Yeild to maturity(YTM)

Under this part we are asked to calculate bond value using given inputs as follows:-

YTM = 10% coumpounded semiannualy (5% for 6 months)

Coupon rate= 8% compounded semiannually (4% for 6 months)

Duration= 2 years i.e. 4 semi annual years

Bond Value will be calculated as sum total of present value of all future cashflows i.e. interest and redemption value

= Σ (CF/ (1 + r)t

Here

CF = 8%/2 of $1,000 = $40 (Semi annual interest payment)

r = YTM = 5% (as interest is payable semiannualy)

t = 4 (As amount will be paid 4 times)

= Interest + Redemption value

= Σ [40/ (1 +0.05)1 +40/ (1 +0.05)2 +40/ (1 +0.05)3 + 40/ (1 +0.05)4] + 1.000 / (1 +0.05)4

= (141.84) + (822.70)

=$964.54

Therefore The price of the portfolio is $964.54

Since YTM is greater than coupon rates, bond value will be lower than par value and hence bond is currently overvalued.

b) Macaulay Duration of the portfolio Calculation

Macaulay Duration of bond or modified duration is use for calculating the weighted average time period at which bond holder will receive payments.

Formula -

Dm = (1 + i) - (1 +i i) + n(r - i)

i r[( 1 + i)n -1] + i

where

i is the YTM i.e. 5% in our case (10% / 2),

n is the period of payments i.e. 4 in our question

r is Coupon rate i.e. 4% in our example (8% / 2)

Dm = ( 1 + 0.05)/ 0.05 - ([ 1 + 0.05 ) + 4 (0.04 - 0.05)] / {0.04[( 1 + 0.05)4 - 1] + 0.05}

Dm = 21 - [ (1.05) + 4 (-0.01) ] / 0.04( 1.21550625 - 1 ) + 0.05

Dm = 21 - 1.01 / (0.00862025 + 0.05

Dm = 21 -  1.01 / 0.05862025

Dm = 21 - 17.23

Dm =  3.77 or 1.88 semiannually (3.77 / 2)

The Macaulay Duration is 3.77 years.