Question

You are thinking about buying a bond and you want to consider your interest rate exposure. The bond in question is a semiannual note issued by Bank of America that has a $1,000 face value, four years left until maturity and pays a coupon rate of 5.265%. It is currently yielding 6.724%. Because of a slowing economy, you expect a 60 basis point parallel downward shift in the yield after the next Fed meeting. Calculate the following:

Price, duration, modified duration and convexity (manually, though you can confirm your answers using the Excel functions or one of the online calculators).

the approximate dollar and percentage change in price due to duration and convexity

the actual dollar and percentage change in price

Answer 1

Soln : Face Value = $1000, Semi annual coupon rate = 5.265% = 5.265*1000/100 = 52.65, Current yield = 6.724%, Time to maturity = 4 years

Let P be the price, then P = (52.65/2)/(1+6.724%/2)¹ + (52.65/2)/(1+6.724%/2)² + ....+(52.65/2)/(1+6.724%/2)⁸ + 1000/(1+6.724%/2)⁸  

On solving this we get , P = $949.56

Now, we need to calculate the duration or we can say Maculay duration:

Duration = Here, w_t = CF_t/(1+r)^t /Bond price , please refer the table :

t	1	2	3	4	5	6	7	8
Y = CF/(1+r)^t	25.47	24.64	23.84	23.06	22.31	21.59	20.89	787.77
w= Y/P	0.0268	0.0259	0.0251	0.0243	0.0235	0.0227	0.0220	0.8296
Price, P	949.56
t*w	0.0268	0.0519	0.0753	0.0972	0.1175	0.1364	0.1540	6.6369
Total	7.2959

Duration, D = 7.30

Modified duration = - D/(1+r/2) = -7.30 /(1+6.724%/2) = 7.059

Convexity =

Please refer this table here :

X = t^2+t	2	6	12	20	30	42	56	72
X*Y	50.937	147.842	286.066	461.270	669.399	906.676	1169.580	56719.210

Convexity = 60410.981/949.56*(1+0.034)² = 59.55

Now change in price = P*(-modified duration*0.60% + 0.6%²/2 *convexity) = $1.052

Percentage change = 0.11%