In: Finance
As a professional financial consultant, you try to advise on the position of a diversified portfolio. Your task is to explain to the management team on the bond sensitivity, which requires you to:
(a) Calculate the duration of a five-year, $3,000 Treasury Bond with a 10 per cent semi-annual coupon selling:
(i) at par?
(ii) selling with a yield to maturity of 20 per cent?
(b) Conclude on the relationship between the duration and yield to maturity based on part (a) results. Justify your answer with support.
(Notes: Please select consistent decimal point for the calculation of PVIFi,n in the financial table.)
Part a)i)
When the bond is trading at par, then YTM = coupon rate.
Coupon per 6month = 5% = 0.05; YTM per 6month = 5% = 0.05; t = 10 6month period
Duration = {(1+0.05)/0.05} - {(1+0.05)+[10*(0.05-0.05)]}/{0.05*([(1+0.05)^10]-1) +0.05} = (1.05/0.05) - [1.05+(10*0)]/{0.05*[(1.05^10)-1] +0.05} = 21 - 1.05/{[0.05*(1.6289-1)]+0.05} = 21 - 1.05/{[0.05*0.6289]+0.05} = 21 - 1.05/(0.031445+0.05) = 21 - (1.05/0.081445) = 21 - 12.89 = 8.11
Part a)ii)
Coupon per 6month = 5% = 0.05; YTM per 6month = 20%/2 = 10% = 0.1; t = 10 6-month period
Duration = {(1+0.1)/0.1} - {(1+0.1)+[10*(0.05-0.1)]}/{0.05*([(1+0.1)^10]-1) +0.1} = (1.1/0.1) - [1.1+(10*-0.05)]/{0.05*[(1.1^10)-1] +0.1} = 11 - (1.1-0.5)/{[0.05*(2.5937-1)]+0.1} = 11 - 0.6/{[0.05*1.5937]+0.1} = 11 - 0.6/(0.07969+0.1) = 11 - (0.6/0.17969) = 11 - 3.339 = 7.661
Part b)
If YTM increases, duration will fall and vise versa.