Question

Bond A Maturity Coupon Yield 2 years 6% 6% Price 100

Coupon is paid semi-annually and your investment horizon is 1.5 years. A) What is the modified duration of this bond? B) What is the convexity of this bond? C) If the yield after 6 months is expected to fall to 5.5% and then to 4.5% from the beginning of year 2, what would be your holding period return, price appreciation/depreciation, re-investment income and coupon income from this bond?

Answer 1

Solution

Modified Duration = D / (1+r)

where D= Duration

r = Interest Rate

For Calculation of Duration is the aggregate of weighted cash flows divided by Aggregate weights.

For Bond A , Let us calculate duration for 2 years, semi annually, as in semi annually it will be for 4 periods and the coupon rate of 6% p. a shall be 3 % semi annually, the bonds will redeemed at par.

Periods (a)	Cash Flows @ 3 % (b)	Discounting Factor @ 6 % (c)	Discounted Value (d)= (b)* (c)	Weighted Discount Value (e)= (a)*(d)	(t2+t) (f)= (a2+a )	g= (f)*(c)
1	3	0.9433	2.83	2.83	2	5.66
2	3	0.8899	2.67	5.34	6	16.02
3	3	0.8396	2.52	7.56	12	30.24
4	3	0.7920	2.38	9.52	20	47.6
4	100	0.7920	79.20	316.8	20	1584
			$\sum$Discounted Value 89.6	$\sum$Weighted Discounted Values 342.05		1683.52

Duration = $\sum$ Weighted Discount/ $\sum$ Discounted Value = 342.05/89.6 = 3.84 period , that is if we convert this in yearly , it means 3.84/2 = 1.92 years

Now Modified Duration is = Duration / (1+r) = 1.92/ (1+0.06) = 1.92/1.06 = 1.81 years.

Modified Duration is 1.81 Years

B) Convexity of Bond

Convexity = [1/ P * (1+ y)²]* $\sum$ _t=4[ CF_t/(1+y)^t(t²+t)

where , P = Current Market Price of Bond , which is 89.6

y= YTM of the Bond; 6%

t= no of periods , 4

Now, using the above table's data in the above formula ,

Convexity= [1/ 89.6(1+0.06)²](1683.52) = [1/89.6*1.1236]* 1683.52= [1/100.67]*1683.52= 16.72 years²

Convexity = 16.72 years²

C) If YTM falls by 5.5% after 6 months then current price of the bond will increase and if during the beginning of 2nd year YTM falls further 4.5% then current price will further fall.

If YTM changes to 5.5%, the duration will change , we will use the duration of the period that is 3.84

Change in Price = - D * Change in YTM

= -3.84 * (0.06-0.055)

=-3.84 * 0.005= -0.0192 or -1.92%

New Price = 89.6* (1-0.0192)= 89.6*0.9808= 87.88

If YTM changes to 4.5%, the duration will change , we will use the duration of the period that is 3.84

Change in Price = - D * Change in YTM

= -3.84 * (0.06-0.045)

=-3.84 * 0.015= -0.0576 or -5.76%

New Price = 89.6* (1-0.0576)= 89.6*0.9424= 84.44

Price at the end of 2 nd year = 84.44

Price at the beginning = 89.60

• Holding Period Return =[( Price at the end - Price at the beggining )/ Price at the beginning ]* 100 = [(84.44-89.60)/89.60]*100= -5.76%
• Price depreciation of 5.16, as the closing price is less than the current price.
• Reinvestment Income shall be if interest rate decline to 4.5 % at the end of year 1 , the

Period	Cash Flows	Reinvestment Rate	Period	Amount at period end
1	6	6%	4	7.57
2	5.5	5.5%	3	6.42
3	4.5	4.5%	2	4.92
4	4.5	4.5%	1	4.70
4	100			100
				123.61

• Coupon rate shall increase .