Question

The yield is 5%, term 8 years semi, coupon 8%. Determine the effective duration and effective convexity (formula in book and notes). If the YTM falls by 40 basis points (bps) calculate the duration-convexity based price change and actual-calculator based price change. The formula for duration-convexity is %∆Price = - Duration X ∆yield + ½ X Convexity X (∆ yield) ^2

Answer 1

Coupon	8%
YTM	5%
Par Value	100
Time	8
Frequency	2

Price of Bond = ∑ C/(1+r)^t + F/(1+r)^T

where, C= coupon

r = yield

F = Final payment (Par value)

t = time period from t=1 to t=T

Duration = ∑ w*t

Where, w = {C/(1+r)^t }/ Bond price

t = time until payment

Modified Duration = Duration / (i+r)

Convexity = [1 / (P *(1+r)²)] * Σ [(C/ (1 + r)^t ) * t * (1+t)]

Where, P = Bond Price

r= Yield

t = Number of Periods (time period)

T = Time to Maturity

Time period	Cash Flow	Discounting factor	PV	Weight (w)	time until payments (t)	w*t	Convexity calc
1	4	0.975609756	3.902439	0.032633863	0.5	0.016317	0.01553069
2	4	0.951814396	3.807258	0.031837915	1	0.031838	0.045455679
3	4	0.928599411	3.714398	0.03106138	1.5	0.046592	0.088694007
4	4	0.905950645	3.623803	0.030303786	2	0.060608	0.144217898
5	4	0.883854288	3.535417	0.029564669	2.5	0.073912	0.211050582
6	4	0.862296866	3.449187	0.02884358	3	0.086531	0.288264209
7	4	0.841265235	3.365061	0.028140078	3.5	0.09849	0.374977833
8	4	0.820746571	3.282986	0.027453734	4	0.109815	0.47035547
9	4	0.800728362	3.202913	0.026784131	4.5	0.120529	0.573604232
10	4	0.781198402	3.124794	0.026130859	5	0.130654	0.683972526
11	4	0.762144782	3.048579	0.025493521	5.5	0.140214	0.800748323
12	4	0.743555885	2.974224	0.024871728	6	0.14923	0.92325749
13	4	0.725420376	2.901682	0.024265101	6.5	0.157723	1.050862184
14	4	0.707727196	2.830909	0.023673269	7	0.165713	1.182959307
15	4	0.690465557	2.761862	0.023095872	7.5	0.173219	1.318979018
16	104	0.673624934	70.05699	0.585846514	8	4.686772	37.91796592

Price of Bond = Sum of all PV = $119.5825

Duration = sum of (w*t) = 6.25 years

Modified Duration = 6.25/(1+5%/2) = 6.10 years

Convexity = Sum of all Convexity calc = 46.09

If YTM falls by 40 bps, the duration-convexity based price change is:

%∆Price = - Modified Duration * ∆yield + 1/2 * Convexity * (∆ yield)²

%∆Price = - 6.10 X 0.004 + ½ X 46.09 X 0.004²

%∆Price = -0.0240

%∆Price = -2.40%