Question

Assume a 2-year Euro-note, with a $100,000 face value, a coupon rate of 10%, and a convexity of 4.53. If today’s YTM is11.5% and term structure is flat. Coupon frequency and compounding frequency are assumed to be annual. Please show all steps.

a.What is the Macaulay duration of this bond?

b.What is the exact price change in dollars if interest rates increase by 10 basis points (a uniform shift)?

c.Use the duration model to calculate the approximate price change in dollars if interest rates decrease by 10 basis points.

Answer 1

Answer (a)

Year (T)	Cash flow	Present value @ 11.50%	Weight % (W)	(W) * (T)
1	$10,000	$8,968.610	0.0920	0.0920
2	$110,000	$88,479.559	0.9080	1.8159
Total		$97,448.169	1.000	1.908

Macaulay duration = 1.908

Working notes:

• Cash flow in year 1 = $100,000 * 10% = $10,000
• Cash flow in year 2 = ($100,000 * 10%) + $100,000 = $110,000
• Present value = Cash flow / (1 + Yield to maturity)^Year
• Weight % = Cash flow / Sum of cash flows in all years

Answer (b)

One basis point = 0.01%

10 basis point = 0.10%

New yield to maturity = 11.50% + 0.10% = 11.60%

Calculation of price:

Year	Cash flow	Present value @ 11.60%
1	$10,000	$8,960.573
2	$110,000	$88,321.065
Total		$97,281.638

Change in price = $97,281.638 - $97,448.169 = -$166.531

Answer (c)

Change in yield = -0.10%

Original price = $97,448.169

Modified duration = Macaulay duration / (1 + Original YTM) = 1.908 / 1.115 = 1.7112

Change in price = -Modified duration * Change in yield * Oringial price

= -1.7112 * (-0.10%) * $97,448.169

= $166.75