Question

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A 10-Year maturity bond making annual coupon payments with a coupon rate of 5% and currently...

A 10-Year maturity bond making annual coupon payments with a coupon rate of 5% and currently selling at a yield to maturity of 4% has a convexity of 145.4.

Compute the modified duration of the bond.

Based on the information above, compute the approximated new price using the Duration & Convexity adjustment if the yield to maturity increases by 75 basis points.

What is the percentage error?

Solutions

Expert Solution

No of periods = 10 years

Coupon per period = Coupon rate * Face value

Coupon per period = 5% * $1000

Coupon per period = $50

Bond price at YTM = 4%

Bond Price = Coupon / (1 + YTM)period + Face value / (1 + YTM)period

Bond Price = $50 / (1 + 4%)1 + $50 / (1 + 4%)2 + ...+ $50 / (1 + 4%)10 + $1000 / (1 + 4%)10

Using PVIFA = ((1 - (1 + Interest rate)- no of periods) / interest rate) to value coupons

Bond Price = $50 * ((1 - (1 + 4%)-10) / 4%) + $1,000 / (1 + 4%)10

Bond Price = $405.54 + $675.56

Bond Price = $1081.11

Bond price at 75 bps increase in Yield to Maturity i.e. YTM = 4.75%

Bond Price = Coupon / (1 + YTM)period + Face value / (1 + YTM)period

Bond Price = $50 / (1 + 4.75%)1 + $50 / (1 + 4.75%)2 + ...+ $50 / (1 + 4.75%)10 + $1000 / (1 + 4.75%)10

Using PVIFA = ((1 - (1 + Interest rate)- no of periods) / interest rate) to value coupons

Bond Price = $50 * ((1 - (1 + 4.75%)-10) / 4.75%) + $1,000 / (1 + 4.75%)10

Bond Price = $390.82 + $628.72

Bond Price = $1019.54

Change in Bond price = (Bond Price at 4.75% yield - Bond Price at 4% yield) / Bond Price at 4% yield

Change in Bond price = ($1019.54 - $1081.11) / $1081.11

Change in Bond price explained by YTM = -5.6949%

Computing Modified Duration

Illustrating for Time period 0.5

Discount factor = 1 / (1 + YTM / 2)(Time period * 2)

Discount factor = 1 / (1 + 4% / 2)(0.5 * 2)

Discount factor = 0.9615

Present value of Cashflow = Discount factor * Cashflow

Present value of Cashflow = 0.9615 * $50

Present value of Cashflow = $48.08

Weight = Present value of Cashflow / Total(Present value of Cashflow)

Weight = $48.08 / $1081.11

Weight = 4.45%

Weighted average of Time = Weight * Time period

Weighted average of Time = 4.45% * 1

Weighted average of Time = 0.0455

Time period Yield to Maturity Discount Factor Cashflow Present value of Cashflow Weight

Weighted average of Time

1 4.00% 0.9615 $50 $48.08 4.45% 0.0445
2 4.00% 0.9246 $50 $46.23 4.28% 0.0855
3 4.00% 0.8890 $50 $44.45 4.11% 0.1233
4 4.00% 0.8548 $50 $42.74 3.95% 0.1581
5 4.00% 0.8219 $50 $41.10 3.80% 0.1901
6 4.00% 0.7903 $50 $39.52 3.66% 0.2193
7 4.00% 0.7599 $50 $38.00 3.51% 0.2460
8 4.00% 0.7307 $50 $36.53 3.38% 0.2703
9 4.00% 0.7026 $50 $35.13 3.25% 0.2924
10 4.00% 0.6756 $1,050 $709.34 65.61% 6.5612
Total $1,500 $1,081.11 100.00% 8.1909

Macaulay Duration = 8.1909

Modified Duration = Macaulay Duration / (1 + YTM)

Modified Duration = 8.1909 / (1 + 4%)

Modified Duration = 7.8759

Change in Bond price expalined by Modified duration & Convexity

Change in Bond price = - Modified Duration * Change in yield + 0.5 * Convexity * (Change in yield)2

Change in Bond price = - 7.8759 * (0.75%) + 0.5 * 145.4 * (0.75%)2

Change in Bond price explained by Duartion & Convexity = -5.4979%

%Error = Change in Bond price explained by YTM - Change in Bond price explained by Duartion & Convexity

%Error = - 5.6949% - (- 5.4980%)

%Error = -0.1969%


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