In: Finance
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?
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%