A 30-year maturity bond making annual coupon payments with a coupon rate of 12% has a duration of 11.54 years and a convexity of 192.4. The bond currently sells at a yield to maturity of 8%. Use a financial calculator or spreadsheet to find the price of the bond if its yield to maturity falls to 6% or rises to 13%. What prices for the bond at these new yields would be predicted by the duration rule and the duration-with-convexity rule?

To calculate the price of the bond when the yield to maturity changes, we can use the following formula:

P = C * (1 - (1 + r)^-n) / r + F / (1 + r)^n

Where: P = price of the bond C = annual coupon payment = 12% of the face value. We'll assume a face value of 100, so C = 12. r = yield to maturity n = number of years to maturity F = face value of the bond = 100

First, let's calculate the price of the bond when the yield to maturity falls to 6%:

P = 12 * (1 - (1 + 0.06)^-30) / 0.06 + 100 / (1 + 0.06)^30 = $167.60

Next, let's calculate the price of the bond when the yield to maturity rises to 13%:

P = 12 * (1 - (1 + 0.13)^-30) / 0.13 + 100 / (1 + 0.13)^30 = $91.41

Now, let's calculate the predicted prices using the duration rule and the duration-with-convexity rule.

The duration rule predicts the change in price as follows:

ΔP/P = -D * Δr

Where: ΔP/P = percentage change in price D = duration Δr = change in yield

For a yield change to 6%, Δr = 0.06 - 0.08 = -0.02. So, the predicted price change is:

ΔP/P = -11.54 * -0.02 = 0.2308 or 23.08%

So, the predicted price is 1.2308 * P = 1.2308 * $100 =$123.08

For a yield change to 13%, Δr = 0.13 - 0.08 = 0.05. So, the predicted price change is:

ΔP/P = -11.54 * 0.05 = -0.577 or -57.7%

So, the predicted price is (1 - 0.577) * P = 0.423 * $100 =$42.30

The duration-with-convexity rule predicts the change in price as follows:

ΔP/P = -D * Δr + 0.5 * C * (Δr)^2

Where: C = convexity

For a yield change to 6%, the predicted price change is:

ΔP/P = -11.54 * -0.02 + 0.5 * 192.4 * (-0.02)^2 = 0.2308 + 0.0385 = 0.2693 or 26.93%

So, the predicted price is 1.2693 * P = 1.2693 * $100 =$126.93

For a yield change to 13%, the predicted price change is:

ΔP/P = -11.54 * 0.05 + 0.5 * 192.4 * (0.05)^2 = -0.577 + 0.2405 = -0.3365 or -33.65%

So, the predicted price is (1 - 0.3365) * P = 0.6635 * $100 =$66.35

So, the actual prices of the bond when the yield to maturity falls to 6% and rises to 13% are $167.60 and$91.41, respectively. The duration rule predicts prices of $123.08 and$42.30, respectively, and the duration-with-convexity rule predicts prices of $126.93 and$66.35, respectively.