Question

Both Bond Bill and Bond Ted have 10.4 percent coupons, make semiannual payments, and are priced at par value. Bond Bill has 5 years to maturity, whereas Bond Ted has 22 years to maturity. Both bonds have a par value of 1,000. a. If interest rates suddenly rise by 3 percent, what is the percentage change in the price of these bonds? (A negative answer should be indicated by a minus sign. Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) b. If rates were to suddenly fall by 3 percent instead, what would be the percentage change in the price of these bonds? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Answer 1

Answer:

Ans,

a) If interest rates suddenly rise by 3 percent, Bill´s bond would drop by -20.02%  and Ted´s bond would go down by -36.07%

.

b) If rates were to suddenly fall by 3 percent, Bill´s bond would rise by 26.79%

and Ted´s bond would rise too by 86.47%

.

Explanation:

Hi, first let´s go ahead and establish the stable scenario, for that we are going to use the information of the problem but we need to add the discount rate of the bond or yield, which is the missing information. All this so this concept can be explained in a better way, so for this example we´ll say that the yield of both bonds is 10% compounded semi-annually, the same units as the coupon. Now we have to use the following formula.

$Price=\frac{Coupon((1+Yield)^{n}-1) }{Yield(1+Yield)^{n} } +\frac{FaceValue}{(1+Yield)^{n} }$

Where:

Coupon = (%Coupon/2)*FaceValue= (0.104/2)*1,000=52

Yield = we are going to assume 10% annual, that is 5% semi-annual

n = Payment periods (For Bill n=5*2=10, for Ted, n=22*2=44)

So, let´s see what is the price of each bond if the yield was 10% annual compounded semi-annually.

$Price(Bill)=\frac{52((1+0.05)^{10}-1) }{0.05(1+0.05)^{10} } +\frac{1,000}{(1+0.05)^{10} } =1,015.44$

In Ted´s case, that is:

$Price(Ted)=\frac{52((1+0.05)^{44}-1) }{0.05(1+0.05)^{44} } +\frac{1,000}{(1+0.05)^{44} } = 1,035.33$

Now, if the interest rate (Yield) suddenly goes up by 3%, this is what happens to Bill´s Bond

$Price(Bill)=\frac{52((1+0.08)^{10}-1) }{0.08(1+0.08)^{10} } +\frac{1,000}{(1+0.08)^{10} } = 812.12$

If yield goes down by 3%, this is the new price of Bill´s bond.

Price(Bill)=\frac{52((1+0.02)^{10}-1) }{0.02(1+0.02)^{10} } +\frac{1,000}{(1+0.02)^{10} } =  1,287.44

Now, in the case of Ted, this is what happens to the price if the yield goes up.

$Price(Ted)=\frac{52((1+0.08)^{44}-1) }{0.08(1+0.08)^{44} } +\frac{1,000}{(1+0.08)^{44} } = 661.84$

If it goes down by 3%, this would be the price for Ted´s bond.

$Price(Ted)=\frac{52((1+0.02)^{44}-1) }{0.02(1+0.02)^{44} } +\frac{1,000}{(1+0.02)^{44} } = 1,930.56$

Now, in percentage, what we need to use is the following formula.

$Change=\frac{(VariationValue-BaseValue)}{BaseValue} x100$

For example, in the case of Bill´s bond, which yield went up by 3%, this is what we should do.

$Change=\frac{(812.12-1,015.44)}{1,015.44} x100=-20.02Percent$

So, the price variation is -20.02% if the yield rises by 3%.

This are the results of the prices and calculations for you to answer this question. Best of luck.

                         Bill        Ted                       % (Bill)       %(Ted)

Base Price     $1,015.44    $1,035.33    

(+) 3% Yield  $812.12          $661.84      -20.02%          -36.07%

(-) 3% Yield  $1,287.44     $1,930.56       26.79%            86.47%