Question

A newly issued bond has a maturity of 10 years and pays a 8.0% coupon rate (with coupon payments coming once annually). The bond sells at par value.

A.) What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7. (Round your answers to 3 decimal places.)

Convexity ______

Duration _________ years

B.) Find the actual price of the bond assuming that its yield to maturity immediately increases from 8.0% to 9.0% (with maturity still 10 years). Assume a par value of 100. (Round your answer to 2 decimal places.)

Actual Price of The Bond _____________

C.)What price would be predicted by the modified duration rule ΔP/P=−D*Δy? What is the percentage error of that rule? (Negative answers should be indicated by a minus sign. Round your answers to 2 decimal places.)

Percentage Change Price ________

Percentage error __________

D.) What price would be predicted by the modified duration-with-convexity rule ΔP/P=−D*Δy⁢+0.5 ×Convexity×(Δy)^2? What is the percentage error of that rule? (Negative answers should be indicated by a minus sign. Round your answers to 2 decimal places.)

Percentage Change Price ________

Percentage error __________

Answer 1

Question A :Duration and Convexity

Duration is utilized as the measurement of risk as it provides the average length of time by when the bond holder will receive their paid money back. Lenghthier duration indicates longer time to receive the payments and hence its riskier.

Given Information:

Price of the bon=Face value (F)=$100

Period of maturity(n)-10 yrs

Number of periods in a year (annual coupon payments)=p=1

Coupon rate (C)=7.5%= 7.5%*$100=$7.5

Yield= r= 7.5% (Bond is yielding at par, therefore coupan rate = yield)

Solution

a. What are the convexity and duraton of the bond?

Macaulay's duration is given by the fowlloing formula:

Macaulay's duration=      ∑nt=1 t * C    + n * F   

                                                        (1+r)t*p         (1+r)n*p   

                                                            B

Substitute the given value,

Macaulay's duration= ∑10t=1   t * 7.5             +   10*       100       

                    (1+7.5%)t+2           (1+7.5%)10+1

                                                                      100

By expansion of the substituded formula and calculating the duration we arrive at,

Period (t)	Cash flow	PV=Cash flow/(1+r/p)(t*p)	t*PV
1	7.5	6.9767	6.9767
2	7.5	6.4899	12.9799
3	7.5	6.0372	18.1116
4	7.5	5.6160	22.4640
5	7.5	5.2241	26.1209
6	7.5	4.8597	29.1582
7	7.5	4.5206	31.6446
8	7.5	4.2052	33.6421
9	7.5	3.9118	35.2068
10	7.5	3.6389	36.3895
10	100	48.5193	485.1939
		Sum of t*PV	737.8887
		Macaulay's duration=sum/B	7.3788

Macaulay's duration=7.3788

Modified duration is given by the formula:

Modified duration = Macaulay's duration/(1+r)=7.3788/(1+7.5%)=6.8640=6.864 (rounded off to 3 decimal places as said in the question)

Convexity is given by the formula

Convexity==      ∑nt=1 (t2+t) * C    + (n2+n) *    F   

                              (1+r)t*p                      (1+r)n*p   

                                                 B*(1+r)2

By expansion of the substituted formula and calculating the convexity we arrive at,

Period (t)	Cash flow	PV=Cash flow/(1+r/p)(t*p)	(t2+t)*PV
1	7.5	6.9767	13.9534
2	7.5	6.4899	38.9399
3	7.5	6.0372	72.4464
4	7.5	5.6160	112.3200
5	7.5	5.2241	156.7256
6	7.5	4.8597	204.1078
7	7.5	4.5206	253.1570
8	7.5	4.2052	302.7792
9	7.5	3.9118	352.0688
10	7.5	3.6389	400.2849
10	100	48.5193	5337.1332
		Sum of (t2+t)*PV	7243.9168
		Convexity=sum/(B*(1+r)2)	62.6839

Convexity=62.6839=62.684(rounded off to 3 decimal places as said in the question)

Answer A :

Modified duration=6.864

Convexity=62.684

Question B

Find the actual price of the bond assuming that its yield to maturity immediately increases from 8% to 9% ( with maturity still 10 years)Assume a par value of 100

Solution: Given information:

Face value=$100

Coupon=7.5%=$7.5

Bond price is given by the formula

Bond price = C * (1-(1+r)-n*p) +    F    

                               r                  (1+r)n*p

By substituting the values

Bond price=7.5* (1-(1+8.5%)-10*1)   +            100            

                                   8.5%                    (1+8.5%)8.5*1

                    =$93.4387

Answer: Actual price of the bond= 93.44 ( rounded off to 2 decimal points)

Question C:

What price would be predicted by the modified duration rule    P=-D*     Y? what is the percentage error of that rule?

Answer C:

Change in the interest rate=8.5%-7.5%=1.5%

By substitution

Change in bond price=(-D*(change in YTM))*B, where Bis the original bond price

=(-608640* (1.5%))* 100=-10.2961

%price change= (-10.2961/B)*100=(-1.2961/100)*100=-10.30%

Change in bond price=10.2961

New bond price-original bond price=-10.2961

New bond price =-10.2961+original bond price=-10.2961+100=$89.7039

% error for the duration rule=

100*( Bond price using duration rule) - 1)   =100* (89.7039 - 1 ) =-10.30%

          (bond price using bond price formula                93.4387

Question D: what price would be predicted by the modified duration with convexity

Answer D:

Change in the bond price using duration –convexity rule:

Change in bond price=

(-D*(change in Y)+ 0.5*c*(change in Y2)*B

Where B is the original bond price

By substitution

Change in bond price= ( -6.864* (1.5%) + 0.5 * 62.684 * (1.5%) *100=

-9.5909

%Price change= (-9.5909*100=(-9.5909/100)*100=-9.59%(rounded off to 2 decimal)

Change in bond price= -9.59%

New bond price-original bond price= -9.5909

New bond price= -9.5909+100= $90.4090

% change erroe for duration rule=

100* (bond price using duration –convexity rule)   -1   )   = 100*   (90.4090   -1)

         (bond price using bond price formula)    93.4387

=-9.5909=-9.59%( rounded off to 2 decimal points)

% price change =-9.59.

Percentage error =-9.59