Question

A bond with 14 years to maturity paying a 4% coupon is selling to yield 5.5%. Calculate the bond's duration when yields to maturity change by 50bp, the bond's convexity and the % change in the bond's price.

Answer 1

Hello,

Let us assume the par value of bond = $ 1000

Coupon rate = 4 % annual

and yield = 5.5% . The formula for bond duration is following -

   In the above formula, C= coupon rate , r = yield and t = Maturity in years

Using the above formula for 5.5% yield and 4% annual coupon rate,

Numerator = (40/1.055)*(1)+(40/(1.055^2))*(2)+(40/(1.055^3))*(3)+(40/(1.055^4))*(4)+(40/(1.055^5))*(5)+(40/(1.055^6))*(6)+(40/(1.055^7))*(7)+(40/(1.055^8))*(8)+(40/(1.055^9))*(9)+(40/(1.055^10))*(10)+(40/(1.055^11))*(11)+(40/(1.055^12))*(12)+(40/(1.055^13))*(13)+(1040/(1.055^14))*(14) = $9162.23

Denominator =(40/1.055)+(40/(1.055^2))+(40/(1.055^3))+(40/(1.055^4))+(40/(1.055^5))+(40/(1.055^6))+(40/(1.055^7))+(40/(1.055^8))+(40/(1.055^9))+(40/(1.055^10))+(40/(1.055^11))+(40/(1.055^12))+(40/(1.055^13))+(1040/(1.055^14)) = $856.1553

So, Duration D = Numerator/Denominator = $9162.23/$856.1553 = 10.70

Now, if the yield change by 50bp

Case 1 - Yield increases by 50 bp

So, in our formula r = yield = 6%

Again using the formula

Numerator= (40/1.06)*(1)+(40/(1.06^2))*(2)+(40/(1.06^3))*(3)+(40/(1.06^4))*(4)+(40/(1.06^5))*(5)+(40/(1.06^6))*(6)+(40/(1.06^7))*(7)+(40/(1.06^8))*(8)+(40/(1.06^9))*(9)+(40/(1.06^10))*(10)+(40/(1.06^11))*(11)+(40/(1.06^12))*(12)+(40/(1.06^13))*(13)+(1040/(1.06^14))*(14) = $8632.52

Denominator =(40/1.06)+(40/(1.06^2))+(40/(1.06^3))+(40/(1.06^4))+(40/(1.06^5))+(40/(1.06^6))+(40/(1.06^7))+(40/(1.06^8))+(40/(1.06^9))+(40/(1.06^10))+(40/(1.06^11))+(40/(1.06^12))+(40/(1.06^13))+(1040/(1.06^14)) = $814.10

So, Duration D = Numerator/Denominator = $8632.52/$814.10 = 10.603

Case 2 - Yield decreases by 50 bp

So, in our formula r = yield = 5%

Again using the formula

Numerator = (40/1.05)*(1)+(40/(1.05^2))*(2)+(40/(1.05^3))*(3)+(40/(1.05^4))*(4)+(40/(1.05^5))*(5)+(40/(1.05^6))*(6)+(40/(1.05^7))*(7)+(40/(1.05^8))*(8)+(40/(1.05^9))*(9)+(40/(1.05^10))*(10)+(40/(1.05^11))*(11)+(40/(1.05^12))*(12)+(40/(1.05^13))*(13)+(1040/(1.05^14))*(14) = $9729.04

Denominator =(40/1.05)+(40/(1.05^2))+(40/(1.05^3))+(40/(1.05^4))+(40/(1.05^5))+(40/(1.05^6))+(40/(1.05^7))+(40/(1.05^8))+(40/(1.05^9))+(40/(1.05^10))+(40/(1.05^11))+(40/(1.05^12))+(40/(1.05^13))+(1040/(1.05^14)) = $901.013

So, Duration D = Numerator/Denominator = $9729.04/$901.013 = 10.80

The formula for bond convexity is following -

where P = bond price , y = yield to maturity , t = maturity in years and CF_t = cash flow at time t

Case 1 - When yield = 5.5 %

Inputting the values in the convexity formula (like we did in duration claculation) , we get

Convexity = 130.91

Case 2 - When yield = 6 %

Inputting the values in the convexity formula (like we did in duration claculation) , we get

Convexity = 128.11

Case 3 - When yield = 5 %

Inputting the values in the convexity formula (like we did in duration claculation) , we get

Convexity = 133.74

The formula for % change in bond's price is -

Case 1 - When the yield increases to 6 % i.e. increase by 50 bp

% change in bond price = (-10.70 * 0.0050) + (1/2 * 130.91 * (0.0050)²) = -0.0535 + 0.001636 = -0.05186

This means that bond price reduces by 5.2% approximately

Case 2 - When the yield decreases to 5 % i.e. decrease by 50 bp

% change in bond price = (-10.70 * -0.0050) + (1/2 * 130.91 * (-0.0050)²) = 0.0535 + 0.001636 = 0.055136

This means that bond price increases by 5.5% approximately