In: Finance
Consider Bond C – a 4% coupon bond that has 10 years to maturity. It makes semi-annual payments and has a YTM of 7%. If interest rates suddenly drop by 2%, what is the percentage change of the bond? What does this problem tell you about the relationship between interest rate and bond price? Consider another bond – Bond D, which is a 10% coupon bond. Similar to Bond C, it has 10 years to maturity. It also makes semi-annual payments and have a YTM of 7%. If interest rates suddenly drop by 2%, what is the percentage change of the bonds? Comparing the percentage change of bond C and bond D, what does this tell you about the interest rate risk of bonds with higher coupon rates?
Bond C
n = 10
Lets assume Face Value = $100
Coupon = 4% of FV = 4% of $100 = $4
YTM = 7%
Since the bond is Semi-annual,
n = 20
Coupon = 2%
Bond price = ( Coupon / (1+YTM)^n ) + ( Face Value / (1+YTM)^n )
Therefore,
Bond price = ( 2 / (1+0.07)^1 ) + ( 2 / (1+0.07)^2 ) + ( 2 / (1+0.07)^3 ) +.....+ ( 2 / (1+0.07)^20 ) + ( 100 / (1+0.07)^20 )
By solving, we get Bond Price = $ 78.68
Bond Price = $ 78.68
Now if interest rate drops by 2%, new YTM = 5%
Bond price = ( Coupon / (1+YTM)^n ) + ( Face Value / (1+YTM)^n )
Therefore,
Bond price = ( 2 / (1+0.05)^1 ) + ( 2 / (1+0.05)^2 ) + ( 2 / (1+0.05)^3 ) +.....+ ( 2 / (1+0.05)^20 ) + ( 100 / (1+0.05)^20 )
By solving, we get Bond Price = $ 92.21
Bond Price = $ 92.21
Therefore, we can say, when expected interest rate (YTM) is lower, price of bond is higher. This shows an inverse relationship between interest rate and bond price.
Bond D
n = 10
Lets assume Face Value = $100
Coupon = 10% of FV = 10% of $100 = $10
YTM = 7%
Since the bond is Semi-annual,
n = 20
Coupon = 5%
Bond price = ( Coupon / (1+YTM)^n ) + ( Face Value / (1+YTM)^n )
Therefore,
Bond price = ( 5 / (1+0.07)^1 ) + ( 5 / (1+0.07)^2 ) + ( 5 / (1+0.07)^3 ) +.....+ ( 5 / (1+0.07)^20 ) + ( 100 / (1+0.07)^20 )
By solving, we get Bond Price = $ 121.32
Bond Price = $ 121.32
Now if interest rate drops by 2%, new YTM = 5%
Bond price = ( Coupon / (1+YTM)^n ) + ( Face Value / (1+YTM)^n )
Therefore,
Bond price = ( 5 / (1+0.05)^1 ) + ( 5 / (1+0.05)^2 ) + ( 5 / (1+0.05)^3 ) +.....+ ( 5 / (1+0.05)^20 ) + ( 100 / (1+0.05)^20 )
By solving, we get Bond Price = $ 138.97
Bond Price = $ 138.97
As it can be seen from above two example, bonds that have lower coupon rate, will have higher interest rate risk than similar bonds that have higher coupon rates.