Price of Bond

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Assume the economy interest rate is 8%. What is the present value of 20-year bond with a face value of $1000 and an annual coupon rate of 10%. What is the present value of a 20-year bond with a face value of $1000 and a annual coupon rate is 4%. Answer the question again assuming that the economy interest rate is now 12%. All else equal, does the price of the bond increase or decrease with an increase in its coupon rate. what happen to bond prices when there is an increase in the economy interest rate.

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Solution:
In the case when the interest rate is 8%
When Coupon Rate = 10% then the coupon payment or Cash inflow = $ 1000 * 10% = $100
Bond Price= cash Inflow 1/(1+i)1 + cash Inflow 2/(1+i)2 + cash Inflow 3/(1+i)3 + …+ cash Inflow 20+Face Value./(1+i)20
Bond Price = $100/(1+.08)sup>1 + $100/(1+.08)2 +$100/(1+.08)3 + $100/(1+.08)4 +.... + $100+$1000/(1+.08)20
= $92.59 + $85.73 + $79.38 + $73.52 + $68.07 + $63.05 + $58.37 + $54.05 + $50.02 + $46.33 + $42.90 + $39.71 + $36.77 + $34.04 + $31.52 + $29.19 + $27.02 + $25.02 + $23.17 + $236.05
= $1196.5


When Coupon Rate is 4% then the coupon payment or Cash inflow will be $ 1000 * 4% = $40
Bond Price = cash Inflow 1/(1+i)1 + cash Inflow 2/(1+i)2 + cash Inflow 3/(1+i)3 + …+ cash Inflow 20+face value/(1+i)20
Bond Price = $40/(1+.08)1 + $40/(1+.08)2 +$40/(1+.08)3 + $40/(1+.08)4 +.... + ($40+$1000)/(1+.08)20
= $37.03 + $34.29 + $31.77 + $29.41 + $27.22 + $25.22 + $23.35 + $21.62 + $20.01 + $18.53 + $17.16 + $15.88 + $14.71 + $13.61 + $12.61 + $11.67 + $10.81 + $10.01+ $9.26 + $223.17
= $607.27
In the case when the interest rate is 12%
When the Coupon Rate is 10%, then the coupon payment or the cash inflow will be $ 1000 * 10% = $100

Bond Price = cash Inflow 1/(1+i)1 + cash Inflow 2/(1+i)2 +(cash Inflow 3)/(1+i)3 +⋯+ cash Inflow 20+Face Value/(1+i)20
Bond Price $100/(1+.12)1 + $100/(1+.12)2 +$100/(1+.12)3 + $100/(1+.12)4 +...+ $100+$1000/(1+.12)20
= $89.28 + $79.74 + $71.22 + $63.57 + $56.75 + $50.68 + $45.24 + $40.40 + $36.06 + $32.20 + $28.75 + $25.67 + $22.92 + $20.46 + $18.27 + $16.31 + $14.56 + $13.00 + $11.61 + $114.03
= $850.87


When the Coupon Rate is 4%, then the coupon payment or the Cash inflow will be =$ 1000 * 4% = $40


Bond Price = cash Inflow 1/(1+i)1 + cash Inflow 2/(1+i)2 + cash Inflow 3/(1+i)3 +…+ cash Inflow20+Face Value/(1+i)20
Bond Price = $40/(1+.12)1 + $40/(1+.12)2 +$40/(1+.12)3 + $40/(1+.12)4 +.... + ($40+$1000)/(1+.12)20
= $35.71 + $31.89 + $28.47 + $25.42 + $22.69 + $20.27 + $18.09 + $16.16 + $14.42 + $12.87 + $11.50 + $10.26 + $9.16 + $8.18 + $7.30 + $6.52 + $5.82 + $5.80 + $5.20 + $4.64 + $107.8
= $403.6

Relationship between Bond Price and Coupon Rate

Yes, the price of the bond increases with an increase in the coupon rate. If we increase the coupon rate from 4% to 10% the price of the bond increases. There is a direct relationship between bond price and coupon rate.

Relationship between Bond Price and Interest Rate or Discount Rate

As the economic interest rate increase, there is a decrease in the price of the bonds. Similarly, as the economic interest rate decrease, there is an increase in the price of the bonds. There is an inverse relationship between bond price and discount rate (interest rate).

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