I purchased a $1,000 five percent coupon bond that matures in 10 years.
How much would my bond be worth if interest rates fall to 4% the day after I purchase the bond? What would the bond be worth in one year if interest rates fell to 4% at that point?
To calculate the current value of a bond when interest rates change, we can use the formula for the present value of a bond:
PV = C × [1 - (1 + r)^(-n)] ÷ r + F / (1 + r)^n
Where:
PV = Present value of the bond
C = Coupon payment (in this case, 5% of $1,000 = $50)
r = Interest rate (as a decimal)
n = Number of years until maturity
F = Face value of the bond ($1,000)
Let's calculate the current value of the bond if interest rates fall to 4% the day after you purchased it:
PV = $50 × [1 - (1 + 0.04)^(-10)] ÷ 0.04 + $1,000 / (1 + 0.04)^10
PV ≈ $50 × [1 - 0.67556] ÷ 0.04 + $1,000 / 1.48864
PV ≈ $50 × 0.32444 ÷ 0.04 + $671.15
PV ≈ $405.51 + $671.15
PV ≈ $1,076.66
Therefore, if interest rates fall to 4% the day after you purchased the bond, its value would increase to approximately $1,076.66.
Now let's calculate the value of the bond after one year if interest rates fall to 4% at that point:
PV = $50 × [1 - (1 + 0.04)^(-9)] ÷ 0.04 + $1,000 / (1 + 0.04)^9
PV ≈ $50 × [1 - 0.65533] ÷ 0.04 + $1,000 / 1.37485
PV ≈ $50 × 0.34467 ÷ 0.04 + $727.81
PV ≈ $862.18 + $727.81
PV ≈ $1,589.99
Therefore, in one year, if interest rates fall to 4%, the bond would be worth approximately $1,589.99.