# Pricing a government bond

I am reading the "Bond" article on investopedia on stumble on the way they price a government bond.

Say that the interest rate at time $$t=0$$ is $$r=10\%$$. I buy a government bond with face value 1000\$and maturity date 10 years from now. The yearly coupon must thus be 100\$ and the price of this bond at $$t=0$$ is 1000\$. Now say that the interest falls afterwards to $$r' = 5\%$$. I want to sell my bond to benefit from this, what is its new price? My reasoning is that the price of the bond is equal to its present value of $$\frac{100}{1+r'} + \ldots + \frac{100}{(1+r')^{10}} + \frac{1000}{(1+r')^{10}} = 1386$$ dollars, but according to investopedia (https://www.investopedia.com/terms/b/bond.asp paragrapph "Pricing Bonds") the new price is in fact 2000\$, because 5% of 2000\\$ is equal to the coupon paid by my bond.

Can you help me sort this out?

• Your calculation is correct. Investopedia entry seems confused. It is true that as maturity extends indefinitely, the price of the 10pct bond in a 5pct environment tends to 2000. Maybe that’s what they were assuming ?
– dm63
Aug 2 '19 at 20:05
• Yes, the Investopedia article (not at all clearly written) says nothing the example bond's maturity, but the calculation seems to be for perperuals. Aug 2 '19 at 21:44

To make this clear. The article considers a bond that is paying a coupon infinitely, so there is no expiration. In this case, the value of the bond is the sum of the discounted coupons $$\frac{100}{(1+r)}+\frac{100}{(1+r)^2}+\frac{100}{(1+r)^3}+\dots$$ which eqals $$\frac{100}{r}$$. So in case of $$r=0.10$$ the bond price is $$\frac{100}{0.1}=1000$$ and in case of $$r=0.05$$ the bond price is $$\frac{100}{0.05}=2000$$.