Both Bond Sam and Bond Dave have 7 percent coupons, make semiannual payments, an

Question

Both Bond Sam and Bond Dave have 7 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 2 years to maturity, whereas Bond Dave has 18 years to maturity.

If rates were to suddenly fall by 2 percent instead, what would the percentage change in the price of Bond Sam be then?

If rates were to suddenly fall by 2 percent instead, what would the percentage change in the price of Bond Dave be then?

Both Bond Sam and Bond Dave have 7 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 2 years to maturity, whereas Bond Dave has 18 years to maturity.

Explanation / Answer

Bonds are priced at the maturity value, discounted by the interest rate. In math speak, this is

Pi=Pf/(1+r)^n

where

Pi=present value (what you're looking for in this problem)

Pf=maturity value

r=interest rate

n=number of years

Since the payments are semiannual, you need to multiply n by two and divide r by 2, since it's discounted for each payment, and not actually the number of years (ok, that was worded weird, sorry)

Pi=Pf/(1+r/2)^2n

Soooo, their payments are the same, but the interest rates rose, so the value of those future payments is going to be less, right? They're asking what the percentage change was.

(Pi2-Pi1)/Pi1

Filling that in:

[Pf(2)/(1+r(2)/2)^2n - Pf(1)/(1+r(1)/2)^2n] / Pf(1)/(1+r(1)/2)^2n

Since Pf is in both of the top equations, you can separate it out. Then Pf is in the num and denom, so it cancels, leaving:

[1/(1+r(2)/2)^2n - 1/(1+r(1)/2)^2n] / [1/(1+r(1)/2)^2n]

You can then switch it around to clean up some

[(1+r(1)/2)^2n]/(1+r(2)/2)^2n - [(1+r(1)/2)^2n]/(1+r(1)/2)^2n)

Ok, so that still looks ugly. But both the numerators and denominators are to the 2n, soo

[(1+r(1)/2)/(1+r(2)/2)]^2n - [(1+r(1)/2)/(1+r(1)/2)]^2n

Since the last piece is the same num and denom, it equals 1, and 1 to any power equals 1.

[(1+r(1)/2)/(1+r(2)/2)]^2n - 1

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