A $1,000 corporate bond with 20 years to maturity pays a coupon of 7% (semi-annual) and the market required rate of return is a) 6.6% b) 13%. What...

Using the zero-arbitrage assumption, we price the bond at the net present value of the cashflows it provides.

It provides the following cashflows:

$35 (1/2 * 7% * $1000) twice a year for the next 20 years

$1000 in 20 years

The net present value then depends upon our discount rate, which is assumed to be the same as the market interest rate, given as 6.6% in part A and 13% in part B.

NPV = 1000/(1+r)^20 + 35/(1+r)^(1/2) + 35/(1+r) + 35/(1+r)^3/2 + ...

NPV = 1000/(1+r)^20 + sum_{i = 1}^{40} 35/(1+r)^(i/2)

(Note: This formula assumes that the coupons start being paid 6 months from now and there is a coupon payment at the end of the 20 year period along with the principal. There may be some adjustments you need to make to this if the coupon payments start right away, and/or if the last one isn't paid.)

We can actually compute this geometric sum as a general formula, even if we don't know r yet:

NPV = 1000/(1+r)^40 + 35 (1 - (1+r)^(-40))/r

The r we want to us is actually half the given rate of return, because we are paying coupons twice a year. There are 40 paying periods in 20 years.
Put in r = 0.066/2 for part A and we get:

NPV = 1000/(1.033^40) + 35 (1 - 1.033^(-40))/0.033
NPV = 272.89 + 35 * (1 - 0.27289)/0.033
NPV = 272.89 + 771.18
NPV = 1044.07

Now let's put in r = 0.13/2 for part B:

NPV = 1000/(1.065^40) + 35 (1 - 1.065^(-40))/0.065
NPV = 80.54 + 35 (1 - 0.08054)/0.065
NPV = 80.54 + 495.09
NPV = 575.63

Let's do a sanity check here: For higher discount rates, we expect the bond to be worth less in today's money. And that is indeed what we find. For yield rate above discount rate, we expect to find a net present value larger than the face value—also what we found.