Oak Tree Corporation issued 500, 7%, 8 year bonds at 102 on 1/1/07. Interest is paid semi-annually. What is the total cost of borrowing for the bonds?

What are the steps to this problem? I thought I'd multiply the 500 by 1,000, because my teacher makes all bonds 1,000. Then I multiply 7% and then multiply (8 year bonds multiplied by 2 for 16) and just subtract by 500,000 to get the cost of borrowing the bonds.

(8 year bonds multiplied by 2 for 16)

Here is a problem. Even when the bonds pay semi-annually, interest is still quoted annually. Interest is always annual unless something says otherwise. The semi-annual interest payment is actually at 3.5%. If you're in the habit of dividing it in two first, and then using 16 periods, you'd have to use 3.5% to do that. Since 7% is already annual, multiply by 8.

I'm not really sure what your instructor is meaning by the "cost." Theoretically, since they are getting a premiium for them, the interest is reduced by that premium. I'm not sure if I've seen a problem ask for that.

Even when the bonds pay semi-annually, interest is still quoted annually. Interest is always annual unless something says otherwise. The semi-annual interest payment is actually at 3.5%. If you're in the habit of dividing it in two first, and then using 16 periods, you'd have to use 3.5% to do that.
... a tip o' the cap to Morgaine for bringing up that important point.

Usually, the 'cost of borrowing', in this context, refers to the issuer's effective interest rate paid on the debt, obtained from a comparison of what they received from the bond sale, to what they'll have to pay later in coupon payments and maturity redemption.

You get there by finding the discount rate that equates the present value of all those outflows, with the issuer's net proceeds.

BTW, scale doesn't matter, as any 'size' multipliers fall away in the math, so let's just work with a single bond. Put another way, when you find the 'cost of borrowing' for one bond, you've found the borrowing cost for the entire bond issue. Also, per Morgaine's point, one "period" = 6 months.

The net proceeds on one bond was $1,020 (sold at 102). Oak Tree's obligatory cash outflows are

16 coupon payments of $35 each, every six months; i.e., one per period.
A single maturity redemption of $1,000 eight years from now (16 periods).


Now you just find the discount rate that makes the PV of those outflows equal to $1,020. In other words, find the r such that

1,020 \ = \ \frac{35}{(1+r)} \ + \ \frac{35}{(1+r)^2} \ + \ ... \ + \frac{1,035}{(1+r)^{16}}

Note that I treated the last coupon payment and the maturity payoff as a single amount, since they occur on the same day, and thus are both discounted by exactly 16 semi-annual periods.

There are various ways you can find the correct r. One pretty decent method is to lay out the equation in Excel and use trial-and-error. Or if you're familiar with Excel's "IRR" function, or its "goal seek" utility, good ol' XL will do the high-speed trial-and-error work for you.

Finally, once you've found r, remember to double it in order to re-express it as an annual rate.

For a quick 'reasonableness' test of your proposed answer, note that since the bonds sold for slightly above par, the effect 'cost of borrowing' should be close to, and slightly below, the bonds' coupon rate.

Cheers!