3. Selling at a Discount

Let's take a look at a scenario where a bond would sell at a discount. Now, if you'll remember from the last video, we talked about discounts and premiums and face value. So a bond would sell at a discount if, basically, it's not as good as what you could get out in the market. So what that looks like is if you have a coupon rate that's less than the prevailing market rate.

If on our bond, we have a coupon rate of 10%, but out in the open market you could get 12%, our bond is not very attractive because you could go out in the market and find another bond for 12% while ours is stupid and sits here at 10%. So the way we make up for that is we sell it at a discount. Let's take a look at an example. If we had a $100,000, three-year, 10% semiannual bond, our schedule would look like this.

If it's semiannual and it's three years, that means we have six periods. So we have 1, 2, 3, 4, 5, 6 periods. At the end, we know we're gonna get the face value back, so we put 100,000 here at the end. Now, for each of these payments, we have to calculate 100,000 times 10%, but that's annual and we're doing semi, so we're gonna do 6 out of 12 months, and that gives us a $5,000 payment each time. So each of these is gonna be 5,000. Now, to find out what the bond is actually gonna sell for, what we do is we bring all of these $5,000 payments back to the present, and then we also bring this $100,000 payment back to the present.

And this should look familiar because what we're doing is, down here, present value of an ordinary annuity, and up here, present value of a lump sum. And we bring all of these back into the present using the 12% market rate. We promised to pay 10%, and that's what we did here. That's how we found our payments. We had to do that because it's printed on the bond. We will pay you 10% semiannually, so we have to do that. Now, when you're pricing the bond, you as an investor would price the bond using this 12% rate because this is the market rate.

So what we end up doing is we say 100,000 times the present value of $1 for 6 periods at 6%, 'cause we're doing the market rate times semiannual, so half of the market rate is 6, and this factor is 0.70496.

So the present value of the face value at the end, the present value of the lump sum, is $70,496. Then we also have to do the present value of the payments. So that's present value of 5,000 times present value of an ordinary annuity, 6 periods at 6%. This factor is 4.91732. So 5,000 times 4.91732 is 24,586.6. We'll call this 24,587. Now, when you add the two of these up, this is the present value of the face amount at the end, and this is the present value of the payments.

So when you add those two together, you get $95,083. And just as a gut check, you know you're on the right path because 95,083 is less than the face value of 100,000. So this bond did indeed sell at a discount. Somebody's gonna pay 95,083, and at the end they're gonna get 100,000 back. And the reason this makes sense is because the market rate is 12%. So no one would pay 100,000 upfront to get 100,000 back because then all they would be earning would be this 10% on these payments.

Now somebody would pay 95,000 upfront to get 100,000 back because this extra, whatever it is, like $4,917, the difference between 100,000 and this 95, that extra money that they're getting makes it come out to be a 12% return. So the effective return is the market rate, which of course it is because we brought all this stuff back at the market rate. Now, if you're the company that sells this, you only got cash of 95,083, right? But you also got a bond payable of 100,000. So we stuff this extra amount into an account called discounts.

Discounts on bonds payable. And this is a contra account to take away from bonds payable. So essentially, no matter what your bond says, if your bond says 1% or if your bond says 50%, the cost to borrow money is always going to come out to be the market rate. Because the price of the bond is going to adjust based on the market rate. So the cost to borrow money is always gonna be the market rate because the market is going to dictate the selling price of the bond by doing this present value of cash flows. They'll bring back this annuity, and they'll bring back this future face value, and they'll figure out, based on the market rate, here's the most amount of money I'm going to pay for it.

And in this case, it's this 95,083. So the company has to take this 95,083 because nobody's gonna pay them more. Because the market rate is 12%, they're not gonna be able to borrow money at 10. So we put this extra amount in this discounts account, and we keep bonds payable at 100,000 because what do we have to do at the end? We have to pay them back 100,000. We promised we would, so we have to put this in at 100,000. Now over time, we amortize this discount, and I'll show you how to do that in another video 'cause it's kinda gnarly.

But for now, know that this is a contra account whose purpose is to take away from the value of bonds payable.