4. Selling for a Premium

Let's take a look at a scenario where a bond sells at a premium. Now, we already did one where a bond sells at a discount, and so premium is gonna be the opposite of that. If you have a bond that is a three-year 10% semiannual bond and the market rate is 8%, right off the bat, you can compare your coupon rate, because it's printed on the bond itself, against the market rate and see that your bond is better than the market. So if your bond is better than what anyone else could buy out on the market, then people are going to want your bond more, and so they'll be willing to pay more money for it.

If the coupon rate is greater than the market rate, then it's gonna sell for a premium. And that makes logical sense because the instrument that you're buying, this bond, it has a 10% rate, and the best you could do elsewhere in the market would be 8%. So people are going to be willing to pay more for it. Let's draw out this schedule and see how this looks. So we're gonna have 1, 2, 3, 4, 5, 6. We have six periods because it's three years, but it's semiannual. So don't forget to split that up to six different periods. And also, if we have N equals 6, our coupon rate is going to be 5% because we need to do 10 divided by 2, and in the same way, our market rate because it's semiannual, that's gonna be 4%.

So please don't forget to adjust for semiannual compounding. Let's go ahead and put the cash payments on here, and we'll assume this is a $100,000 bond. So if it's a $100,000 bond and it's semiannual 10%, then that means the payments are going to be 5% payments twice a year to add up to 10%. So our payments are going to be $5,000 payments. So we have six $5,000 payments.

And then at the end, we know that we get the face value back. So this is the schedule of all of the payments. You have your six $5,000 payments semiannually, and then like every bond, at the end, you get your face value back. If the face value printed on the bond is 100,000, that's what you get back at the end. And remember, while you're doing this, to always use the coupon rate. This schedule that we're drawing out here is the schedule of this bond. This bond is printed. It cannot be changed. Market rate is gonna go up and down, but this rate is on the bond, so it cannot be changed.

The question becomes: how much would a person pay for this bond? And to state that question a different way, the question is: how much would a person pay for six payments of $5,000 and one payment of $100,000 at a market rate of 8%? Because that's what this bond is. This bond is six payments of 5 and 100,000 at the end. So what we're trying to do is value this schedule of payments, and the way that we do that is we bring all of these payments back to the present value using the market rate because that's the rate you could get elsewhere.

The way this ends up looking is you take your 5,000, and because they are regular payments, it becomes a present value of an ordinary annuity problem. So you take your present value of an ordinary annuity factor, and you do 6 periods for 4%. Remember to use the market rate. When you're discounting the cash flows, you have to use the market rate. Coupon rate is only used to find the payments that we promised to pay.

This factor is 5.24214. 5,000 times 5.24214 comes out to be 26,211, which tells us that the present value of six payments of $5,000 at 4%, which is 8% semiannual, is $26,211. But this is only one portion. We don't wanna forget this $100,000 that we're gonna get back either. For that, we do 100,000 times the present value of a single dollar, again, for 6 periods at 4%, which is 3 years at 8% compounded semiannual. So we double the periods and half the percentage rate.

This factor is 0.79031, which tells me that 100,000 times that is gonna be 79,031. So if you add these two guys together, you get 105,242. Now, if you think about this, this is saying that you are going to be able to sell your $100,000 bond for $105,000. You may think that that's insane. Why would anyone buy a $100,000 bond for $105,000? But that is because these payments here, these $5,000 payments, are at a 10% rate.

If they were at the 8%, all of these payments would be $4,000 payments. So the person is technically buying a $100,000 bond for 105,000, but the stream of payments is calculated at a higher interest rate than the market rate. So when we bring all of this stuff back at the market rate, then that's how you end up with this premium. That's how you end up with the 105,000 instead of the 100,000. If a person paid 105,242 for this bond, their effective rate, the rate that they would effectively be earning, is 8%.

To record this bond, to record the issuance of this bond at 105, we're gonna do cash at 105,242 because if you think about it, somebody paid the 105,242. So that's an easy one. Then we know that at the end, we're only gonna pay back the face value. I don't care what they paid for it. That doesn't matter to me. On the bond, it's printed, "I owe you $100,000 and this stream of payments." So I don't care that you paid 105. My liability to you is 100. So that difference then becomes the premium, and the premium account serves to increase the value of bonds payable.

Just like we had the discount account over here, and we said that that was a contra account that served to take away value from bonds payable, premium is just the opposite of discount. It serves to add value to bonds payable. On a balance sheet, you may see something like bonds payable 100,000, premium 5,242, and then in a different problem, you could see discount like that. So premium adds and discount takes away from. They're both kind of adjunct accounts to this bonds payable account.