ABC Company issues 1,000 bonds with a face value of $1,000 mat...

Question

ABC Company issues 1,000 bonds with a face value of $1,000 maturing in eight years. The bonds pay 8% interest semi-annually and the current market rate of interest is 12%. What is the total amount of cash received from the bond issuance?

Accepted Answer

Alright. Let's see how you guys did. This stuff's pretty tricky, right? ABC Company issues a 1,000 bonds with a face value of a $1,000 each maturing in 8 years. The bonds pay 8% interest semi annually and the current market rate of interest is 12%. What is the total amount of cash received from the bond issuance? So when it says what's the total cash received from the bond issuance, they're asking what's the present value of these bonds? And we're going to do very similar calculations as we did in the previous problem. Now, they they told you the principal of the bonds a little differently. They told you that each bond is worth $1,000 and that they issued a 1,000 bonds. So what's the total principal? The total principal because there's a 1,000 of these bonds. Well, the bonds are going to be a 1,000 bonds with a $1,000 principal value. This is going to be 1,000,000 worth of bonds. The principal of these bonds is a 1,000,000, right? There's a 1,000 bonds worth a $1,000 each. Well, we're issuing a $1,000,000 worth of bonds. Okay? They're maturing in 8 years, but again, we're dealing with semi annual periods. They pay 8% interest. So which interest rate is this? Remember, we we're dealing with 2 interest rates. 8% interest is going to be our stated rate. This is the stated rate of interest, 8%. And what is this 12% over here? This is the market rate, right? It told us the current market rate is 12%. Okay? So we've got our 2 interest rates there. Now, why don't we go ahead and build our timeline and fill in our cash flows. So let's go ahead right here. We'll draw our timeline. And again, we're going to truncate it because I'm not going to draw 16 periods here. 15/16. All right. So right here, 0 is right now and each of these is a semi annual period. Right? We're talking about semi annual periods. So it's not going to be 8 years. It's going to be 16 8, multiply it by 2 because we're talking about 6 month periods rather than a year. So it's going to go all the way to 16, 6 month periods, okay? Now what we need to do is find out what those interest payments are going to be and we know what the principal payment is of a $1,000,000 at the end. So let's go ahead and find the interest real quick. Interest and remember, this is the cash interest that we're going to be paying. Well, what do we need to do? We need to take the principal payment, the principal amount, which is a $1,000,000 and multiply it by the stated rate. That's what tells us the cash interest, right? Because these bonds, regardless of what the market rate is, we say, hey, these bonds are paying 8% interest and that's the amount of interest they're going to pay in cash, right? So here we go. We've got the $1,000,000 in the $1,000,000 in principle for the 1,000 bonds at $1,000 each times the 8% interest, 0.08. That's gonna give us the amount of interest that we pay. Okay? So 1,000,000 times point 0 8 that comes out to 80,000. Okay? And that's 80,000 per year. I almost made a mistake here, right? Because we're talking about semi annual periods. So we need to take that 80,000 and divide it by 2 because we're paying every 6 months. And this is one of the big tricks there. I actually almost made that mistake right now. We have to be careful, right? The 8% is an annual rate, but we're talking about semi annual periods. So we need to divide that annual interest by 2 to find the semi annual payment. Okay? So it's 40,000 in interest paid every semi annual period, every 6 months. And then finally, at the end, we're going to make the principal payment, that lump sum principal payment. So our timeline is gonna look like this, 40,000 each period. And then at the very end in the 16th period, the 16th semiannual period, we're also going to make a $1,000,000 principal payment, okay? So those are all the cash flows related to this bond. Now we need to find the present value of these bonds of these cash flows, right? So we got the present value of the annuity, which is the interest payments and then we've got the present value of the lump sum principal payment at the end, right? So this is the principal and this was the interest, okay? Those are the only cash flows related to a bond is interest and principal. So we need to find what the present value of those are today. Okay? So let's go ahead and start with the present value of the interest. Okay? So remember when we do this, we're talking about about an annuity, right? This is the same payment at equal time periods. So the present value of the interest, well, that's going to be 40,000 times our present value factor. And we're going to have to get this factor from the table. So let's go ahead and find out what our n and our r are gonna be when we go through the table. So n, like we see in our timeline, it's gonna be 16, right? We had the 8 years times 2 for each semiannual period gives us 16 for our n, right? We're going to the table with an n of 16 because it's 8 years, but we're talking about semiannual period, so it's 16. What about r? What's gonna be r in this case? Remember whenever we use sem whenever we're going to the table, we use the market rate of interest. That's what goes to the table. Our market rate of interest was 12% over here. So it's going to be 12%, but since it's semiannual periods, 12% is the annual rate. We need to divide it by 2 and it gives us 6% is our semiannual rate, okay? So we're going to go to the table with an n of 16 and an R of 6%, okay? So let's go ahead and do that for our annuity first and then we'll come back and do it for the lump sum. So our annuity is our interest payments and we're looking in our annuity table, right? We need to go to the annuity table for the interest payments because it is an annuity of payments. So what is it in this case? We're looking for 16 periods and an interest rate of 6%. Right? So 16 and 6 percent, they meet up right here at 10.106. That is the number in our table. So that's what is our present value factor for this, this stream. So that's what I'm going to put in right here. 10.106 is our present value factor. So let's go ahead and find what the present value of this interest is and it's going to be 40,000 times 10.106 and that comes out to 404,240. That is the present value of the interest payments, okay? But we're not done here yet, right? The present value of the interest. So I'll write that in. 404240, right? 404,240 is the present value of the interest. Now, let's do the present value of the principal. And that's going to be the 100,000, excuse me not 100,000, 1,000,000 and that's going to be times the present value factor again. But remember this time we have to go to the other present value table because this is a lump sum. It's not an annuity, this is a one time payment of $1,000,000 after 16 periods. So we're going to have the same n and r. N equals 16 periods, r equals the semi annual 6% rate. So we'll go to our lump sum table. Let's erase the previous problem here. And we're going to have 16 periods, 6% interest. What does that give us? It gives us a present value factor of 0.394. So that's the number we're gonna bring up here, and let's go ahead and put that in. 0.394 is our present value factor. So we've got 1,000,000 times 0.394 4, and that's worth 394,000. All right? So we're just about done. We've done all the hard work already, and all that's left to do is to add these two numbers together. We found the present value of all the cash flows. First, we found the present value of all the interest payments at 404,240. Then we found the present value of the principal payment as 394,000. And now, all we got to do is add the two numbers together to find the present value of the bond. Right? The present value is gonna be those together. So 404,240 plus 394,000 that comes out to 798,000 240. And that is the present value of the bond, and that's the amount of cash that this is gonna bring in. Okay? The and present value of the bonds, right? We sold 1,000 bonds worth $1,000 each, okay? So if you guys got this right, I'm really proud of you. This is very tough. This is the toughest calculation you would have to make in this class. And if you got this at this point, you should be feeling really good. I don't expect you to get this on your first try. And even if it's the same numbers, it's worth doing the problem again and again to make sure that you get these concepts down. Alright? So once you're comfortable with using your present value tables and dealing with valuing bonds, let's go ahead and move on to the next lesson.

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ABC Company issues 1,000 bonds with a face value of $1,000 maturing in eight years. The bonds pay 8% interest semi-annually and the current market rate of interest is 12%. What is the total amount of cash received from the bond issuance?

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