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TVM Tables and Bonds Payable

Brian Krogol
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All right, let's check out this example. Your company inc issues $100,000 of 10% bonds in 10 do in 10 years. The bonds pay interest semiannually. And the current market rate of interest is 8%. What is the present value current selling price of the bonds? Alright. So whenever we talk about the present value of bonds, that's what they're worth today. What people are willing to pay for them today based on these future cash flows. Okay. So it all comes down to these time value of money calculations. So what do we have here? We're issuing $100,000 worth of bonds and they pay 10% interest. So notice this 10%. This is the stated interest rate, Right? And they're due in 10 years. So this is the number of years. But notice what happened in this problem is we're dealing with semiannual. So we're gonna have to divide these things by two because of what we talked about above, right? Because the periods are half not full years. But notice we also have one more interest rate here. 8% is the market rate. Okay, so this is where things start getting a little complicated, You have to remember which rate does what the market rate is. The one that we use to go to our present value table. However, the stated rate, that's the rate that the bond actually pays the bond says, I'm gonna pay 10% interest. So that's how we calculate the cash payments. There's gonna be cash interest payments and that's based on this 10% interest. So let's find out what that cash interest is gonna be The cash interest each semiannual period, well, it's going to be 100,000 times the 10% right? 0.10. But remember its semiannual, right? So like we said, we have to divide by two when we get when we're dealing with semiannual periods. So we're gonna multiply this by half because it's not 10%. We're dealing with half the amount of time, half periods. Okay. Half year periods. So let's go ahead and find out what the cash interest is every six months. 100,000 times 60.1 Uh times half, right, divided by two. So that's gonna tell us that we're gonna pay 5000 in interest interests. Let me get that right interest every Six months, right? Every semiannual period we're gonna pay 5000 in interest, which is $10,000 per year, the 10% per year. Okay, so let's go ahead and see this on a timeline. So I'm gonna cut out some of these periods here because they're about to be, they're gonna be the same and then we'll get to the end there. So notice when we make our timeline here, we're not gonna do it in years, we're going to do it in semi annual periods. So in half years. So this is really right now is zero, this is one, this is two and this is not one year, this is one semiannual period, right semiannual, this is to semiannual periods from now. So that's technically one year from now, is the two in this case. Right. Are you following me? We have to stay in semiannual periods because of what we said above, since there's 10, since we're talking about 10 years, we're talking about 20 semiannual periods. So our timeline is going to go all the way to 20 semiannual periods here. And the reason we do this is because we're paying the cash out every semiannual period And in this case it's 5000 being paid out each semiannual period. So 5000 in interest every six months For those 20 semiannual periods over the next 10 years. Okay, so that's gonna be the 5000 is going to be that annuity that we talked about those interest payments. Right? So this here is our annuity But we also have the principal payment right at the end of the of the 10 years, which is the 20 semi annual periods at the end of that period. We're gonna have to pay $100,000, right? We're gonna have to repay them The $100,000 that they lent to us, they lent to us $100,000. Uh we sold these bonds that said in 10 years we're gonna pay you $100,000 plus the interest. So we gotta find what those were today. This is the principle and that's going to be a lump sum. So every time we deal with payable it comes down to this we're gonna find the present value of the annuity. Which means we need to find the cash payment of interest that's going to happen each period and we have to find the present value of the principle which is a lump sum at the end of the the life of the bond. Okay. So we're pretty much done with all the tough math here now that we've got it all visualized on our timeline, we're almost ready to go to our table. Okay. So there's one more thing we have to do before we go to our table is we need to find out what R. N. And R. Are are gonna be. Okay so here is the annuity and we're gonna find the present value of the annuity and then we need to find the present value of the the principal payment as well. So what's going to be our n. And what's going to be our our in these cases when we go to the table? Well N. Is going to be equal to 20. Right? We've got 10 years times the two semi annual periods per year comes out to 20 for R. N. And how about our our our interest payments are interest rate. So let me get out of the way here our interest rate. What are we gonna use for our interest rate. They gave us to interest rates in this problem. Remember when we first introduced interest rates, we always said that the R. is gonna be the market interest rate, right? This is the market interest rate that we use when we calculate our when we go to our present value table, always remember that we use the market interest rate when we go to the table and we use the stated interest rate to calculate the cash interest. Okay, so notice how we use the stated rate already up here now it's time to use the market interest rate of 8%. So since it's 8% we're gonna use 8%. But since it's semi annual periods, well it's not 8% per it's 8% per year. So it's 4% per six months. So there we go. We've got our N and R. R. We've got end of 20 are of four. We're ready to go to our table. Okay. And we're gonna go to our table for 22 equations. So let's write those equations in real quick. Our first one is for the annuity and the annuity. We're gonna find the present value of the annuity Is going to be equal to 5000. The amount of the annuity payment, right? The interest payment times the present value factor From that, from the table for 20 and 4%. So let's go ahead and do the annuity 1 1st and then we'll come back and we'll do the lump sum payment. So let's go down to our table And let's find what the present value factor for an annuity is for 20 periods at 4% interest. So remember we're using the annuity one for the interest payments. So let me erase this previous problems data And what are we doing here? Well we said it's 4% interest per semiannual period for 20 semiannual periods. And that gets us right here 13.590 13.590. So that's what we're gonna use in our problem here. So let's go ahead and write that in for our present value factor. Let me erase that and put it in here to save space 13.590. So what's the present value of the annuity? Well that's gonna be 5000 times 13.590. It comes out to 67,950. That is the present value of the annuity today. So that's the present value of just the interest payments. But remember that's not the only oops not 590 67,950. That's the present value of the interest payments. But we also need the present value of the principal. So we're gonna use the other table to find that the present value of the principal Is equal to the 100,000 that we're trying to find what that lump sum is worth today times the present value factor. Right? So we're gonna need to go to the table again. But we've already done the hard work. We know what our end is. We know what our our is we're ready to go to the table. So remember this time we use the lump sum right because this is one payment of principal. The 100,000 is just one payment that's happening uh 10 years from now in 20 semiannual periods. Okay so let's erase this from the last problem. And what do we have? Our end was 20 because there's 20 semiannual periods. Our our our interest rate is 4%. So we go down here and we find that we're gonna use 0.456 is our present value factor. So let's go ahead and bring that up here and we're going to put equals and for a present value factor whoop sees Oh I'm writing on top of the other question. So our present value factor is 0.456. So let's see what that comes out to 100,000 times .456. It comes out to 45,600. So that is the present value of our Of our principal payment. So that principal payment of of $100,000 that's happening 10 years from now. Well that's worth $45,600 today wow this has been a lot of work so far but we're finally onto our final step. And all we gotta do is add the present value of our interest payments and the present value of our principal payment. And that will tell us the present value of the bond 9 67,050 plus 45,600. That tells us that the bond today is worth $113,550. This is the present value of the bond. Okay. So that means that if we went to the market and we said hey right now we're gonna sell $100,000 worth of bonds that pay 10% interest for the next 10 years semiannually well the market would be willing to pay us $113,000 and $550 for those bonds. Why aren't they willing to just pay us $100,000? Well that's because we're paying more interest than the market, right? The market is only paying 8% interest but we're saying hey check us out we're gonna pay 10% interest were even better than the market. So people are gonna be more willing to pay us uh because we pay out more interest per period than other similar bonds on the market. So that's what makes our value be more than 100,000. Okay so these bonds are gonna sell for 113,550 and we're going to deal more with the accounting side of this once we get to those calculations, okay so there we go this is pretty tricky. And like I said this is about as tricky as these calculations are gonna get an accounting okay? And you're in this first accounting course. So I would even suggest before going on to the next practice problem. Double check that you underst and everything that went on in this video. It came down to finding the present value of the annuity and finding the present value of the principal and adding those together. Okay? So let's go ahead and once you're ready, move on to the next one and you guys can try a practice problem yourself. Alright, let's do that.