Using Time Value of Money Tables - Online Tutor, Practice Problems & Exam Prep

To determine the present value of annuities and lump sums, utilize present value tables based on interest rates and periods. For lump sums, the formula is PVFV=FV(1+r)n, while for annuities, multiply the annuity payment by the present value factor from the annuity table. Understanding these concepts is crucial for evaluating cash flows, especially in bond transactions involving both annuity interest payments and lump sum principal repayments.

Time Value of Money Tables:Equations

Alright. So just like I promised, we're going to learn how to find the present value of annuities and the present value of lump sums using tables. Okay? Let's check it out. So instead of having those formulas that we've been working with so far and like I said, those formulas for annuities are beyond the scope of this course. Well, we're going to use tables that make it a lot easier to find the present value based on the interest rate and the number of periods. Remember, those were 2 of our variables that we used in finding present value. So those are going to be important when we use the tables. So in this course, we're generally going to be focused on present value, okay? We're going to be taking some future sums of money like those interest payments and those principal payments from bonds and we're going to find what those are worth today. Okay? And that's the key, we're finding those present values of those future sums of money. Specifically, like I said, we're finding what those future interest payments are. And like we saw in our previous example, the future interest payments, they form an annuity. Remember that an annuity is getting equal amounts of money, so the same amount of money each period, so in equal amount of time. So annual interest each year, you would be getting the same interest payment. Okay? That's what makes an annuity. And then there's also going to be principal payment at the very end. And that's going to be a lump sum. Right? There's just going to be the one principal payment at the end when they have to pay back the bond, they have to pay back the liability to the person who bought the bond. Well, that's going to be at the end as a lump sum. They're going to pay off the value of the bond. Okay? So we're going to be dealing with both, annuities and lump sums. So let's go ahead and see how we're going to use the tables to do our present values of lump sums and present value of annuity.

So remember when we use our formula for lump sums, where we had some future amount of money and we wanted to know what it is worth today. That was like the example where we were saving up for a European vacation and we wanted to have $12,000 in the future, well, how much did we need to invest in the bank today? Well, we can use a table to save us time rather than have that clunky formula with exponents and stuff. So remember, we had the future value, so our formula was:

P V = F V ∕ ( 1 + r ) n

Notice how I highlighted the 1+r)n, because instead of doing the dividing by 1+r)n, we're going to have our future value and then we're going to multiply by the present value factor. And these present value factors, they come from the table, okay? So you'll see in the last page of this lesson, I've included the present value tables for lump sums and for annuities, okay? So we'll go over those before we're done here and before we get to the example. So notice, it's very simple. Instead of doing all of those calculations, what we have is the future value times the present value factor. And we're just going to get some number out of the table and multiply it by the future value and that's it. We found our present value.

Same thing with the annuities. Except in this case, we're going to use the annuity payment instead of the future value. So say you were going to get $10,000 a year for 5 years. Well, that would be the annuity payment is $10,000. We wouldn't add them all together and say $10,000 for 5 years, that's $50,000 total dollars. No, no, no. We're going to take the amount of each payment, which is $10,000, and that's going to be the amount in this formula, annuity payment and we're going to find the number in the table that we're going to multiply it by. Okay? So let's pause here and then we're going to go to the tables and we're going to do a quick explanation of how to read the tables and we'll come back to the example. So flip over to the other page where you've got the tables listed. Alright? Let's do that.

Time Value of Money Table:Present Value of Lump-Sum

All right. So here we go. This is the first of our two tables and this is for finding the present value of $1. When they say the present value of $1, this is the table for lump sums. Okay? The table for lump sum payments. So remember, we're looking for the present value factor and all these numbers in here, these are the present value factors all listed throughout. I missed when I was circling. All these different numbers, these are the present value factors. So it's all about being able to pick out which is the correct number. And notice, as we go down the table, down this way, we've got different n's. Remember we had those variables n for number of periods and r for interest rates. Well, this way going left to right, those are our interest rates, r, right? So we've got the number of periods going down and the interest rate going left to right. So what we're going to do is we're going to analyze our problem and we're going to find out what our interest rate is, what our number of periods is and then we'll go to the table. Say that we find out that there's 10 periods and the interest rate is 7%. So we would go to 10 periods and 7%. Let me do it in a different color. 10 periods and 7%. And we would go in our table and we would find what that present value factor is. So then we would use our formula, which is fine. Take the future value, say we wanted to know what a $1,000 10 years from now is worth at an interest rate of 7%. Well, we would take that $1,000 and multiply it by this present value factor in the table. $1,000 times 0.508 will tell us what a $1,000 10 years from now is worth at a rate of 7%. Okay? So that's how we're going to read this table. The important factors that we need are our number of periods and our interest rate. Okay? Those are the key to being able to read this table. Alright? Let's pause and let's move on to the next table.

Time Value of Money Table:Present Value of Annuity

So here we have a very similar-looking table, except notice the title. It says present value of an ordinary annuity of $1. Okay? So since this is an ordinary annuity of $1, this is your table for annuities. Okay? So it's very easy to tell which table is for annuities because it's going to say annuity in the title, alright? But why does it say of $1? Well, that's because this is what $1, if the annuity was just $1, this is what it would be worth. And that's why we multiply it by the annuity payment because it's usually not just $1 it's going to be some bigger amount of money. Okay? So in the same way, we read this table the same way. We're going to go into our problem and we're going to analyze what is our n. Notice we've got n's going down again, the number of periods. And we've got interest rates going across. Okay? So we're going to have to analyze our problem to find the interest rate and the number of periods. And then we would dive into this table to find what our present value is that we're going to multiply by. Okay? So now that you've seen the tables, one of the biggest tricks is knowing which table to use. Right? When you're dealing with an annuity, you want to make sure you're using the annuity table. When you're dealing with a lump sum, which is just one amount of money, right? That's when you use the other table, the lump sum table above. Present value of $1. So the lump sum is just $1. Right? Here, we're talking about an annuity. Getting $1 for several periods at equal intervals. Alright? So I think the best way to learn this is with an example. So let's go back to our first page of the lesson and let's do an example related to this.

Using Time Value of Money Tables

All right. Let's go ahead and get some practice using our present value tables. All right? Your grandmother is giving you a graduation gift, but has given you 2 options. She's either going to give you $5,000 one year from now. So she doesn't want to cough up any money right now. She'll give you $5,000 one year from now. Or she'll give you $1,000 per year for the next 6 years. And the first payment is occurring 1 year from now. Right? And that's how we typically expect our annuities is that they start 1 year from now.

Okay? So notice what we've got. In the first option, she's going to give you $5,000. One payment, one time. This is a lump sum. Right? Lump sum and we're dealing with What what are the what are the factors in this first one? So let's deal with these one at a time. And I guess we'll draw a timeline just to see what's going on here. So this is right now 0, and this is 1, 1 year in the future, where she's going to give us $5,000 and we need to bring that $5,000 back in time to today to see what it's worth today. Right? We're trying to see which is the better option. Are we going to take option 1 or option 2? It's basically going to be the one with the higher present value. So this $5,000 1 year from now, it tells us the interest rate here is 10%. So 10% is our r. So when we go to our table, we're going to have r equal to 10% and we're going to have N equal to 1, right? Because it's 1 year in the future at a rate of 10%.

Okay? So remember, when we use our formula, we want to take that future value of 5,000 and multiply it by the present value factor that we get out of the table. So PV=5000×present value factor. So now let's go down to the table and find the present value factor for a rate of 10% and an n of 1. Okay? I'm going to scroll down to the table and I'm going to ask you which table are we going to use? Are we going to use the top table, which is the present value of $1 or the bottom table the present value of an ordinary annuity of $1? We're going to use the top table, right? Because this is a lump sum. She's paying us one amount of money at one point in time and that's it. There's not going to be a stream of payments. Remember, an annuity involves a stream of payments. So here, what we need to do is use our variables. It told us that it was one period at an interest rate of 10%, so we need to find that number. And that's pretty easy. It's right here. 0.909. That is our present value factor in this case. 0.909. So let's go back up to our problem and let's fill in our 0.909 into our present value factor. Okay? So here we are. 5000 times 0.909. This is going to tell us what the present value of $5,000 at an interest rate of 10% for 1 period is. 5000 times 0.909, that comes out to $4,545, right? $4,545 that is what $5,000 is worth 1 year, today if the $5,000 was received 1 year from now. Okay? So that's the present value of option Draw it a little longer because there's a few more payments going on. Draw it a little longer because there's a few more payments going on.

123 4, 5 and then 6, right? And here we are today. So what she gonna do? She's gonna give us a 1,000 per year for 6 years. So what do we have? Each year starting next year, a1000, a 1000 each year, right? So it's gonna be 6 payments of $1,000 rather than one payment of $5,000. That already sounds pretty enticing, right? We're gonna get it sounds like we're gonna get $6,000 rather than $5,000 Sounds pretty nice. But let's go ahead and see what the present value of this annuity is, right? In this case, we have an annuity, right? We have a stream of payments rather than just one payment, and they're happening at equal intervals each year, a $1,000 each year in this case for 6 years. So we want to bring that and find the present value of this annuity. Okay? So we're going to use our formula that we had above, it's very simple, right? We're gonna do PV=Pt×present value factor and our annuity payment in this case, what is the annuity payment? It's $1,000, right? We don't add them all together. It's just the $1,000 that we're going to get each period. So it's going to be a 1,000 times the present value factor. So let's go ahead and go to our table and find out what that's going to be. So notice, what is our R and our N in this case? Before we go to the table, we need to know what our R and our N is. R is the same, right? The interest rate didn't change from each option, so the r is still 10% and n, what's n in this case? It's not just 1, right? There's not just 1 year going on.

You have won the lottery! You are given two options for your payout:

1. You can receive $540,000 today

2. You can receive $50,000 annually each year for the next twenty years

Assuming the interest rate is 6%, which is the better option?

Alright. Let's see how you guys did. You have won the lottery. You are given 2 options for your payout. And this is typically how lottery payouts go. So this is a very useful, time value of money application for when you win the lottery. Alright? So you either receive $540,000 today or you can receive $50,000 annually each year for the next 20 years. And that's usually how they do it. They'll say, "Lottery, $1,000,000." But really what it means is that they're not going to just give you $1,000,000. They're going to give you $50,000 for 20 years. And that's what a $1,000,000 payout is for a lottery. Okay? Still sounds pretty good. Hey, I would take it. But when you know the time value of money, you know that that's not exactly worth $1,000,000. Alright? So let's go ahead and see what's the better option. Should you take the $540,000 today or the $50,000 annually for the next 20 years? So we've got an interest rate in this case of 6%. So what do you see about option 1? We already know the present value of option 1, right? They told us. It's $540,000 today. So what is the present value of $540,000 today? It's $540,000, right? If I handed you $540,000 right now, you'd say, "this is worth $540,000," because it's in your hands right now. It's already the present. So the big deal here is to see if option 2 is worth more or less than the option 1. The lump sum payment of $540,000. So let's go ahead and deal with option 2 here. So we're going to have a timeline, and yes, it's going to be a really long timeline because it's 20 years. What I'm going to do is like a dot dot dot here, and then we'll get to the end. So we've got 0, 1, 2, 3, ... because nothing changes in all those years, 19, 20. Okay? So what's going to happen is you're going to get $50,000 per year. 50 k, I'm going to put 50 k for 1,000, and that's going to be every year. So year 4, 5, 6, all of those other years, you're going to be getting $50,000, and we want to know what that stream of payments is worth today. What does it look like to you? Looks like an annuity, right? You're getting the same amount of money in equal periods, equal time passing between each payment. So we have an annuity of $50,000 for 20 years. So we're going to use our table. Let's find the present value of this annuity using our table. So remember, what is the annuity payment? It's $50,000 in this case and we're going to multiply that by our present value factor which comes out of the table, right? So before we go to the table, we need to know 2 things. We need to know our n, number of periods and we need to know our r, interest rate. In this case, n is going to be 20, right? We're dealing with 20 equal payments over 20 years. And what's our r in this case? R is the 6%, right? The 6% assumed interest rate that they told us in the problem. Alright? So knowing those two things, we're ready to go to our table. So let's go ahead and go to that sheet and which table are we going to use? Are we going to use the present value of $1 or the present value of the ordinary annuity of $1? We're going to use the ordinary annuity, right? We just talked about how this is an annuity, so we use our annuity table to find the value. So in this case, we've got an n is going to be 20, right? We've got 20 periods and it's 6%, right? Our interest rate was 6%, so we need to find that on the table, and that's going to be right here. 11.470. That is our present value factor for an annuity at 20 years at a rate of 6%. So let's go ahead and move back to our problem and we will see, the finale here. So $50,000 times 11.470, it comes out to $573,500. Okay. So notice what we've got here. We've got option 1, $540,000 today, or option 2 is a present value of these annuity payments of $573,500. So in essence, option 2 is the better option, right? You're going to get more present value out of taking these annuity payments over the 20 years. And this is usually what happens when you have the lottery and they make these payments to you. It's generally going to be some sort of equation whereby paying you over the next 20 years, you're going to end up getting more money than just taking a lump sum now. But what do people love? Money now. So they generally take that cut that they can have that money right now. But now you know what's actually worth more, right? Taking those payments over the 20-year period. So you can always use your present value tables once you win the lottery, and you can go ahead and figure out what's your best payment schedule once you find out what they're going to pay you. Alright? So congrats here, let's go ahead and move on to the next page.

Time Value of Money Tables:Bonds Payable

Alright, so now let's relate everything we've been talking about with these tables, back to bonds payable, right? Like I said, this is going to be the main way that we use time value of money in this class is with bonds payable. So remember, with a bond, there are 2 streams of cash flow. The first one is the annual interest payments, right? Sometimes we're going to deal with semiannual interest payments, which just means 2 per year, every 6 months. Right? It's the same thing as saying every 6 months we're going to pay interest is semiannual interest. So notice these are going to be happening every 6 months or every year consistently and they're going to be the same amount of money. So these are going to be an annuity, but we're also going to deal with the principal payment. At the end of the bond's life, they're going to have to pay back the principal of the bond, and that's going to be a lump sum payment. They're going to pay back the entire amount of the bond's principal as a lump sum, once the bond matures, which is usually going to be 5 years from now, 10 years from now, something like that. Okay?

Now, a very important thing when we're dealing with bonds is that we're going to have 2 interest rates that we're dealing with when we're finding the interest payments of the bond. But we're also going to have the market interest rate. Okay? So the first one is what we call the stated interest rate. And the stated interest rate is used to calculate the actual cash payments of interest. Okay? The bond has some interest rate attached to it. You're going to sell a bond that says, "Hey, we're going to pay 6% interest." Well, that is going to be the stated interest rate of the bond. Now, that stated interest rate does not have to be the same as the market interest rate. You can sell bonds at whatever interest rate you want, right? You can say, "Hey, I'm paying 5% interest right now. Do you want to buy my bond?" Well, the value that people place on your interest rate, well it's got to compare to the market interest rate, right? What if you're offering 5% on bonds when everyone else offering similar bonds is offering 8% interest? And yours is only 5% interest? Well, people are more likely going to buy the 8% bonds, right? Because they pay more. So we're going to see how this works in an example.

One more note that I want to mention is that if the bond pays interest semiannually, well, what we're going to have to do before we go to the tables because we're still going to be using our tables, right? Notice, we're going to be dealing with annuities where we have our annuity table and lump sums because we have our lump sum table as well, right? So, up to this point, we’ve been dealing with things that happen annually. But what if they pay semiannually? Well, the interest rates that you are given are always annual interest rates. You are always given annual interest rates. But if interest is paid semiannually, what we're going to have to do is we're going to have to divide the interest rate by 2. So this is our r. Remember, r is going to be divided by 2 to find the semiannual rate. And we're going to have to multiply the number of years by 2. This is our n, right? Our n is going to be multiplied by 2 before we go to the tables. And that's usually the biggest trick before you get to the table is having to divide by 2 for the r and multiply by 2 for the n, right? That makes sense because if there's 10 years, well that means there are 20 six-month periods, right? And if it's 10% interest per year, well then it means there's 5% interest per half-year, per 6 months. Okay? So let's go ahead and pause here and then we're going to do an example related to this.

TVM Tables and Bonds Payable

All right. Let's check out this example. Your company, Inc. issues $100,000 of 10% bonds due in 10 years. The bonds pay interest semi-annually, and the current market rate of interest is 8%. What is the present value, the current selling price of the bonds? 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 are issuing $100,000 worth of bonds, and they pay a 10 percent interest. So note this 10%, this is the stated interest rate, right? And they're due in 10 years. So this is the number of years, but note what happened in this problem is we're dealing with semi-annual. So we're going to have to divide these things by 2, as mentioned above, right? Because the periods are half, not full years. But note we also have one more interest rate here. 8% is the market rate. Okay? So this is where things start getting a little complicated, as 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 going to pay a 10% interest, so that's how we calculate the cash payments. There's going to be cash interest payments, and that's based on this 10% interest. So let's find out what that cash interest is going to be. The cash interest each semi-annual period will be $100,000 times 10%, right? 0.10. But remember, it's semi-annual. Right? So as we said, we have to divide by 2 when we're dealing with semi-annual periods. So we're going to 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 6 months. $100,000 times 0.1, times half, right, divided by 2. So that's going to tell us that we're going to pay $5,000 in interest every 6 months. Right? Every semi-annual period, we're gonna pay $5,000 in interest, which totals $10,000 per year, the 10% per year. Okay? So let's go ahead and see this on a timeline. So I'm going to cut out some of these periods here because they're going to be the same, and then we'll get to the end there. So remember, when we make our timeline here, we're not going to do it in years. We're going to do it in semi-annual periods, so in half years. So this is now 0. This is 1. This is 2. And this is not 1 year. This is 1 semi-annual period. Right? Semiannual, this is 2 semi-annual periods from now. So that's technically 1 year from now is the 2, in this case, right? Are you following me? We have to stay in semi-annual 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 that we're paying the cash out every semi-annual period. And in this case, it's $5,000 being paid out each semi-annual period. So $5,000 in interest every six months for those 20 semi-annual periods over the next 10 years. Okay? So that's going to be the $5,000 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 tenure, which is the 20 semiannual 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 us $100,000. We sold these bonds that said in 10 years we're going to pay you a $100,000 plus the interest. So we got to find what those were worth today. This is the principal, and that's going to a lump sum. So every time we deal with bonds payable, it comes down to this. We're going to 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 principal, which is a lump sum at the end of 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 our n and our r are going to be. Okay? So here is the annuity, and we're going to find the present value of the annuity, and then we need to find the present value of the principal as well. So what's going to be our n and what's going to be our r 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 2 semiannual periods per year, comes out to 20 for our n. And how about our r? Our interest payment, our interest rate. So let me move out of the way here. Our interest rate, what are we going to use for our interest rate? They gave us 2 interest rates in this problem. Remember when we first introduced interest rates, we always said that the r is going to 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 note how we used 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 semiannual periods, well, it's not 8% per, it's 8% per year, so it's 4% per 6 months. So there we go. We've got our n and our r, we've got n of 20, r of 4. We're ready to go to our table. Okay? And we're going to go to our table for 2 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 $5,000, the amount of the annuity payment, right? The interest payment times the present value factor from that from the table for 20 at 4%. So let's go ahead and do the annuity 1 first, 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. Let me erase this previous problem's data. And what are we doing here? Well, we said it's 4% interest per semiannual period for 20 semi-annual periods, and that gets us right here, 13.590. 13.590, so that's what we're going to 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 going to be $5,000 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 issue. 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 going to use the other table to find that. The present value of the principal is equal to the $100,000, and we're trying to find out what that lump sum is worth today times the present value factor. Right? So we're going to need to go to the table again, but we've already done the hard work. We know what our n is, we know what our r 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 10 years from now in 20 semiannual periods. Okay. So let's erase this from the last problem. And what do we have? Our n was 20 because there are 20 semiannual periods. Our interest rate is 4%. So we go down here, and we find that we're going to use 0.456 as our present value factor. So let's go ahead and bring that up here, and we're going to put it equals, and for our present value factor. Oops. 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 0.456, it comes out to $45,600. So that is the present value of our principal payment. So that principal payment 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 on to our final step. And all we got to 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, $67,950 plus $45,600 that tells us that the bond today is worth $113,550. This is the present value of the bond, okay?

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?

Alright. Let's see how you guys did. This stuff's pretty tricky, right? ABC company issues 1,000 bonds with a face value of $1,000 each, maturing in 8 years. The bonds pay 8% interest semiannually and the current market rate of interest is 12 percent. 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 told you the principal total principal? The total principal because there's a 1,000 of these bonds, well, the bonds are going to be 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 million, right? There are 1,000 bonds worth $1,000 each. Well, we're issuing $1,000,000 worth of bonds. Okay? They're maturing in 8 years, but again, we're dealing with semiannual periods. They pay 8% interest. So which interest rate is this? Remember, 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. Alright. So right here, 0 is right now and each of these is a semiannual period. Right? We're talking about semiannual periods. So it's not going to be 8 years, it's going to be 16 half-years, right? We need to take the 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 $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 amount, which is $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 times the 8% interest, 0.08, that's going to give us the amount of interest that we pay, okay? So a million times 0.08 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 semiannual 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 semiannual periods. So we need to divide that annual interest by 2 to find the semiannual payment. Okay? So it's $40,000 in interest paid every semiannual 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 going to look like this, $40,000 each period. And then at the very end, in 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've 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 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 going to be when we go through the table. So n, like we see in our timeline, it's going to be 16, n. Right? We're going to the table with an n of 16 because it's 8 years 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 periods. So it's 16. What about r? What's going to 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%, 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 gonna 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. 404,240. 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 $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 = 16 periods, r = 6% rate. So we'll go to our lump sum table. To 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 0.394. So that's the number we're gonna bring up here, and let's go ahead and put that in. 0.3 9 four is our present value factor. So we've got $1,000,000 times 0.394 and that's worth $394,000. Alright, 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 going to be those together. So $404,240 plus $394,000 that comes out to $798,240 and that is the present value of the bond, and that's the amount of cash that this is going to bring in. Okay? And the 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.

Here’s what students ask on this topic:

How do you use time value of money tables to find the present value of a lump sum?

To find the present value of a lump sum using time value of money tables, you need to know the future value, the interest rate, and the number of periods. First, locate the appropriate table for the present value of $1. Then, find the intersection of the interest rate (r) and the number of periods (n) to get the present value factor. Multiply the future value by this factor to get the present value. The formula is:

$\mathrm{PV}=\mathrm{FV}\times \mathrm{PVF}$

where PV is the present value, FV is the future value, and PVF is the present value factor from the table.

What is the difference between the present value of a lump sum and an annuity?

The present value of a lump sum refers to the current worth of a single future payment, while the present value of an annuity refers to the current worth of a series of equal payments made at regular intervals. For lump sums, you use the present value of $1 table, and for annuities, you use the present value of an ordinary annuity of $1 table. The key difference lies in the nature of the cash flows: a single payment versus multiple periodic payments.

How do you read a present value table for annuities?

To read a present value table for annuities, first identify the number of periods (n) and the interest rate (r) from your problem. The table will have periods listed vertically and interest rates listed horizontally. Find the intersection of the row corresponding to the number of periods and the column corresponding to the interest rate. This intersection gives you the present value factor. Multiply the annuity payment by this factor to find the present value of the annuity.

Why is it important to adjust the interest rate and periods for semi-annual payments when using time value of money tables?

When dealing with semi-annual payments, it is crucial to adjust the interest rate and periods because the tables are typically based on annual rates and periods. For semi-annual payments, divide the annual interest rate by 2 and multiply the number of years by 2. This adjustment ensures that the calculations accurately reflect the more frequent compounding and payment intervals. For example, if the annual interest rate is 6% and the bond term is 10 years, use 3% (6%/2) and 20 periods (10 years × 2) in the table.

How do you determine which present value table to use for a given problem?

To determine which present value table to use, identify whether the problem involves a lump sum or an annuity. If the problem involves a single future payment, use the present value of $1 table (lump sum). If the problem involves a series of equal payments at regular intervals, use the present value of an ordinary annuity of $1 table. The key is to recognize the type of cash flow: a single payment versus multiple periodic payments.