The Hewlett Packard 10BII The Hewlett Packard 10BII is a very easy to use financial. In this tutorial, I will demonstrate how to use the financial functions to handle time value of money problems and make financial math easy. I will keep the examples rather elementary, but understanding the basics is all that is necessary to learn the calculator.
The orange key is referred to as Shift because it is used to shift to the orange-colored function below the key that is pressed next.
Initial Setup Before we get started, letâ€™s change the calculator settings. The 10BII comes from the factory set to assume monthly compounding, but itâ€™s better to set it to assume annual compounding and then make manual adjustments when you enter numbers. Why? Well, the compounding assumption is hidden from view and in my experience people tend to forget to set it to the correct assumption. To fix this, press 1 (once per year) then Shift and finally PMT. To check that it has taken, press Shift and then C (clear all). You should see 1 p yr on the screen. Problem solved. Now, just make sure that you always enter the total number of periods (not necessarily years) into N, the per period interest rate into I/YR, and the per period payment into PMT. By default the 10BII displays only two decimal places. To change the display, press Shift =, and finally the numeric key that corresponds to the number of digits you would like to see displayed. I would press Shift = 5 to display 5 decimal places. That's it; the calculator is ready to go.
Example 1 - Lump Sums We'll begin with a very simple problem that will provide you with most of the skills to perform financial math on the 10BII: Suppose that you have $100 to invest for a period of 5 years at an interest rate of 10% per year. How much will you have accumulated at the end of this time period? In this problem, the $100 is the present value (PV), N is 5, and i is 10%. Before entering the data you need to make sure that the financial registers (each key is nothing more than a memory register) are clear. Otherwise, you may find that numbers left over from previous problems will interfere with the solution to this one. Press Shift C to clear the memory. Now all we need to do is enter the numbers into the appropriate keys: 5 into N, 10 into I/YR, -100 into PV. Now to find the future value simply press the FV key. The answer you get should be 161.05.
A Couple of Notes 1.
Every time value of money problem has either 4 or 5 variables. Of these, you will always be given 3 or 4 and asked to solve for the other. In this case, we have a 4-variable problem and were given 3 of them (N, i, and PV) and had to solve for the 4th (FV). To solve these problems you simply enter the variables that you know in the appropriate keys and then press the other key to get the answer. The order in which the numbers are entered does not matter.
When we entered the interest rate, we input 10 rather than 0.10. This is because the calculator automatically divides any number entered into I/YR by 100. Had you entered 0.10, the future value would have come out to 100.501 — obviously incorrect.
Notice that we entered the 100 in the PV key as a negative number. This was on purpose. Most financial calculators (and spreadsheets) follow the Cash Flow Sign Convention. This is simply a way of keeping the direction of the cash flow straight. Cash inflows are entered as positive numbers and cash outflows are entered as negative numbers. In this problem, the $100 was an investment (i.e., a cash outflow) and the future value of $161.05 would be a cash inflow in five years. Had you entered the $100 as a positive number no harm would have been done, but the answer would have been returned as a negative number. This would be correct had you borrowed $100 today (cash inflow) and agreed to repay $161.05 (cash outflow) in five years.
We can change any of the variables in this problem without needing to re-enter all of the data. For example, suppose that we wanted to find out the future value if we left the money invested for 10 years instead of 5. Simply enter 10 into N and solve for FV. You'll find that the answer is 259.37.
Example 2 — Present Value of Lump Sums Solving for the present value of a lump sum is nearly identical to solving for the future value. One important thing to remember is that the present value will always (unless the interest rate is negative) be less than the future value. Keep that in mind because it can help you to spot incorrect answers due to a wrong input. Let's try a new problem: Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you believe that you can earn an average annual rate of return of 8% per year, how much money would you need to invest today as a lump sum to achieve your goal? In this case, we already know the future value ($100,000), the number of periods (18 years), and the per period interest rate (8% per year). We want to find the present value. Enter the data as follows: 18 into N, 8 into I/YR, and 100,000 into FV. Note that we enter the $100,000 as a positive number because you will be withdrawing that amount in 18 years (it will be a cash inflow). Now press PV and you will see that you need to invest $25,024.90 today in order to meet your goal. That is a lot of money to invest all at once, but we'll see on the next page that you can lessen the pain by investing smaller amounts each year.
Example 3 — Solving for the Number of Periods
Sometimes you know how much money you have now, and how much you need to have at an undetermined future time period. If you know the interest rate, then we can solve for the amount of time that it will take for the present value to grow to the future value by solving for N. Suppose that you have $1,250 today and you would like to know how long it will take you double your money to $2,500. Assume that you can earn 9% per year on your investment. This is the classic type of problem that we can quickly approximate using the Rule of 72. However, we can easily find the exact answer using the HP 10BII calculator. Enter 9 into I/YR, 1250 into PV, and 2500 into FV. Now solve for N and you will see that it will take 8.04 years for your money to double. One important thing to note is that you absolutely must enter your numbers according to the cash flow sign convention. If you don't make either the PV or FV a negative number (and the other one positive), then you will get No Solution on the screen instead of the answer. That is because, if both numbers are positive, the calculator thinks that you are getting a benefit without making any investment. If you get this error, just press C to clear it and then fix the problem by changing the sign of either PV or FV.
Example 4 â€” Solving for the Interest Rate Solving for the interest rate is quite common. Maybe you have recently sold an investment and would like to know what your compound average annual rate of return was. Or, perhaps you are thinking of making an investment and you would like to know what rate of return you need to earn to reach a certain future value. Let's return to our college savings problem from above, but we'll change it slightly. Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you have $20,000 to invest today, what compound average annual rate of return do you need to earn in order to reach your goal? As before, we need to be careful when entering the PV and FV into the calculator. In this case, you are going to invest $20,000 today (a cash outflow) and receive $100,000 in 18 years (a cash inflow). Therefore, we will enter -20,000 into PV, and 100,000 into FV. Type 18 into N, and then press I/YR to find that you need to earn an average of 9.35% per year. Again, if you get No Solution instead of an answer, it is because you didn't follow the cash flow sign convention. Note that in our original problem we assumed that you would earn 8% per year, and found that you would need to invest about $25,000 to achieve your goal. In this case, though, we assumed that you started with only $20,000. Therefore, in order to reach the same goal, you would need to earn a higher interest rate. When you have solved a problem, always be sure to give the answer a second look and be sure that it seems likely to be correct. This requires that you understand the calculations that the calculator is doing and the relationships between the variables. If you don't, you will quickly learn that if you enter wrong numbers you will get wrong answers. Remember, the calculator only knows what you tell it, it doesn't know what you really meant. Please continue on to part II of this tutorial to learn about using the HP 10BII to solve problems involving annuities and perpetuities.
In the previous section we looked at the basic time value of money keys and how to use them to calculate present and future value of lump sums. In this section we will take a look at how to use the HP 10BII to calculate the present and future values of regular annuities and annuities due. A regular annuity is a series of equal cash flows occurring at equally spaced time periods. In a regular annuity, the first cash flow occurs at the end of the first period. An annuity due is similar to a regular annuity, except that the first cash flow occurs immediately (at period 0).
Example 5 — Present Value of Annuities Suppose that you are offered an investment which will pay you $1,000 per year for 10 years. If you can earn a rate of 9% per year on similar investments, how much should you be willing to pay for this annuity? In this case we need to solve for the present value of this annuity since that is the amount that you would be willing to pay today. Press Shift C to clear the financial keys. Enter the numbers into the appropriate keys: 10 into N , 9 into I/YR , and 1000 (cash inflow) into PMT . Now press PV to solve for the present value. The answer is -6,417.6577. Again, this is negative because it represents the amount you would have to pay (cash outflow) to purchase this annuity.
Example 6 — Future Value of Annuities Now, suppose that you will be borrowing $1000 each year for 10 years at a rate of 9%, and then paying back the loan immediate after receiving the last payment. How much would you have to repay? All we need to do is to put a 0 into PV to clear it out, and then press is -15,192.92972 ( a cash outflow).
to find that the answer
Example 7 — Solving for the Payment Amount We often need to solve for annuity payments. For example, you might want to know how much a mortgage or auto loan payment will be. Or, maybe you want to know how much you will need to save each year in order to reach a particular goal (saving for college or retirement perhaps). On the previous page, we looked at an example about saving for college. Let's look at that problem again, but this time we'll treat it as an annuity problem instead of a lump sum: Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you believe that you can earn an average annual rate of return of 8% per year, how much money would you need to invest at the end of each year to achieve your goal? Recall that we previously determined that if you were to make a lump sum investment today, you would have to invest $25,024.90. That is quite a chunk of change. In this case, saving for college
will be easier because we are going to spread the investment over 18 years, rather than all at once. (Note that, for now, we are assuming that the first investment will be made one year from now. In other words, it is a regular annuity.) Let's enter the data: Type 18 into N , 8 into I/YR , and 100,000 into FV . Now, press PMT and you will find that you need to invest $2,670.21 per year for the next 18 years to meet your goal of having $100,000.
Example 8 — Solving for the Number of Periods Solving for N answers the question, "How long will it take..." Let's look at an example: Imagine that you have just retired, and that you have a nest egg of $1,000,000. This is the amount that you will be drawing down for the rest of your life. If you expect to earn 6% per year on average and withdraw $70,000 per year, how long will it take to burn through your nest egg (in other words, for how long can you afford to live)? Assume that your first withdrawal will occur one year from today (End Mode). Enter the data as follows: 6 into I/YR , -1,000,000 into PV (negative because you are investing this amount), and 70,000 into PMT . Now, press N and you will see that you can make 33.40 withdrawals. Assuming that you can live for about a year on the last withdrawal, then you can afford to live for about another 34.40 years.
Example 9 — Solving for the Interest Rate Solving for I/Y works just like solving for any of the other variables. As has been mentioned numerous times in this tutorial, be sure to pay attention to the signs of the numbers that you enter into the TVM keys. Any time you are solving for N, I/YR, or PMT there is the potential for a wrong answer or error message if you don't get the signs right. Let's look at an example of solving for the interest rate: Suppose that you are offered an investment that will cost $925 and will pay you interest of $80 per year for the next 20 years. Furthermore, at the end of the 20 years, the investment will pay $1,000. If you purchase this investment, what is your compound average annual rate of return? Note that in this problem we have a present value ($925), a future value ($1,000), and an annuity payment ($80 per year). As mentioned above, you need to be especially careful to get the signs right. In this case, both the annuity payment and the future value will be cash inflows, so they should be entered as positive numbers. The present value is the cost of the investment, a cash outflow, so it should be entered as a negative number. If you were to make a mistake and, say, enter the payment as a negative number, then you will get the wrong answer. On the other hand, if you were to enter all three with the same sign, then you will get an error message, Let's enter the numbers: Type 20 into N , -925 into PV , 80 into PMT , and 1000 into FV . Now, press I/YR and you will find that the investment will return an average of 8.81% per year. This particular problem is an example of solving for the yield to maturity (YTM) of a bond.
Example 10 — Annuities Due
In the examples above, we assumed that the first payment would be made at the end of the year, which is typical. However, what if you plan to make (or receive) the first payment today? This changes the cash flow from from a regular annuity into an annuity due. Normally, the calculator is working in End Mode. It assumes that cash flows occur at the end of the period. In this case, though, the payments occur at the beginning of the period. Therefore, we need to put the calculator into Begin Mode. To change to Begin Mode, press Shift MAR (note that the key says BEG/END in orange). The screen will now show BEGIN at the bottom. Note that nothing will change about how you enter the numbers. The calculator will simply shift the cash flows for you. Obviously, you will get a different answer. Let's do the college savings problem again, but this time assuming that you start investing immediately: Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you believe that you can earn an average annual rate of return of 8% per year, how much money would you need to invest at the beginning of each year (starting today) to achieve your goal? As before, enter the data: 18 into N , 8 into I/YR , and 100,000 into FV . The only thing that has changed is that we are now treating this as an annuity due. So, once you have changed to Begin Mode, just press PMT . You will find that, if you make the first investment today, you only need to invest $2,472.42. That is about $200 per year less than if you make the first payment a year from now because of the extra time for your investments to compound. Be sure to switch back to End Mode after solving the problem. Since you almost always want to be in End Mode, it is a good idea to get in the habit of switching back. Press Shift MAR . When in End Mode, the bottom of the screen will be blank.
Example 11 â€” Perpetuities Occasionally, we have to deal with annuities that pay forever (at least theoretically) instead of for a finite period of time. This type of cash flow is known as a perpetuity (perpetual annuity, sometimes called an infinite annuity). The problem is that the HP 10BII has no way to specify an infinite number of periods using the N key. Calculating the present value of a perpetuity using a formula is easy enough: Just divide the payment per period by the interest rate per period. In our example, the payment is $1,000 per year and the interest rate is 9% annually. Therefore, if that was a perpetuity, the present value would be: $11,111.11 = 1,000 Ăˇ 0.09 If you can't remember that formula, you can "trick" the calculator into getting the correct answer. The trick involves the fact that the present value of a cash flow far enough into the future (way into the future) is going to be approximately $0. Therefore, beyond some future point in time the cash flows no longer add anything to the present value. So, if we specify a suitably large number of payments, we can get a very close approximation (in the limit it will be exact) to a perpetuity.
Let's try this with our perpetuity. Enter 500 into N (that will always be a large enough number of periods), 9 into I/YR , and 1000 into PMT . Now solve for PV and you will get $11,111.11 as your answer. Please note that there is no such thing as the future value of a perpetuity because the cash flows never end (period infinity never arrives). In the previous section we looked at the basic time value of money keys and how to use them to calculate present and future value of lump sums and regular annuities. In this section we will take a look at how to use the HP 10BII to calculate the present and future values of uneven cash flow streams. We will also see how to calculate net present value (NPV), internal rate of return (IRR), and the modified internal rate of return (MIRR).
Example 12 â€” Present Value of Uneven Cash Flows In addition to the previously mentioned financial keys, the 10BII also has a key labeled CFj to handle a series of uneven cash flows. Suppose that you are offered an investment which will pay the following cash flows at the end of each of the next five years:
How much would you be willing to pay for this investment if your required rate of return is 12% per year? We could solve this problem by finding the present value of each of these cash flows individually and then summing the results. However, that is the hard way. Instead, we'll use the cash flow key (CFj). All we need to do is enter the cash flows exactly as shown in the table. Again, clear the financial keys first. Now, press 0 CFj, 100 CFj, 200 CFj, 300 CFj, 400 CFj, and finally 500 CFj. Now, enter 12 into I/YR and then press Shift NPV. We find that the present value is $1,000.17922.
Example 13 â€” Future Value of Uneven Cash Flows Now suppose that we wanted to find the future value of these cash flows instead of the present value. There is no key to do this so we need to use a little ingenuity. Realize that one way to find the future value of any set of cash flows is to first find the present value. Next, find the future value of that present value and you have your solution. In this case, we've already determined that the present value is $1,000.17922. Clear the financial keys (Shift C) then enter -1000.17922 into PV. N is 5 and I/YR is 12. Now press FV and you'll see that the future value is $1,762.65753.
Pretty easy, huh? (Ok, at least its easier than adding up the future values of each of the individual cash flows.)
Example 14 — Net Present Value (NPV) Calculating the net present value (NPV) and/or internal rate of return (IRR) is virtually identical to finding the present value of an uneven cash flow stream as we did in Example 12. Suppose that you were offered the investment in Example 3 at a cost of $800. What is the NPV? IRR? To solve this problem we must not only tell the calculator about the annual cash flows, but also the cost. Generally speaking, you'll pay for an investment before you can receive its benefits so the cost (initial outlay) is said to occur at time period 0 (i.e., today). To find the NPV or IRR, first clear the financial keys and then enter -800 into CFj, then enter the remaining cash flows exactly as before. For the NPV we must supply a discount rate, so enter 12 into I/YR and then press Shift PRC (note that below the PRC key is NPV in orange). You'll find that the NPV is $200.17922.
Example 15 — Internal Rate of Return Solving for the IRR is done exactly the same way, except that the discount rate is not necessary because that is the variable for which we are solving. This time, you'll press Shift CST and find that the IRR is 19.5382%.
Example 16 — Modified Internal Rate of Return The IRR has been a popular metric for evaluating investments for many years — primarily due to the simplicity with which it can be interpreted. However, the IRR suffers from a couple of serious flaws. The most important flaw is that it implicitly assumes that the cash flows will be reinvested for the life of the project at a rate that equals the IRR. A good project may have an IRR that is considerably greater than any reasonable reinvestment assumption. Therefore, the IRR can be misleadingly high at times. The modified internal rate of return (MIRR) solves this problem by using an explicit reinvestment rate. Unfortunately, financial calculators don't have an MIRR key like they have an IRR key. That means that we have to use a little ingenuity to calculate the MIRR. Fortunately, it isn't difficult. Here are the steps in the algorithm that we will use: 1. Calculate the total present value of each of the cash flows, starting from period 1 (leave out the initial outlay). Use the calculator's NPV function just like we did in Example 3, above. Use the reinvestment rate as your discount rate to find the present value. 2. Calculate the future value as of the end of the project life of the present value from step 1. The interest rate that you will use to find the future value is the reinvestment rate. 3. Finally, find the discount rate that equates the initial cost of the investment with the future value of the cash flows. This discount rate is the MIRR, and it can be interpreted as the compound average annual rate of return that you will earn on an investment if you reinvest the cash flows at the reinvestment rate.
Suppose that you were offered the investment in Example 3 at a cost of $800. What is the MIRR if the reinvestment rate is 10% per year? Let's go through our algorithm step-by-step: 1. The present value of the cash flows can be found as in Example 3. Clear the TVM keys and then enter the cash flows (remember that we are ignoring the cost of the investment at this point): press Shift C to clear the cash flow keys. Now, press 0 then CFj, 100 CFj, 200 CFj, 300 CFj, 400 CFj, and finally 500 CFj. Now, enter 10 into the I/YR key and then press Shift NPV. We find that the present value is $1,065.26. 2. To find the future value of the cash flows, enter -1,065.26 into PV, 5 into N, and 10 into I/YR. Now press FV and see that the future value is $1,715.61. 3. At this point our problem has been transformed into an $800 investment with a lump sum cash flow of $1,715.61 at period 5. The MIRR is the discount rate (I/YR) that equates these two numbers. Enter -800 into PV and then press I/YR. The MIRR is 16.48% per year. So, we have determined that our project is acceptable at a cost of $800. It has a positive NPV, the IRR is greater than our 12% required return, and the MIRR is also greater than our 12% required return. Many, perhaps most, time value of money problems in the real world involve other than annual time periods. For example, most consumer loans (e.g., mortgages, car loans, credit cards, etc) require monthly payments. All of the examples in the previous pages have used annual time periods for simplicity. On this page, I'll show you how easy it is to deal with non-annual problems.
General Considerations The first thing to understand is that all of the principles that you have learned to apply for annual problems still apply for non-annual problems. In truth, nothing has changed at all. If you try to think in terms of "periods" rather than years, you will be ahead of the game. A period can be any amount of time. Most common would be daily, monthly, quarterly, semiannually, or annually. However, a time period could be any imaginable amount of time (e.g., seven weeks, hourly, three days). The first, and most important, thing to think about when dealing with non-annual periods is the number of periods in a year. The reason that this is so important is because you must be consistent when entering data into the HP 10BII. The numbers entered into the N , I/YR and PMT keys must agree as to the length of the time periods being used. So, if you are working on a problem with monthly compounding, then N should be the total number of months, I/YR should be the monthly interest rate, and PMT should be the monthly annuity payment.
An Example Very often in a problem, you are given annual numbers but then told that "payments are made on a monthly basis," or that "interest is compounded daily." In these cases, you must adjust the numbers given in the problem. Let's look at an example:
You are considering the purchase of a new home for $250,000. Your banker has informed you that they are willing to offer you a 30-year, fixed rate loan at 7% with monthly payments. If you borrow the entire $250,000, what is the required monthly payment? Notice that we are told that the loan term is 30 years and the interest rate is 7% per year (that is implied, not explicitly stated). So, you might be forgiven for expecting that a period is one year. However, on further reading you see that the payments must be made every month. Therefore, the length of a period is one month, and you must convert the variables to a monthly basis in order to get the correct answer. Since there are 12 months in a year, we calculate the total number of periods by multiplying 30 years by 12 months per year. So, N is 360 months, not 30 years. Similarly, the interest rate is found by dividing the 7% annual rate by 12 to get 0.5833% per month. Note that we do not make any adjustments to the PV ($250,000) because it occurs at a single point in time, not repeatedly. The same logic would apply if there was an FV in this problem. When you solve for the payment, the calculator will automatically give you the monthly (per period to be exact) payment amount. In this problem, then, we would solve for the payment amount by entering 360 in N , 0.5833 into I/YR , and 250,000 into PV . When you press PMT you will find that the monthly payment is $1,663.26. One thing to be careful about is rounding. For example, when calculating the monthly interest rate, you should do the calculation in the calculator and then immediately press the I/YR key. Do not do the calculation and then write down the answer for later entry. If you do, you will be truncating the interest rate to the number of decimal places that are shown on the screen, and your answer will suffer from the rounding. The difference may not be more than a few pennies, but every penny matters. Try sending your lender a payment that is consistently three cents less than required and see what happens. It probably won't be long before you get a nasty letter. Adjust First, Not After, Solving the Problem !!!! You might be tempted to think that you could treat the problem as an annual one, and then adjust your answer to be monthly. Don't do that! The math simply doesn't work that way. To prove it, let's input annual numbers, and then convert the annual payment to monthly by dividing by 12. Enter 30 into N , 7 into I/YR , and 250,000 into PV . When you press PMT , you will find that the annual payment would be $20,146.60. However, you have to make monthly payments so if we divide that by 12 we get a monthly payment of $1,678.88. Do you see the problem? If you do the problem this way, you get an answer that is $15.63 too high every month. So, when you make the adjustments matters. Always adjust your variables before solving the problem. The reason for the difference is the compounding of interest. If you have read through my tutorial on the Mathematics of Time Value of Money, then you know that the more frequently interest is compounded, the smaller the payment has to be in order to grow to a particular future value.
Payments per Year Setting You may have noticed that the HP 10BII can semi-automatically adjust for payment frequency for you by using the P/YR setting. I strongly recommend that you avoid this feature because I
think it causes more problems than it solves. The reason is that this setting is hidden away, and if you forget to change it you will probably get a wrong answer. It can be difficult to spot problems caused by this setting. Regardless of my feelings about this setting, I'm going to tell you how to use it. If you look at the PMT key you will notice that the second function of this key is P/YR, which means "payments per year." If you set this value to, say, 12 then the calculator will assume monthly compounding and adjust the interest rate appropriately. However, and this is very important, it will not adjust the number of periods or the payment amount! That makes this feature virtually worthless. Let's do the problem again, but using this "feature." First, set the payments per year to 12 (monthly) by pressing: 12 Shift PMT . Now, we can enter the data. 360 into N (again, you still have to enter the total number of periods), 7 into I/YR , and 250,000 into PV . Now, solve for the payment by pressing PMT and you will find that the monthly payment is $1,663.26. The answer is correct, but what did you save by using that "shortcut?" Nothing at all. In fact, it takes an extra keystroke or two to use this feature. Furthermore, if you forget to change the setting when you do the next problem, you will get the wrong answer unless that problem also happens to use monthly compounding. My recommendation is to follow the simple steps that I outlined above: Set P/YR to 1 and then forget about it forever. Always make N the total number of periods, I/YR the interest rate per period, and PMT the payment per period. I hope that you have found this tutorial to be helpful.