CHAPTER 5 TIME VALUE OF MONEY

1. What is Time Value of Money (TVM)? Time Value of Money (TVM) is an important concept in financial management. It can be used to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings, and annuities. TVM is based on the concept that a dollar that you have today is worth more than the promise or expectation that you will receive a dollar in the future. Money that you hold today is worth more that money you hold in the future because you can invest it and earn interest. In other words, value of $1 today is worth more than value of $1 in the future, for example, in year 1. After all, you should receive some compensation for foregoing spending. This is called the opportunity cost, that is, the opportunity that you give up from making a certain decision (e.g., an investment). For example, by investing in a certain financial instrument, you are giving up an opportunity of enjoying a purchase of a new car. You need certain compensations for such opportunity that you give up and that can be translated into interest or return. For a numerical example, you can invest $100 for one year at a 6% annual interest rate and accumulate $106 at the end of the year. You can say that the future value of the $100 is $106 given a 6% interest rate and a one-year period. It also follows that the present value of the $106 you expect to receive in one year is only $100. A key concept of TVM is that a single sum of money or a series of equal, evenly-spaced payments or receipts promised in the future can be converted to an equivalent value today (called, Present Value). Conversely, you can determine the value to which a single sum or a series of future payments will grow to at some future date (called, Future Value). You can calculate the fourth value if you are given any three of: Interest Rate, Number of Periods, Payments, and Present or Future Value. Interest Rate Interest is a charge for borrowing money, usually stated as a percentage of the amount borrowed over a specific period of time. Interest rate includes two things: 1) opportunity cost, and 2) inflation. Simple interest is computed only on the original amount borrowed. It is the return on that principal for one time period. In contrast, compound interest is calculated each period on the original amount borrowed plus all unpaid interest accumulated to date. Compound interest is always assumed in TVM problems. Number of Periods Periods are evenly-spaced intervals of time. They are intentionally not stated in years since each interval must correspond to a compounding period for a single amount or a payment period for an annuity. A compounding period can be any period, such as a year, semi-year, quarter, month, or day. Payments Payments are a series of equal, evenly-spaced cash flows. In TVM applications, payments must represent all outflows (negative amount) or all inflows (positive amount). If you receive

1

payments, those are inflows (positive amount) and if you make payments, those are outflows (negative amount). Present Value Present Value is an amount today that is equivalent to a future payment, or series of payments, that has been discounted by an appropriate interest rate. The future amount can be a single sum that will be received at the end of the last period, as a series of equally-spaced payments (an annuity), or both. Since money has time value, the present value of a promised future amount is worth less the longer you have to wait to receive it. Future Value Future Value is the amount of money that an investment with a fixed, compounded interest rate will grow to by some future date. The investment can be a single sum deposited at the beginning of the first period, a series of equally-spaced payments (an annuity), or both. Since money has time value, we naturally expect the future value to be greater than the present value. The difference between the two depends on the number of compounding periods involved and the going interest rate. 2. Simple Future Value The first TVM concept to be presented is how to calculate the future value of a lump sum or an individual cash flow. In other words, what is a certain sum of money worth in the future at a particular rate of interest? An example would be that you deposit $1,000 into a bank account today that pays 5% interest annually. In this case, $1,000 is a lump sum or an individual cash flow which represents present value (PV), and 0 represents today. The timelines presented below often helps us understand the problem clearly.

One year from today, the bank account will have a balance of $1,000 plus interest earned over that one year. Since the interest rate is 5%, you will earn $50 (5% of $1,000 = $1,000×0.05). Thus, the balance of the account after one year will be $1,050 ($1,000 + $50). This can be depicted as follows: FV1 = $1,000 × (1 + 0.05)1. If we decide to keep the money in the account for another year and earn another 5% interest for the second year, the balance will grow to $1,102.50 (= $1,050 + $52.50; $52.50 = 5% of $1,050 = $1,050×0.05). This can be depicted as follows: FV2 = $1,000 × (1 + 0.05)2.

2

The general calculation of the future value can be formularized as follows: FVn = PV × (1 + i)n, where FV represents the future value; PV represents the present value; i represents the interest rate, and n represents the number of periods between the future value and the present value. As can be seen from the example above, the interest is compounded, meaning that you earn interest on interest; for the second year, you earn 5% interest not only on $1,000 (original deposit), but also on $50 (interest income from the previous period – Year1). In this course, we use ‘Time Value of Money Tables’ to calculate the future value (and other values). The tables are designed to save the user from working through the mathematics, using the formula. For this type of problem, the table provides a factor that we multiply by the known present value to solve for the future value. Because we are solving for a future value, the factor is called a future value interest factor (FVIF). FVIFi,n is equal to (1 + i)n, so that the equation of FVn = PV × (1 + i)n can be written as FVn = PV × (FVIFi,n); (FVIFi,n) is the FVIF for a given interest rate (i) and a given number of periods (n). To solve the preceding problem using ‘Time Value of Money Tables’ (specifically ‘Simple Future Value Table’) we can write the equation as FV2 = $1,000 × (FVIF5%,2). From ‘Simple Future Value Table’ (Appendix), we can find FVIF, using the two given information; 5% of interest rate (i) and 2 of time periods (n). Each row of the table represents the number of periods while each column represents the interest rate. Therefore, we find the intersection of 5% of i in columns and 2 of n in rows, which shows 1.1025 as FVIF. Thus, by inputting 1.1025 into the equation, FV2 = $1,000 × 1.1025, we can calculate the future value of $1,102.50. 3. Simple Present Value Now, we know the future value and want to calculate the present value. We will explain the concept with an example. What is $1,320 worth today at a 10% annual interest rate? In other words, how much do you have to invest today at a 10% annual interest rate to receive $1,320 after one year. We call the process of calculating the present value of a future value, discounting. We can visualize this problem set as below.

First, this problem can be solved using the following equation:

3

PV = FVn / (1 + i)n = $1,320 / (1 + 10%)1 = $1,200 Using ‘Simple Present Value Table’, we can write the equation as PV = FVn × (PVIFi,n) = $1,320 × (PVIF10%,1) = $1,320 × 0.9091 = $1,200 (PVIFi,n) is equal to 1 / (1 + i)n. (PVIF10%,1) can be found from ‘Simple Present Value Table’ by looking at the intersection of 10% of interest rate (i) in columns and 1 of time period (n) in rows – 0.9091. 4. Annuity Up to this point, we calculated the future and present value of a single lump sum, but now we are going to find the future value of a series of payments called annuity payments. An annuity is defined as a series of payments of a fixed amount for a specified number of periods of equal length. In other words, if there is the same amount of payment more than once over the same length of interval, that entire cash flow is considered as an annuity. For example, if you deposit $1,000 every year for 5 years, the entire cash flow is considered as an annuity; the same amount of payments of $1,000 and the length of interval is a year. The timelines of this example can be drawn as below:

If the third year’s deposit changes to $2,000, there is no longer one annuity, but are two annuities.

Examples of an annuity in our daily lives include the car payments you make to pay off a car loan, the mortgage payments made to pay off a home mortgage, or even the lease payments you make on an apartment rental to fulfill a rent contract. Of course, these examples call for monthly payments (that is, compounding period is a month), but for the time being we will look at solving problems with annual payments (that is, compounding period is a year). Later, we will deal with monthly payments or other periodic payments (that is, compounding periods other than annual). There are two types of annuity: 1) an ordinary annuity, and 2) an annuity due. In this course, we will consider only ‘an ordinary annuity.’ For an ordinary annuity, each payment or cash flow happens at the end of each time period. In other words, the first payment happens at the end of

4

the first time period, the second payment happens at the end of the second time periods, and so on, thus the last payment happens at the end of the last time period. In such case, the timeline of the entire annuity begins one period ahead of the first payment and ends at the last payment. For the example of the 5-year with $1,000 deposit, if we see this annuity as an ordinary annuity, we can find out the beginning and ending lines of the annuity as below.

This ordinary annuity begins at year 0 and ends at year 5. We can determine the number of time period (n) of this ordinary annuity, using either of the following two methods. First, it is the number of payments – so, you can simply count them. For the example, there are five payments of $1,000, thus the number of time periods of the annuity (n) is five. The second method is to use the beginning and ending lines; n = Ending Year – Beginning Year. So, for the example, n is 5 (= 5 – 0). It is often easy to count the number of payments with relatively a small number of payments, such as the above example. However, if we have to deal with a large number of payments, such as a fixed amount of annual investment from your 23rd birthday to 65th birthday, or monthly payments for 30 years (i.e., a 30-year mortgage payment), the second method will be very useful. Knowing the correct beginning and ending lines is critical to calculate the future and present values of an annuity. It is because when we apply the future value concept to an annuity, we basically calculate the value of the entire annuity at the ending line. For the example above, we will be calculating the value of the entire annuity (including all five $1,000s over the five years) at the end of Year 5. When we apply the present value concept to an annuity, we basically calculate the value of the entire annuity at the beginning line. For the example above, we will be calculating the value of the entire annuity at Year 0. Now, we will show how to calculate the future and present values of an annuity (i.e., an ordinary annuity). 5. Future Value of an Annuity Let’s consider the following example to illustrate the future value of an annuity. If you deposit $50,000 at the end of each year for the next four years, earning 10% annual interest, what will be the value of your account at the end of Year 4? First, let’s draw the timeline of this example.

5

We can first determine that the entire cash flow is an annuity. Second, we will consider this annuity as an ordinary annuity (because in this course, we will consider all annuities as an ordinary annuity), we can draw the beginning line of the annuity at Year 0 and the ending line at Year 4. Now, our concern is to calculate the value of the annuity at the end of Year 4 (that is, future value of the annuity). That is one of the two things that we can calculate by using the annuity concept; again, we can calculate the value of the entire annuity either at the beginning line (when we apply the present value concept) or ending line (when we apply the future value concept). Thus, if we apply the future value concept to this annuity, we can solve the problem. Using ‘Time Value of Money Table’ (specifically, ‘Future Value of an Ordinary Annuity Table’) and the following formula, we can accomplish the goal. The formula is

FVAn = PMT × (FVIFAi,n)

where FVAn represents the future value of an annuity at the time period of n; PMT represents the payment per period, and (FVIFAi,n) represents the future value interest factor of an annuity from the table, using the interest rate of i and the number of time period of n. Now, using this formula and the table, we can solve the future value of the annuity of the example as follows:

FVA4 = $50,000 × (FVIFA10%,4) = $50,000 × 4.6410 = $232,050

Thus, the future value of this annuity at Year 4 is $230,050. 6. Present Value of an Annuity Now, we calculate the present value of an annuity. Let’s say that you plan to withdraw $1,000 annually from an account at the end of each of the next five years. If the account pays 12% interest annually, what must you deposit today to have just enough to cover the five withdrawals? This can be viewed on a timeline as follows.

6

The timeline shows five annual payments of $1,000 each which consists of an ordinary annuity and “FVA=?” indicates we are looking for the present value of this annuity including all five payments. The beginning line of this ordinary annuity is Year 0 and the ending line is Year 5. Therefore, when we apply the present value concept to this annuity, we calculate the value of the entire annuity at the beginning line which happens to be Year 0. Now, we use the following formula to solve the problem.

PVAn = PMT × (PVIFAi,n) where PVAn represents the present value of an annuity at the time period of n; PMT represents the payment per period, and (PVIFAi,n) represents the present value interest factor of an annuity from the table, using the interest rate of i and the number of time period of n. By plugging numbers in this formula, we can solve the problem.

PVA5 = $1,000 × (PVIFA12%,5) = $1,000 × 3.6048 = $3,604.80 Thus, the present value of this annuity (at Year 0) is $3,604.80. 7. Future Value of a Deferred Annuity Now, let’s consider the following example. Suppose that you would like to make an investment of $3,000 annually starting at the end of year 1 for next 10 years (10 investments) at 12% return rate. Then, put the entire money into a safer investment for additional 10 years at 8% return rate. What will the future value of this investment at the end of year 20? Let’s visualize this on a timeline.

Now, we have to realize that we cannot calculate the future value of this annuity at Year 20 with one step. It is because, by using the annuity concept, we can calculate the value of this annuity at only two time periods, beginning (Year 10 in this case) and ending lines (Year 10 in this case).

7

In such case, we can achieve the ultimate goal of this problem (that is, to calculate the future value at Year 20) with two steps. Those two steps are visualized on the timeline below.

With the first step, we can calculate the future value of the annuity at Year 10 (which is the ending line of the annuity). Then, we will have one lump sum value of the annuity at Year 10. Once we figure out such value, then we can calculate the simple future value of such one lump sum at Year 20 which is the second step. Therefore, in the first step, we use the future value of an annuity concept with the formula of FVAn = PMT × (FVIFAi,n), then in the second step, we use the simple future value concept with the formula of FVn = PV × (FVIFi,n). Let’s see how the calculations can be done. First, the annuity has $3,000 of PMT for each year for 10 years and the interest rate is 12%. By plugging these numbers into the formula, we can calculate the future value of this annuity at the ending line of this annuity (which is, again, Year 10): FVA10 = $3,000 × (FVIFA12%,10) = $3,000 × 17.5487 = $52,646.10. Let’s see this on the timeline.

Now, we took care of the annuity part, and have one lump sum of $52,646.10 at Year 10. With the second step, we can calculate the future value of this one lump sum at Year 20 which is the solution of this problem. For the second investment (from Year 10 to Year 20), the number of time period is 10 (n) and the interest rate is 8% (i); FV10 = $52,646.10 × (FVIF8%,10) = $52,646.10 × 2.1589 = $113,658.98. So, the solution is $113,658.98. We have to understand that the second part of this problem (that is, the second investment), the timeline begins at Year 10 and ends at Year 20. The second investment does not concern Year 0 to Year 10. Therefore, FV10 represents the future value at the tenth year from the beginning of this part (which is Year 10, not Year 0). So, FV10 means the future value at Year 20. In the same vein, PV in the second investment is the value at Year 10, not Year 0. So, PV is $52,646.10. When we need two steps to solve the problem for an annuity, we call it a deferred annuity.

8

8. Present Value of a Deferred Annuity Now, let’s try to calculate the present value of a deferred annuity. Suppose there is an investment that promises to pay you $100 annually beginning at the end of year 5 and continuing until the end of year 10. If you decide 7 percent is an appropriate rate of return, what is the present value of these cash flows? The timeline for this problem can be drawn as below.

Now, we have to realize that we cannot calculate the present value of this annuity at Year 0 with one step. It is because, by using the annuity concept, we can calculate the value of this annuity at only two time periods, beginning (Year 4 in this case) and ending lines (Year 10 in this case). In such case, we can achieve the ultimate goal of this problem (that is, to calculate the value at Year 0) with two steps. Those two steps are visualized on the timeline below.

With the first step, we can calculate the present value of the annuity at Year 4 (which is the beginning line of the annuity). Then, we will have one lump sum value of the annuity at Year 4. Once we figure out such value, then we can calculate the simple present value of such one lump sum at Year 0 which is the second step. Therefore, in the first step, we use the present value of an annuity concept with the formula of PVAn = PMT × (PVIFAi,n), then in the second step, we use the simple present value concept with the formula of PVn = FV × (PVIFi,n). Let’s see how the calculations can be done. First, the annuity has $100 of PMT for each year for 6 years and the interest rate is 7%. By plugging these numbers into the formula, we can calculate the present value of this annuity at the ending line of this annuity (which is, again, Year 4): PVA4 = $100 × (PVIFA7%,6) = $100 × 4.7665 = $476.65. Let’s see this on the timeline.

9

Now, we took care of the annuity part, and have one lump sum of $476.65 at Year 4. With the second step, we can calculate the present value of this one lump sum at Year 0 which is the solution of this problem. For the second part (from Year 4 to Year 0), the number of time period is 4 (n) and the interest rate is still 7% (i); PV0 = $476.65 × (PVIF7%,4) = $476.65 × 0.7629 = $363.64. So, the solution is $363.64. In other words, if you invest $363.64 today and earn 7% annually, then you will be able to withdraw $100 from Year 5 to Year 10, annually. 9. Time Value of Non-Constant Cash Flows Now, it is time to investigate how to value a complex stream of cash flows. Obviously not all cash flow streams meet the definition of an annuity. Here, we deal with a stream of cash flows that are not constant which possibly consists of both individual cash flows and an annuity(ies). Let’s try an example to illustrate the case. You are considering new kitchen equipment for your restaurant that is expected to save you $1,000 the first year, $500 a year for years 2 through 5, and $750 in the sixth year. If you believe 10 percent is a fair discount rate, what is the present value of this non-constant stream of cash flow savings?

These six cash flows are not constant. This stream of cash flows includes three components: 1) one lump sum of $1,000 from Year 1, 2) an annuity of $500 of PMT per year for 4 years from Year 2 to Year 5, and 3) one lump sum of $750 from Year 6. Now, we just have to take care of these three components, one by one. When you work on each component, you consider only that component, ignoring other components. We can see these three components separately as below.

[The first component of $1,000]

10

[The second component of an annuity]

[The third component of $750]

Now, let’s calculate the present values of each of the three components. First, we can calculate the present value (Year 0) of one lump sum of $1,000 from Year 1 as follows: PV0 = FV × (PVIF10%, 1) = $1,000 × (PVIF10%, 1) = $1,000 × 0.9091 = $909.10. Second, to calculate the present value of the ordinary annuity (from Year 2 to Year 5), we have to understand that this annuity is a deferred annuity. That means, we have to go through two steps to solve this part of the problem. First, we have to find out the present value of the four-payments (four annual $500s). The beginning line is Year 1 and ending line is Year 5. Therefore, when we calculate the present value of this annuity, we calculate the value of this annuity at the beginning line (Year 1) as follows: PVA1 = PMT × (PVIFAi,n) = $500 × (PVIFA10%,4) = $500 × 3.1699 = $1,584.95. Next, we have to calculate the present value (at Year 0) of this lump sum of $1,584.95 from Year 1 by using the simple present value concept as follows: PV0 = $1,584.95 × (PVIF10%,1) = $1,584.95 × 0.9091 = $1,440.88. Third, we calculate the present value (Year 0) of a lump sum of $750 from Year 6 as follows: PV0 = FVn × (PVIFi,n) = $750 × (PVIF10%,6) = $750 × 0.5645 = $423.38. Last, we can simply sum all of these three present values to calculate the present value of all three components as follows: Total PV0 = $909.10 + $1,440.88 + $423.38 = $2,773.36. 10. Compounding Periods Other Than Annual Up to this point, we have assumed annual compounding and annual payments. Yet there are many real-world applications using compounding and payments that are not annual. In reality, the interest can be compounded annually, semi-annually, quarterly, monthly or even daily. In this section, we deal with compounding periods other than annual. Let’s suppose that you have $100 to invest for a year and have narrowed down your choices to two banks. The Blue Bank pays 10% annually, compounded annually while the White Bank pays 10% annually, compounded semi-annually (every six months). The difference between the two banks is illustrated in the timelines below.

11

[Blue Bank]

[White Bank]

On the first timeline, $100 earns 10% interest once for one year (the compounding period is annual) while on the second timeline, $100 earns 5% interest twice for one year (the compounding period is semi-annual). As we can see from this example, with the given annual rate (in this example, 10%), we earn more interest income when the interest is compounded more often. Now, let’s see how we can calculate the future value of each case, using the formula. For the Blue Bank case, the present value (PV) is $100, the number of compounding period (n) is 1, and the interest per compounding period (i) is 10%. Thus, the future value can be calculated as follows: FV1 = PV × (FVIFi,n) = $100 × (FVIF10%,1) = $100 × 1.1 = $110. For the White Bank case, the present value (PV) is $100, same as the Blue Bank case. However, the number of compounding period (n) is 2 for the White Bank because we have two six-months for a year. Also, the interest rate per compounding period for the White Bank is 5%. It is the interest rate per six months for the White Bank, so we can calculate it by dividing the annual rate (10%) by the number of compounding period per year (n=2): 10% / 2 = 5%. Then, the future value can be calculated as follows: FV2 = PV × (FVIFi,n) = $100 × (FVIF5%,2) = $100 × 1.1025 = $110.25. Let’s consider another example. If a bank pays 12% annual interest, compounded monthly, your $100 deposit today will grow to $112.68 after one year: FV12 = PV × (FVIFi,n) = $100 × (FVIF1%,12) = $100 × 1.1268 = $112.68. The compounding period is a month, thus over the year, there are 12 compounding periods (n=12). Also, the interest rate per compounding period is 1% per month (=12% / 12 months). The annual interest rate in these examples is called, nominal annual rate. When compounding periods are not annual, the actual annual interest rate paid or received is referred to as the effective annual rate which would be different from nominal annual interest rate. Considering the Blue and White Bank example, we can calculate the effective annual rate for the White Bank as follows: ($110.25 / $100) – 1 = 1.1025 – 1 = 0.1025, 10.25%. Of course, the effective annual

12

rate for the Blue Bank is 10%, the same as the nominal annual interest rate, because the compounding period is annual for the Blue Bank. Let’s try one more example. Suppose you are planning to invest in $1,000 of a certificate of deposit for 10 years beginning today at 8% annual interest rate, compounded quarterly. What is the future value of this investment and effect annual rate? First, the future value of the investment can be calculated as follows: FVn = PV × (FVIFi,n) = $1,000 × (FVIF2%,40) = $1,000 × 2.2080 = $2,208. The number of compounding period (n) is 40: 4 quarters × 10 years = 40 quarters. The interest rate per quarter (compounding period) is 2% (= 8% / 4 quarters). The effect annual rate can be calculated as follows: (FV at Year 1 / PV) – 1 = [(PV × 1.0824) / PV] – 1 = ($1,082.40 / $1,000) – 1 = 1.0824 – 1 = 0.0824, 8.24%. Again, we can see that the effective annual rate is greater than the nominal annual rate. We use only one year, not the entire 10-year period, to calculate the effective annual rate because it is an annual rate. 11. When Payment is Unknown Basically, to this point, we tried to calculate either future value or present value of a time value of money problem. In this section, we discuss a case where we have three values available (Interest Rate, Number of Periods, and Present or Future Value) and thus figure out the fourth value of Payments. Let’s suppose that you are planning to buy a computer for $2,000 on a loan. You have to make monthly payments for one year with the annual interest rate of 12%, compounded monthly. The first payment starts at the end of the first month. How much do you have to pay per month? We can draw the timeline as below.

In this case, we know three values: present value of $2,000, interest rate of 1% (12% / 12 months), and the number of compounding periods of 12 (= 12 months). Then, we should be able to figure out the fourth value, payments (PMT). Clearly, these payments form an ordinary annuity and the present value of the annuity is $2,000. Therefore, by applying the present value concept to this annuity, we can solve the problem as follows: PVA = PMT × (PVIFAi,n) $2,000 = PMT × (PVIFA1%, 12) $2,000 = PMT × 11.2551 Then, divide the both sides by 11.2551; $2,000 / 11.2551 = (PMT × 11.2551) / 11.2551 Then, PMT = $2,000 / 11.2551 = $177.70 (monthly payment) Now, we can construct a table called, ‘Amortization Table’ of this loan payment.

[Amortization Table]

13

Month Payment Interest Principal Balance 0 2000 1 177.70 20 157.70 1842.30 2 177.70 18.42 159.28 1683.02 3 177.70 16.83 160.87 1522.15 4 177.70 15.22 162.48 1359.67 5 177.70 13.60 164.10 1195.57 6 177.70 11.96 165.74 1029.83 7 177.70 10.30 167.40 862.43 8 177.70 8.62 169.08 693.35 9 177.70 6.93 170.77 522.58 10 177.70 5.23 172.47 350.11 11 177.70 3.50 174.20 175.91 12 177.70 1.76 175.94 0 (-0.03)

Total 2132.40 132.371 2000 First, the balance starts as $2,000 today (Month 0). Then, after the first month, you make the first monthly payment of $177.70 as calculated before. This monthly payment incurs for every month for the entire period of 12 months, and consists of two components: interest payment and principal payment. Interest payment is calculated by multiplying the balance by the interest rate per compounding period. Thus, for the first month, the interest payment is calculated by the balance of $2,000 by the interest rate of 1% per month; $20 = $2,000 × 1%. After the bank takes this interest payment of $20 out of the monthly payment of $177.70, the remaining portion of the monthly payment is $157.70 (= $177.70 – $20) that is called ‘Principal Payment’. This principal payment reduces the balance, thus the balance after the first month becomes $1,842.30 (= $2,000 – $157.70). After the second month, you make another monthly payment of $177.70. The interest payment for the second month is calculated by multiplying the balance of $1,842.30 by the interest rate per month of 1% = $18.42. The principal payment is calculated after taking the interest payment ($18.42) from the monthly payment ($177.70); $159.28 (=$177.70 – $18.42). The same procedure is repeated for the remaining 10 months. After making the last monthly payment, the balance should become zero. In the table above, the final balance is -$0.03, not exactly $0, just because the monthly payment used in this problem is a round-up value. 12. When Interest Rate is Unknown In this section, we discuss a case where we have two or three values available (Number of Periods, Present or Future Value, and/or Payments) and thus can figure out the interest rate. Let’s see an example. If you invest $4,000 in a certificate of deposit today, a bank promises the certificate of deposit will be worth $5,000 in five years. What is your interest rate return on this investment? Since we deal with one lump sum in this case, we apply either simply future or present value concept. Let’s use the simple future value; FV = PV × (FVIFi,n)

14

$5,000 = $4,000 × (FVIFi,5) Divide the both sides by $4,000; ($5,000 / $4,000) = ($4,000× (FVIFi,5))/$4,000 Then, (FVIFi,5) = $5,000 / $,4000 = 1.25 Now, we know the future value interest factor (with the number of period of 5) is 1.25. Then, we can figure out the interest rate by looking at ‘Simple Future Value Table.’ We know two values; n of 5 and the interest factor of 1.25. Thus, in the row of 5 (n=5), we should be able to find the interest factor of 1.25 or the closest number to 1.25. From the table, we can see that 1.2763 is the closest value to 1.25, and this interest factor is in the column of 5%. Thus, the solution we are looking for is 5% of interest rate.

15

[Problems]

1. What is the future value of $1,000 invested for five years at the following interest rates? a. 5% b. 8% c. 10%

2. What is the future value of $1,000 invested at a 7 percent rate for the following length of time? a. 2 years b. 5 years c. 10 years

3. What is the present value of $10,000 to be received in four years at the following interest rates? a. 4% b. 7% c. 12%

4. Using an 8 percent interest rate, what is the present value of $10,000 to be received in the following number of years? a. 3 years b. 6 years c. 12 years

5. A bond issued by Fried’s Restaurants pays no interest but will return $1,000 in 15 years. If you buy the bond for $326.39 today, what will be your interest rate of return on the investment?

6. If you invest $4,000 in a certificate of deposit today, a bank promises the certificate of deposit will be worth $5,000 in five years. What is your interest rate return on this investment?

7. Brewer Resort is considering the purchase of a piece of real estate for the future site if a new project. The real estate costs $5 million. A bank has offered to finance the purchase at a 7 percent interest rate with a 10 percent down payment. The loan would be repaid with 15 equal, annual, end-of-year payments. If Brewer borrows the $4.5 million (90 percent of $5 million), what is the amount of each payment?

8. Grace turned 25 years old today and would like to retire by the time of her 60th birthday. In addition to social security and her company pension plan, she plans to invest $3,000 annually into an investment that promises to return 9 percent annually. If her first $3,000 payment is on her 26th birthday and her last $3,000 payment is on her 60th birthday, what will be the value of this investment on her 60th birthday?

16

9. Andy wants to take out a loan to purchase a new home. He is willing to pay up to $10,000 at the end of each of the next 30 years to repay the loan. If the loan interest rate is 6 percent, what is the most he can borrow?

10. An investment costs $20,000 today and will return $3,000 at the end of each of the next 10 years. What is the interest rate of return on this investment?

11. Carl would like to save $100,000 by his 40th birthday to pay for a special mildlife crisis vacation. He plans to achieve this by investing equal annual amounts each year beginning on his 24th birthday and ending and including a payment on his 40th birthday. If the investment pays an 11 percent interest rate, what is the size of each annual payment Carl needs to invest?

12. An investment of $1,000 annually at the end of each year for the next 15 years will be worth $30,000 at the end of 15 years. What is the interest rate return on this investment?

13. A $20,000 loan requires equal annual end-of-year payments for four years. The interest rate is 10 percent. a. What is the amount of each loan payment? b. Construct a loan amortization schedule to include the amount of interest and principal paid each

year as well as the remaining balance at the end of each year.

14. A $100,000 loan requires equal annual end-of-year payments of $38,803.35 for three years. a. What is the annual interest rate? b. Construct a loan amortization schedule to include the amount of interest and principal paid each

year as well as the remaining balance at the end of each year.

15. An investment promises to return $2,000 at the end of each of the next 10 years and then $5,000 at the end of each of the next five years (years 11 through 15). What is the value of this investment today at a 7 percent interest rate?

16. An investment promises to return $8,000 at the end of each of the next eight years and then $3,000 at the end of each of the remaining seven years (years 9 through 15). What is the value of this investment today at a 9 percent interest rate?

17. You plan to invest $10,000 into a bank certificate of deposit for three years. The certificate of deposit pays a 12 percent annual nominal rate. What is the value of your investment in three years if the 12 percent rate is compounded at the following periods? a. annually b. semiannually (every six months) c. quarterly (every three months) d. monthly

17

18. You plan to invest $5,000 into a bank certificate of deposit for five years. The certificate of deposit pays a 6 percent annual nominal rate. What is the value of your investment in five years if the 6 percent rate is compounded at the following periods? a. annually b. semiannually (every six months) c. quarterly (every three months) d. monthly

19. An investment promises to return $1,000 annually with the first $1,000 to be received at the end of 10 years and the last $1,000 to be received at the end of 25 years. What is the value of this investment today at a 7 percent rate of return?

20. An investment promises to return $1,500 annually with the first $1,500 to be received at the end of 5 years and the last $1,500 to be received at the end of 12 years. What is the value of this investment today at a 5 percent rate of return?

21. Andy just won a lottery. The prize is 20 annual payments of $100,000 each with the first payment to be today. What is the value of this prize (the 20 payments of $100,000 each) today at an 8 percent interest rate?

22. You just celebrated your 25th birthday today. You plan to invest $1,000 annually, with the first $1,000 invested today and the last invested on your 59th birthday. a. What is the value of this investment on your 60th birthday if all invested funds earn 6 percent

annually? b. What interest rate do you need to earn for the investment to be worth $150,000 on your 60th

birthday?

23. You just celebrated your 25th birthday today. You plan to invest $2,000 annually, with the first $2,000 invested on your 26th birthday and the last invested on your 60th birthday. a. What is the value of this investment on your 61st birthday if all invested funds earn 6 percent

annually? b. What interest rate do you need to earn for the investment to be worth $300,000 on your 61st

birthday?

24. Mike is planning to provide for his son’s future college tuition. He expects to need $40,000 in 15 years, $42,000 in 16 years, $45,000 in 17 years, and $50,000 in 18 years for this purpose. If he can earn 10 percent annually, what single amount does he need to invest today to provide for his son’s future college tuition?

25. Mike is planning to provide for his son’s future college tuition. He expects to need $40,000 in 15 years, $42,000 in 16 years, $45,000 in 17years, and $50,000 in 18 years for this purpose. He plans to provide for this by investing equal annual end-of-year payments for the next 15 years. If he can earn 10 percent annually, what is the required amount of each payment?

18

26. Ted and Carol are planning to provide for their two daughters’ future college tuition. The oldest daughter is expected to need $8,000 in 8 years, $9,000 in 9 years, $10,000 in 10 years, and $11,000 in 11 years. The youngest daughter is expected to need $14,000 in 14 years, $15,000 in 15 years, $16,000 in 16 years, and $17,000 in 17 years. If Ted and Carol can earn 8 percent annually, what single amount do they need to invest today to provide for their daughters’ future college tuition?

27. Ted and Carol are planning to provide for their two daughters’ future college tuition. The oldest daughter is expected to need $8,000 in 8 years, $9,000 in 9 years, $10,000 in 10 years, and $11,000 in 11 years. The youngest daughter is expected to need $14,000 in 14 years, $15,000 in 15 years, $16,000 in 16 years, and $17,000 in 17 years. Ted and Carol plan to provide for this by investing equal annual end-of-year payments for the next 8 years. If they can earn 8 percent annually, what is the required amount of each payment?

28. Larry plans to retire in his 60s. In addition to social security and his company pension plan, he has a supplemental retirement investment plan. All funds invested in this plan will earn 12 percent annually. From this supplemental investment plan, he hopes to make 20 annual withdrawals of $100,000, with the first withdrawal on his 66th birthday and the last on his 85th birthday. a. What single amount does Larry need to invest on his 30th birthday to provide for the 20

withdrawals of $100,000 each? b. What equal annual payment does Larry need to invest in order to provide for the 20 withdrawals?

The first payment will be on his 31st birthday, and the last payment will be on his 65th birthday.

29. Larry plans to retire in his 60s. In addition to social security and his company pension plan, he has a supplemental retirement investment plan. All funds invested in this plan will earn 12 percent annually. From this supplemental investment plan, he hopes to withdraw $500,000 on his 66th birthday and also make 20 annual withdrawals of $100,000, with the first withdrawal on his 66th birthday and the last on his 85th birthday. a. What single amount does Larry need to invest on his 30th birthday to provide for these

withdrawals? b. What equal annual payment does Larry need to invest to provide for these withdrawals? The first

payment will be on his 31st birthday, and the last payment will be on his 65th birthday.

30. Larry plans to retire in his 60s. In addition to social security and his company pension plan, he has a supplemental retirement investment plan. All funds invested in this plan will earn 12 percent annually. Currently (assume today is Larry’s 30th birthday), Larry has $15,000 invested in this plan. From this supplemental investment plan, he hopes to withdraw $500,000 on his 66th birthday and also make 20 annual withdrawals of $100,000, with the first withdrawal on his 66th birthday and the last on his 85th birthday. a. To provide for these withdrawals, what single amount does Larry need to invest on his 30th

birthday in addition to the $15,000 already there? b. To provide for these withdrawals, what equal annual payment does Larry need to invest in

addition to the $15,000 already there? The first payment will be on his 31st birthday, and the last payment will be on his 65th birthday.

19

Answers to Problem Sets

Time Value of Money

1. a. FV = PV × FVIF (5%, 5) = $1,000 × 1.2763 = $1,276.30

b. FV = PV × FVIF (8%, 5) = $1,000 × 1.4693 = $1,469.30

c. FV = PV × FVIF (10%, 5) = $1,000 × 1.6105 = $1,610.50

2. a. FV = PV × FVIF (7%, 2) = $1,000 × 1.1449 = $1,144.90

b. FV = PV × FVIF (7%, 5) = $1,000 × 1.4026 = $1,402.60

c. FV = PV × FVIF (7%, 10) = $1,000 × 1.9672 = $1,967.20

3. a. PV = FV × PVIF (4%, 4) = $10,000 × 0.8548 = $8,548

b. PV = FV × PVIF (7%, 4) = $10,000 × 0.7629 = $7,629

c. PV = FV × PVIF (12%, 4) = $10,000 × 0.6355 = $6,355

4. a. PV = FV × PVIF (8%, 3) = $10,000 × 0.7938 = $7,938

b. PV = FV × PVIF (8%, 6) = $10,000 × 0.6302 = $6,302

c. PV = FV × PVIF (8%, 12) = $10,000 × 0.3971 = $3,971

5. FV = PV × FVIF (i=?, n=15) $1,000 = $326.39 × FVIF (i=?, n=15) FIVF (i=?, n=15) = $1,000 / 326.39 = 3.0638 From the table of Simple Future Value, i=8% (approximately) 6. FV = PV × FVIF (i=?, n=5) $5,000 = $4,000 × FVIF (i=?, n=5) FIVF (i=?, n=5) = $5,000 / 4,000 = 1.25 From the table of Simple Future Value, i=5% (approximately) 7.

20

The loan is structured as an ordinary annuity where $4,500,000 is the present value of the annuity. PVA = CF × PVIFA (7%, 15)

$4,500,000 = CF × 9.1079

CF = $4,500,000 / 9.1079 = 494.076.57

8. This investment is structured as an ordinary annuity where the value on Grace’s 60th birthday is the future value of the annuity.

FVA = $3,000 × FVIFA (9%, 35) = $3,000 × 215.7108 = $647,132.40

9. This loan is structured as an ordinary annuity where the most Andy can borrow is the present value of the annuity.

PVA = $10,000 × PVIFA (6%, 30) = $10,000 × 13.7648 = $137,648

10. PVA = CF × PVIFA (i=?, n=10)

$20,000 = $3,000 × PVIFA (i=?, n=10)

PVIFA (i=?, n=10) = $20,000 / $3,000 = 6.6667

From the table of Present Value of an Ordinary Annuity, i=8% (approximately) 11. This investment is structured as an ordinary annuity where $100,000 is the future value of the annuity.

FVA = CF × FVIFA (11%, 17)

$100,000 = CF × 44.5008

CF = $100,000 / 44.5008 = $2,247.15

12. FVA = CF × FVIFA (i=?, n=15)

$30,000 = $1,000 × FVIFA (i=?, n=15)

FVIFA (i=?, n=15) = $30,000 / $1,000 = 30

From the table of Future Value of an Ordinary Annuity, i=9% (approximately)

21

13. (a)

PVA = CF × PVIFA (10%, 4)

$20,000 = CF × 3.1699

CF = $20,000 / 3.1699 = $6,309.35

14. (a)

PVA = CF × PVIFA (i=?, n=3)

$100,000 = $38,803.35 × PVIFA (i=?, n=3)

PVIFA (i=?, n=3) = 2.5771

From the table of Present Value of an Ordinary Annuity, i=8% (approximately) 15. The value of the investment today is the present value of a 10 payment ordinary annuity plus the present value of a 5 payment deferred annuity.

There are two annuities. The first one is:

PVA = $2,000 × PVIFA (7%, 10) = $2,000 × 7.0236 = $14,047.20

The second annuity is a deferred annuity as follows:

PVA = $5,000 × PVIFA (7%, 5) = $5,000 × 4.1002 = $20,501 (at the end of Year 10)

PV = FV10 × PVIF (7%, 10) = $20,501 × 0.5083 = $10,420.66

Total PV = $14,047.20 + $10,420.66 = $24,467.86

16. The value of the investment today is the present value of an 8 payment ordinary annuity plus the present value of a 7 payment deferred annuity.

There are two annuities. The first one is:

PVA = $8,000 × PVIFA (9%, 8) = $8,000 × 5.5348 = $44,278.40

The second annuity is a deferred annuity as follows:

22

PVA = $3,000 × PVIFA (9%, 7) = $3,000 × 5.0330 = $15,099 (at the end of Year 8)

PV = FV8 × PVIF (9%, 8) = $15,099 × 0.5019 = $7,578.19

Total PV = $44,278.40 + $7,578.19 = $51,856.59

17. We incorporate the compounding into the future value of a single lump sum equation.

a. FV = $10,000 × FVIF (12%, 3) = $10,000 × 1.4049 = $14,049

b. FV = $10,000 × FVIF (6%, 6) = $10,000 × 1.4185 = $14,185

6% = 12% (Annual Nominal Rate) / 2 (Annual Compounding Periods)

6 = 2 (Annual Compounding Periods) × 3 (Number of Years)

c. FV = $10,000 × FVIF (3%, 12) = $10,000 × 1.4258 = $14,258

3% = 12% (Annual Nominal Rate) / 4 (Annual Compounding Periods)

12 = 4 (Annual Compounding Periods) × 3 (Number of Years)

d. FV = $10,000 × FVIF (1%, 36) = $10,000 × 1.4308 = $14,308

1% = 12% (Annual Nominal Rate) / 12 (Annual Compounding Periods)

36 = 12 (Annual Compounding Periods) × 3 (Number of Years)

18. We incorporate the compounding into the future value of a single lump sum equation.

a. FV = $5,000 × FVIF (6%, 5) = $5,000 × 1.3382 = $6,691

b. FV = $5,000 × FVIF (3%, 10) = $5,000 × 1.3439 = $6,719.50

3% = 6% (Annual Nominal Rate) / 2 (Annual Compounding Periods)

10 = 2 (Annual Compounding Periods) × 5 (Number of Years)

c. Not possible with using the tables.

d. Not possible with using the tables.

19. This investment is structured as a deferred annuity where the value today is the present value of the deferred annuity.

PVA = $1,000 × PVIFA (7%, 16) = $1,000 × 9.4466 = $9,446.60 (at the end of Year 9)

23

PV = $9,446.60 × PVIF (7%, 9) = $9,446.60 × 0.5439 = $5,138.01

20. This investment is structured as a deferred annuity where the value today is the present value of the deferred annuity.

PVA = $1,500 × PVIFA (5%, 8) = $1,500 × 6.4632 = $9,694.80 (at the end of Year 9)

PV = $9,694.80 × PVIF (5%, 4) = $9,694.80 × 0.8227 = $7,975.91

21. The value of the prize today is the present value of an annuity due. It is an annuity due because the first payment of $100,000 is today, the same date as the present value.

PVAD = CF × PVIFA (8%, 20) × (1+i) = $100,000 × 9.8181 × 1.08 = $1,060,354.80

22. The value of this investment on your 60th birthday is the future value of an annuity due. It is an annuity due because the future value is one period after the last payment.

a. FVAD = CF × FVIFA (6%, 35) × (1+i) = $1,000 × 111.4348 × 1.06 = $118,120.89 b. Not possible with using the tables. 23. a. FVAD = CF × FVIFA (6%, 35) × (1+i) = $2,000 × 111.4348 × 1.06 = $236,241

b. Not possible with using the tables.

24. The single amount Mike needs to invest today is the present value of a series of non-constant cash flows.

PV = [$40,000 × PVIF (10%, 15)] + [$42,000 × PVIF (10%, 16)] + [$45,000 × PVIF (10%, 17)] + [$50,000 × PVIF (10%, 18)]

= [$40,000 × 0.2394] + [$42,000 × 0.2176] + [$45,000 × 0.1978] + [$50,000 × 0.1799]

= $9,576 + $9,139.20 + $8,901 + $8,995 = $36,611.20

25. The first step is to find the value of the four uneven cash flows at year 15 or year 0. If at year 15, this value is the future value of the 15-payment annuity. If at year 0, this value is the present value of the 15-payment annuity. Let’s use the later approach since we have already computed the value of the four uneven cash flows at year 0 to answer Problem #24.

24

PVA = CF × PVIFA (10%, 15)

$36,611.20 = CF × 7.6061

CF = $36,611.20 / 7.6061 = $4,813.40

26. The single amount Ted and Carol need to invest today is the present value of a series of non-constant cash flows.

PV = [$8,000 × PVIF (8%, 8)] + [$9,000 × PVIF (8%, 9)] + [$10,000 × PVIF (8%, 10)] + [$11,000 × PVIF (8%, 11)] + [$14,000 × PVIF (8%, 14)] + [$15,000 × PVIF (8%, 15)] + [$16,000 × PVIF (8%, 16)] + [$17,000 × PVIF (8%, 17)]

= [$8,000 × 0.5403] + [$9,000 × 0.5002)] + [$10,000 × 0.4632)] + [$11,000 × 0.4289)] + [$14,000 × 0.3405] + [$15,000 × 0.3152] + [$16,000 × 0.2919] + [$17,000 × 0.2703]

= $4,322.40 + $4,501.80 + $4,632 + $4,717.90 + $4.767 + $4,728 + $4,670.40 + $4,595.10 = $ 36,934.60

27. The first step is to find the value of the eight uneven cash flows at year 8 or year 0. If at year 8, this value is the future value of the 8-payment annuity. If at year 0, this value is the present value of the 8-payment annuity. Let’s use the later approach since we have already computed the value of the eight uneven cash flows at year 0 to answer Problem #26.

PVA = CF × PVIFA (8%, 8)

$36,934.60 = CF × 5.7466

CF = $36,934.60 / 5.7466 = $6,427.21

28. a. This can be solved as the present value of a deferred annuity. The present value is the single amount Larry needs on his 30th birthday. This is a deferred annuity because the payments begin 36 years later after the present value.

PVA = CF × PVIFA (12%, 20) = $100,000 × 7.4694 = $746,940 (at the end of Year 65)

PV = FV × PVIF (12%, 35) = $746,940 × 0.0189 = $14,117.17

b. The first step is to find the value of the twenty $100,000 annuity cash flows at year 65 or year 30. If at year 65, this value is the future value of the 35-payment annuity. If at year 30, this value is the present value of the 35-payment annuity. Let’s use the later approach since we have already computed the value of the twenty $100,000 cash flows at year 30 to answer part a.

PVA = CF × PVIFA (12%, 35)

25

$14,117.17 = CF × 8.1755

CF = $14,117.17 / 8.1755 = $1,726.77

29. a. This can be solved as the present value of a deferred annuity plus the present value of the $500,000. Both present values need to be on Larry’s 30th birthday and then added together for the answer. The annuity is deferred because the payments begin 36 years later after the present value on the 30th birthday.

Total PV

= PV (from #28, part a.) + [$500,000× PVIF (12%, 36)]

= $14,117.17 + [$500,000 × 0.0169] = $14,117.17 + $8,450

= 22,567.17

b. The first step is to find the value of the $500,000 and the twenty $100,000 annuity cash flows at year 65 or year 30. If at year 65, this value is the future value of the 35-payment annuity. If at year 30, this value is the present value of the 35-payment annuity. Let’s use the later approach since we have already computed the value of the $500,000 and the twenty $100,000 cash flows at year 30 to answer part a.

PVA = CF × PVIFA (12%, 35)

$22,567.17 = CF × 8.1755

CF = $22,567.17 / 8.1755 = $2,760.34

30. a. This can be solved as the present value of a deferred annuity plus the present value of the $500,000 minus the $15,000 already invested. Both present values need to be computed to Larry’s 30th birthday, added together and then subtract the $15,000. The annuity is deferred because the payments begin 36 years later after the present value on the 30th birthday.

PV = $22,567.17 (from #29, part a.) - $15,000 = $7,567.17

b. The first step is to find the value of the $500,000 and the twenty $100,000 annuity cash flows at year 30. Then subtract the $15,000 from the sum of these two numbers. This value is the present value of the 35-payment annuity and this value has already been computed to answer part a.

PVA = CF × PVIFA (12%, 35)

$7,567.17 = CF × 8.1755

CF = $7,567.17 / 8.1755 = $925.59

26

APPENDIX

[Simple Future Value Table]

N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.1000 1.1100 1.1200 1.1300 2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.2100 1.2321 1.2544 1.2769 3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 1.2597 1.2950 1.3310 1.3676 1.4049 1.4429 4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 1.3605 1.4116 1.4641 1.5181 1.5735 1.6305 5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 1.4693 1.5386 1.6105 1.6851 1.7623 1.8424 6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 1.5869 1.6771 1.7716 1.8704 1.9738 2.0820 7 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 1.7138 1.8280 1.9487 2.0762 2.2107 2.3526 8 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 1.8509 1.9926 2.1436 2.3045 2.4760 2.6584 9 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 1.9990 2.1719 2.3579 2.5580 2.7731 3.0040

10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 2.1589 2.3674 2.5937 2.8394 3.1058 3.3946 11 1.1157 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.8531 3.1518 3.4785 3.8359 12 1.1268 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.1384 3.4985 3.8960 4.3345 13 1.1381 1.2936 1.4685 1.6651 1.8856 2.1329 2.4098 2.7196 3.0658 3.4523 3.8833 4.3635 4.8980 14 1.1495 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.7975 4.3104 4.8871 5.5348 15 1.1610 1.3459 1.5580 1.8009 2.0789 2.3966 2.7590 3.1722 3.6425 4.1772 4.7846 5.4736 6.2543 16 1.1726 1.3728 1.6047 1.8730 2.1829 2.5404 2.9522 3.4259 3.9703 4.5950 5.3109 6.1304 7.0673 17 1.1843 1.4002 1.6528 1.9479 2.2920 2.6928 3.1588 3.7000 4.3276 5.0545 5.8951 6.8660 7.9861 18 1.1961 1.4282 1.7024 2.0258 2.4066 2.8543 3.3799 3.9960 4.7171 5.5599 6.5436 7.6900 9.0243 19 1.2081 1.4568 1.7535 2.1068 2.5270 3.0256 3.6165 4.3157 5.1417 6.1159 7.2633 8.6128 10.1974 20 1.2202 1.4859 1.8061 2.1911 2.6533 3.2071 3.8697 4.6610 5.6044 6.7275 8.0623 9.6463 11.5231 21 1.2324 1.5157 1.8603 2.2788 2.7860 3.3996 4.1406 5.0338 6.1088 7.4002 8.9492 10.8038 13.0211 22 1.2447 1.5460 1.9161 2.3699 2.9253 3.6035 4.4304 5.4365 6.6586 8.1403 9.9336 12.1003 14.7138 23 1.2572 1.5769 1.9736 2.4647 3.0715 3.8197 4.7405 5.8715 7.2579 8.9543 11.0263 13.5523 16.6266 24 1.2697 1.6084 2.0328 2.5633 3.2251 4.0489 5.0724 6.3412 7.9111 9.8497 12.2392 15.1786 18.7881 25 1.2824 1.6406 2.0938 2.6658 3.3864 4.2919 5.4274 6.8485 8.6231 10.8347 13.5855 17.0001 21.2305 26 1.2953 1.6734 2.1566 2.7725 3.5557 4.5494 5.8074 7.3964 9.3992 11.9182 15.0799 19.0401 23.9905 27 1.3082 1.7069 2.2213 2.8834 3.7335 4.8223 6.2139 7.9881 10.2451 13.1100 16.7386 21.3249 27.1093 28 1.3213 1.7410 2.2879 2.9987 3.9201 5.1117 6.6488 8.6271 11.1671 14.4210 18.5799 23.8839 30.6335 29 1.3345 1.7758 2.3566 3.1187 4.1161 5.4184 7.1143 9.3173 12.1722 15.8631 20.6237 26.7499 34.6158 30 1.3478 1.8114 2.4273 3.2434 4.3219 5.7435 7.6123 10.0627 13.2677 17.4494 22.8923 29.9599 39.1159 31 1.3613 1.8476 2.5001 3.3731 4.5380 6.0881 8.1451 10.8677 14.4618 19.1943 25.4104 33.5551 44.2010 32 1.3749 1.8845 2.5751 3.5081 4.7649 6.4534 8.7153 11.7371 15.7633 21.1138 28.2056 37.5817 49.9471 33 1.3887 1.9222 2.6523 3.6484 5.0032 6.8406 9.3253 12.6760 17.1820 23.2252 31.3082 42.0915 56.4402 34 1.4026 1.9607 2.7319 3.7943 5.2533 7.2510 9.9781 13.6901 18.7284 25.5477 34.7521 47.1425 63.7774 35 1.4166 1.9999 2.8139 3.9461 5.5160 7.6861 10.6766 14.7853 20.4140 28.1024 38.5749 52.7996 72.0685 36 1.4308 2.0399 2.8983 4.1039 5.7918 8.1473 11.4239 15.9682 22.2512 30.9127 42.8181 59.1356 81.4374 37 1.4451 2.0807 2.9852 4.2681 6.0814 8.6361 12.2236 17.2456 24.2538 34.0039 47.5281 66.2318 92.0243 38 1.4595 2.1223 3.0748 4.4388 6.3855 9.1543 13.0793 18.6253 26.4367 37.4043 52.7562 74.1797 103.9874 39 1.4741 2.1647 3.1670 4.6164 6.7048 9.7035 13.9948 20.1153 28.8160 41.1448 58.5593 83.0812 117.5058 40 1.4889 2.2080 3.2620 4.8010 7.0400 10.2857 14.9745 21.7245 31.4094 45.2593 65.0009 93.0510 132.7816 41 1.5038 2.2522 3.3599 4.9931 7.3920 10.9029 16.0227 23.4625 34.2363 49.7852 72.1510 104.2171 150.0432 42 1.5188 2.2972 3.4607 5.1928 7.7616 11.5570 17.1443 25.3395 37.3175 54.7637 80.0876 116.7231 169.5488 43 1.5340 2.3432 3.5645 5.4005 8.1497 12.2505 18.3444 27.3666 40.6761 60.2401 88.8972 130.7299 191.5901 44 1.5493 2.3901 3.6715 5.6165 8.5572 12.9855 19.6285 29.5560 44.3370 66.2641 98.6759 146.4175 216.4968 45 1.5648 2.4379 3.7816 5.8412 8.9850 13.7646 21.0025 31.9204 48.3273 72.8905 109.5302 163.9876 244.6414 46 1.5805 2.4866 3.8950 6.0748 9.4343 14.5905 22.4726 34.4741 52.6767 80.1795 121.5786 183.6661 276.4448 47 1.5963 2.5363 4.0119 6.3178 9.9060 15.4659 24.0457 37.2320 57.4176 88.1975 134.9522 205.7061 312.3826 48 1.6122 2.5871 4.1323 6.5705 10.4013 16.3939 25.7289 40.2106 62.5852 97.0172 149.7970 230.3908 352.9923 49 1.6283 2.6388 4.2562 6.8333 10.9213 17.3775 27.5299 43.4274 68.2179 106.7190 166.2746 258.0377 398.8813 50 1.6446 2.6916 4.3839 7.1067 11.4674 18.4202 29.4570 46.9016 74.3575 117.3909 184.5648 289.0022 450.7359

27

[Simple Present Value Table]

N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 0.9009 0.8929 0.8850 2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264 0.8116 0.7972 0.7831 3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513 0.7312 0.7118 0.6931 4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830 0.6587 0.6355 0.6133 5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209 0.5935 0.5674 0.5428 6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645 0.5346 0.5066 0.4803 7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132 0.4817 0.4523 0.4251 8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665 0.4339 0.4039 0.3762 9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241 0.3909 0.3606 0.3329

10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855 0.3522 0.3220 0.2946 11 0.8963 0.8043 0.7224 0.6496 0.5847 0.5268 0.4751 0.4289 0.3875 0.3505 0.3173 0.2875 0.2607 12 0.8874 0.7885 0.7014 0.6246 0.5568 0.4970 0.4440 0.3971 0.3555 0.3186 0.2858 0.2567 0.2307 13 0.8787 0.7730 0.6810 0.6006 0.5303 0.4688 0.4150 0.3677 0.3262 0.2897 0.2575 0.2292 0.2042 14 0.8700 0.7579 0.6611 0.5775 0.5051 0.4423 0.3878 0.3405 0.2992 0.2633 0.2320 0.2046 0.1807 15 0.8613 0.7430 0.6419 0.5553 0.4810 0.4173 0.3624 0.3152 0.2745 0.2394 0.2090 0.1827 0.1599 16 0.8528 0.7284 0.6232 0.5339 0.4581 0.3936 0.3387 0.2919 0.2519 0.2176 0.1883 0.1631 0.1415 17 0.8444 0.7142 0.6050 0.5134 0.4363 0.3714 0.3166 0.2703 0.2311 0.1978 0.1696 0.1456 0.1252 18 0.8360 0.7002 0.5874 0.4936 0.4155 0.3503 0.2959 0.2502 0.2120 0.1799 0.1528 0.1300 0.1108 19 0.8277 0.6864 0.5703 0.4746 0.3957 0.3305 0.2765 0.2317 0.1945 0.1635 0.1377 0.1161 0.0981 20 0.8195 0.6730 0.5537 0.4564 0.3769 0.3118 0.2584 0.2145 0.1784 0.1486 0.1240 0.1037 0.0868 21 0.8114 0.6598 0.5375 0.4388 0.3589 0.2942 0.2415 0.1987 0.1637 0.1351 0.1117 0.0926 0.0768 22 0.8034 0.6468 0.5219 0.4220 0.3418 0.2775 0.2257 0.1839 0.1502 0.1228 0.1007 0.0826 0.0680 23 0.7954 0.6342 0.5067 0.4057 0.3256 0.2618 0.2109 0.1703 0.1378 0.1117 0.0907 0.0738 0.0601 24 0.7876 0.6217 0.4919 0.3901 0.3101 0.2470 0.1971 0.1577 0.1264 0.1015 0.0817 0.0659 0.0532 25 0.7798 0.6095 0.4776 0.3751 0.2953 0.2330 0.1842 0.1460 0.1160 0.0923 0.0736 0.0588 0.0471 26 0.7720 0.5976 0.4637 0.3607 0.2812 0.2198 0.1722 0.1352 0.1064 0.0839 0.0663 0.0525 0.0417 27 0.7644 0.5859 0.4502 0.3468 0.2678 0.2074 0.1609 0.1252 0.0976 0.0763 0.0597 0.0469 0.0369 28 0.7568 0.5744 0.4371 0.3335 0.2551 0.1956 0.1504 0.1159 0.0895 0.0693 0.0538 0.0419 0.0326 29 0.7493 0.5631 0.4243 0.3207 0.2429 0.1846 0.1406 0.1073 0.0822 0.0630 0.0485 0.0374 0.0289 30 0.7419 0.5521 0.4120 0.3083 0.2314 0.1741 0.1314 0.0994 0.0754 0.0573 0.0437 0.0334 0.0256 31 0.7346 0.5412 0.4000 0.2965 0.2204 0.1643 0.1228 0.0920 0.0691 0.0521 0.0394 0.0298 0.0226 32 0.7273 0.5306 0.3883 0.2851 0.2099 0.1550 0.1147 0.0852 0.0634 0.0474 0.0355 0.0266 0.0200 33 0.7201 0.5202 0.3770 0.2741 0.1999 0.1462 0.1072 0.0789 0.0582 0.0431 0.0319 0.0238 0.0177 34 0.7130 0.5100 0.3660 0.2636 0.1904 0.1379 0.1002 0.0730 0.0534 0.0391 0.0288 0.0212 0.0157 35 0.7059 0.5000 0.3554 0.2534 0.1813 0.1301 0.0937 0.0676 0.0490 0.0356 0.0259 0.0189 0.0139 36 0.6989 0.4902 0.3450 0.2437 0.1727 0.1227 0.0875 0.0626 0.0449 0.0323 0.0234 0.0169 0.0123 37 0.6920 0.4806 0.3350 0.2343 0.1644 0.1158 0.0818 0.0580 0.0412 0.0294 0.0210 0.0151 0.0109 38 0.6852 0.4712 0.3252 0.2253 0.1566 0.1092 0.0765 0.0537 0.0378 0.0267 0.0190 0.0135 0.0096 39 0.6784 0.4619 0.3158 0.2166 0.1491 0.1031 0.0715 0.0497 0.0347 0.0243 0.0171 0.0120 0.0085 40 0.6717 0.4529 0.3066 0.2083 0.1420 0.0972 0.0668 0.0460 0.0318 0.0221 0.0154 0.0107 0.0075 41 0.6650 0.4440 0.2976 0.2003 0.1353 0.0917 0.0624 0.0426 0.0292 0.0201 0.0139 0.0096 0.0067 42 0.6584 0.4353 0.2890 0.1926 0.1288 0.0865 0.0583 0.0395 0.0268 0.0183 0.0125 0.0086 0.0059 43 0.6519 0.4268 0.2805 0.1852 0.1227 0.0816 0.0545 0.0365 0.0246 0.0166 0.0112 0.0076 0.0052 44 0.6454 0.4184 0.2724 0.1780 0.1169 0.0770 0.0509 0.0338 0.0226 0.0151 0.0101 0.0068 0.0046 45 0.6391 0.4102 0.2644 0.1712 0.1113 0.0727 0.0476 0.0313 0.0207 0.0137 0.0091 0.0061 0.0041 46 0.6327 0.4022 0.2567 0.1646 0.1060 0.0685 0.0445 0.0290 0.0190 0.0125 0.0082 0.0054 0.0036 47 0.6265 0.3943 0.2493 0.1583 0.1009 0.0647 0.0416 0.0269 0.0174 0.0113 0.0074 0.0049 0.0032 48 0.6203 0.3865 0.2420 0.1522 0.0961 0.0610 0.0389 0.0249 0.0160 0.0103 0.0067 0.0043 0.0028 49 0.6141 0.3790 0.2350 0.1463 0.0916 0.0575 0.0363 0.0230 0.0147 0.0094 0.0060 0.0039 0.0025 50 0.6080 0.3715 0.2281 0.1407 0.0872 0.0543 0.0339 0.0213 0.0134 0.0085 0.0054 0.0035 0.0022

28

[Future Value of an Ordinary Annuity Table]

N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 2.0100 2.0200 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000 2.1100 2.1200 2.1300 3 3.0301 3.0604 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 3.2781 3.3100 3.3421 3.3744 3.4069 4 4.0604 4.1216 4.1836 4.2465 4.3101 4.3746 4.4399 4.5061 4.5731 4.6410 4.7097 4.7793 4.8498 5 5.1010 5.2040 5.3091 5.4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051 6.2278 6.3528 6.4803 6 6.1520 6.3081 6.4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156 7.9129 8.1152 8.3227 7 7.2135 7.4343 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872 9.7833 10.0890 10.4047 8 8.2857 8.5830 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359 11.8594 12.2997 12.7573 9 9.3685 9.7546 10.1591 10.5828 11.0266 11.4913 11.9780 12.4876 13.0210 13.5795 14.1640 14.7757 15.4157

10 10.4622 10.9497 11.4639 12.0061 12.5779 13.1808 13.8164 14.4866 15.1929 15.9374 16.7220 17.5487 18.4197 11 11.5668 12.1687 12.8078 13.4864 14.2068 14.9716 15.7836 16.6455 17.5603 18.5312 19.5614 20.6546 21.8143 12 12.6825 13.4121 14.1920 15.0258 15.9171 16.8699 17.8885 18.9771 20.1407 21.3843 22.7132 24.1331 25.6502 13 13.8093 14.6803 15.6178 16.6268 17.7130 18.8821 20.1406 21.4953 22.9534 24.5227 26.2116 28.0291 29.9847 14 14.9474 15.9739 17.0863 18.2919 19.5986 21.0151 22.5505 24.2149 26.0192 27.9750 30.0949 32.3926 34.8827 15 16.0969 17.2934 18.5989 20.0236 21.5786 23.2760 25.1290 27.1521 29.3609 31.7725 34.4054 37.2797 40.4175 16 17.2579 18.6393 20.1569 21.8245 23.6575 25.6725 27.8881 30.3243 33.0034 35.9497 39.1899 42.7533 46.6717 17 18.4304 20.0121 21.7616 23.6975 25.8404 28.2129 30.8402 33.7502 36.9737 40.5447 44.5008 48.8837 53.7391 18 19.6147 21.4123 23.4144 25.6454 28.1324 30.9057 33.9990 37.4502 41.3013 45.5992 50.3959 55.7497 61.7251 19 20.8109 22.8406 25.1169 27.6712 30.5390 33.7600 37.3790 41.4463 46.0185 51.1591 56.9395 63.4397 70.7494 20 22.0190 24.2974 26.8704 29.7781 33.0660 36.7856 40.9955 45.7620 51.1601 57.2750 64.2028 72.0524 80.9468 21 23.2392 25.7833 28.6765 31.9692 35.7193 39.9927 44.8652 50.4229 56.7645 64.0025 72.2651 81.6987 92.4699 22 24.4716 27.2990 30.5368 34.2480 38.5052 43.3923 49.0057 55.4568 62.8733 71.4027 81.2143 92.5026 105.4910 23 25.7163 28.8450 32.4529 36.6179 41.4305 46.9958 53.4361 60.8933 69.5319 79.5430 91.1479 104.6029 120.2048 24 26.9735 30.4219 34.4265 39.0826 44.5020 50.8156 58.1767 66.7648 76.7898 88.4973 102.1742 118.1552 136.8315 25 28.2432 32.0303 36.4593 41.6459 47.7271 54.8645 63.2490 73.1059 84.7009 98.3471 114.4133 133.3339 155.6196 26 29.5256 33.6709 38.5530 44.3117 51.1135 59.1564 68.6765 79.9544 93.3240 109.1818 127.9988 150.3339 176.8501 27 30.8209 35.3443 40.7096 47.0842 54.6691 63.7058 74.4838 87.3508 102.7231 121.0999 143.0786 169.3740 200.8406 28 32.1291 37.0512 42.9309 49.9676 58.4026 68.5281 80.6977 95.3388 112.9682 134.2099 159.8173 190.6989 227.9499 29 33.4504 38.7922 45.2189 52.9663 62.3227 73.6398 87.3465 103.9659 124.1354 148.6309 178.3972 214.5828 258.5834 30 34.7849 40.5681 47.5754 56.0849 66.4388 79.0582 94.4608 113.2832 136.3075 164.4940 199.0209 241.3327 293.1992 31 36.1327 42.3794 50.0027 59.3283 70.7608 84.8017 102.0730 123.3459 149.5752 181.9434 221.9132 271.2926 332.3151 32 37.4941 44.2270 52.5028 62.7015 75.2988 90.8898 110.2182 134.2135 164.0370 201.1378 247.3236 304.8477 376.5161 33 38.8690 46.1116 55.0778 66.2095 80.0638 97.3432 118.9334 145.9506 179.8003 222.2515 275.5292 342.4294 426.4632 34 40.2577 48.0338 57.7302 69.8579 85.0670 104.1838 128.2588 158.6267 196.9823 245.4767 306.8374 384.5210 482.9034 35 41.6603 49.9945 60.4621 73.6522 90.3203 111.4348 138.2369 172.3168 215.7108 271.0244 341.5896 431.6635 546.6808 36 43.0769 51.9944 63.2759 77.5983 95.8363 119.1209 148.9135 187.1021 236.1247 299.1268 380.1644 484.4631 618.7493 37 44.5076 54.0343 66.1742 81.7022 101.6281 127.2681 160.3374 203.0703 258.3759 330.0395 422.9825 543.5987 700.1867 38 45.9527 56.1149 69.1594 85.9703 107.7095 135.9042 172.5610 220.3159 282.6298 364.0434 470.5106 609.8305 792.2110 39 47.4123 58.2372 72.2342 90.4091 114.0950 145.0585 185.6403 238.9412 309.0665 401.4478 523.2667 684.0102 896.1984 40 48.8864 60.4020 75.4013 95.0255 120.7998 154.7620 199.6351 259.0565 337.8824 442.5926 581.8261 767.0914 1013.7042 41 50.3752 62.6100 78.6633 99.8265 127.8398 165.0477 214.6096 280.7810 369.2919 487.8518 646.8269 860.1424 1146.4858 42 51.8790 64.8622 82.0232 104.8196 135.2318 175.9505 230.6322 304.2435 403.5281 537.6370 718.9779 964.3595 1296.5289 43 53.3978 67.1595 85.4839 110.0124 142.9933 187.5076 247.7765 329.5830 440.8457 592.4007 799.0655 1081.0826 1466.0777 44 54.9318 69.5027 89.0484 115.4129 151.1430 199.7580 266.1209 356.9496 481.5218 652.6408 887.9627 1211.8125 1657.6678 45 56.4811 71.8927 92.7199 121.0294 159.7002 212.7435 285.7493 386.5056 525.8587 718.9048 986.6386 1358.2300 1874.1646 46 58.0459 74.3306 96.5015 126.8706 168.6852 226.5081 306.7518 418.4261 574.1860 791.7953 1096.1688 1522.2176 2118.8060 47 59.6263 76.8172 100.3965 132.9454 178.1194 241.0986 329.2244 452.9002 626.8628 871.9749 1217.7474 1705.8838 2395.2508 48 61.2226 79.3535 104.4084 139.2632 188.0254 256.5645 353.2701 490.1322 684.2804 960.1723 1352.6996 1911.5898 2707.6334 49 62.8348 81.9406 108.5406 145.8337 198.4267 272.9584 378.9990 530.3427 746.8656 1057.1896 1502.4965 2141.9806 3060.6258 50 64.4632 84.5794 112.7969 152.6671 209.3480 290.3359 406.5289 573.7702 815.0836 1163.9085 1668.7712 2400.0182 3459.5071

29

[Present Value of an Ordinary Annuity Table] N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 0.9009 0.8929 0.8850 2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355 1.7125 1.6901 1.6681 3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869 2.4437 2.4018 2.3612 4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699 3.1024 3.0373 2.9745 5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908 3.6959 3.6048 3.5172 6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553 4.2305 4.1114 3.9975 7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684 4.7122 4.5638 4.4226 8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349 5.1461 4.9676 4.7988 9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590 5.5370 5.3282 5.1317

10 9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446 5.8892 5.6502 5.4262 11 10.3676 9.7868 9.2526 8.7605 8.3064 7.8869 7.4987 7.1390 6.8052 6.4951 6.2065 5.9377 5.6869 12 11.2551 10.5753 9.9540 9.3851 8.8633 8.3838 7.9427 7.5361 7.1607 6.8137 6.4924 6.1944 5.9176 13 12.1337 11.3484 10.6350 9.9856 9.3936 8.8527 8.3577 7.9038 7.4869 7.1034 6.7499 6.4235 6.1218 14 13.0037 12.1062 11.2961 10.5631 9.8986 9.2950 8.7455 8.2442 7.7862 7.3667 6.9819 6.6282 6.3025 15 13.8651 12.8493 11.9379 11.1184 10.3797 9.7122 9.1079 8.5595 8.0607 7.6061 7.1909 6.8109 6.4624 16 14.7179 13.5777 12.5611 11.6523 10.8378 10.1059 9.4466 8.8514 8.3126 7.8237 7.3792 6.9740 6.6039 17 15.5623 14.2919 13.1661 12.1657 11.2741 10.4773 9.7632 9.1216 8.5436 8.0216 7.5488 7.1196 6.7291 18 16.3983 14.9920 13.7535 12.6593 11.6896 10.8276 10.0591 9.3719 8.7556 8.2014 7.7016 7.2497 6.8399 19 17.2260 15.6785 14.3238 13.1339 12.0853 11.1581 10.3356 9.6036 8.9501 8.3649 7.8393 7.3658 6.9380 20 18.0456 16.3514 14.8775 13.5903 12.4622 11.4699 10.5940 9.8181 9.1285 8.5136 7.9633 7.4694 7.0248 21 18.8570 17.0112 15.4150 14.0292 12.8212 11.7641 10.8355 10.0168 9.2922 8.6487 8.0751 7.5620 7.1016 22 19.6604 17.6580 15.9369 14.4511 13.1630 12.0416 11.0612 10.2007 9.4424 8.7715 8.1757 7.6446 7.1695 23 20.4558 18.2922 16.4436 14.8568 13.4886 12.3034 11.2722 10.3711 9.5802 8.8832 8.2664 7.7184 7.2297 24 21.2434 18.9139 16.9355 15.2470 13.7986 12.5504 11.4693 10.5288 9.7066 8.9847 8.3481 7.7843 7.2829 25 22.0232 19.5235 17.4131 15.6221 14.0939 12.7834 11.6536 10.6748 9.8226 9.0770 8.4217 7.8431 7.3300 26 22.7952 20.1210 17.8768 15.9828 14.3752 13.0032 11.8258 10.8100 9.9290 9.1609 8.4881 7.8957 7.3717 27 23.5596 20.7069 18.3270 16.3296 14.6430 13.2105 11.9867 10.9352 10.0266 9.2372 8.5478 7.9426 7.4086 28 24.3164 21.2813 18.7641 16.6631 14.8981 13.4062 12.1371 11.0511 10.1161 9.3066 8.6016 7.9844 7.4412 29 25.0658 21.8444 19.1885 16.9837 15.1411 13.5907 12.2777 11.1584 10.1983 9.3696 8.6501 8.0218 7.4701 30 25.8077 22.3965 19.6004 17.2920 15.3725 13.7648 12.4090 11.2578 10.2737 9.4269 8.6938 8.0552 7.4957 31 26.5423 22.9377 20.0004 17.5885 15.5928 13.9291 12.5318 11.3498 10.3428 9.4790 8.7331 8.0850 7.5183 32 27.2696 23.4683 20.3888 17.8736 15.8027 14.0840 12.6466 11.4350 10.4062 9.5264 8.7686 8.1116 7.5383 33 27.9897 23.9886 20.7658 18.1476 16.0025 14.2302 12.7538 11.5139 10.4644 9.5694 8.8005 8.1354 7.5560 34 28.7027 24.4986 21.1318 18.4112 16.1929 14.3681 12.8540 11.5869 10.5178 9.6086 8.8293 8.1566 7.5717 35 29.4086 24.9986 21.4872 18.6646 16.3742 14.4982 12.9477 11.6546 10.5668 9.6442 8.8552 8.1755 7.5856 36 30.1075 25.4888 21.8323 18.9083 16.5469 14.6210 13.0352 11.7172 10.6118 9.6765 8.8786 8.1924 7.5979 37 30.7995 25.9695 22.1672 19.1426 16.7113 14.7368 13.1170 11.7752 10.6530 9.7059 8.8996 8.2075 7.6087 38 31.4847 26.4406 22.4925 19.3679 16.8679 14.8460 13.1935 11.8289 10.6908 9.7327 8.9186 8.2210 7.6183 39 32.1630 26.9026 22.8082 19.5845 17.0170 14.9491 13.2649 11.8786 10.7255 9.7570 8.9357 8.2330 7.6268 40 32.8347 27.3555 23.1148 19.7928 17.1591 15.0463 13.3317 11.9246 10.7574 9.7791 8.9511 8.2438 7.6344 41 33.4997 27.7995 23.4124 19.9931 17.2944 15.1380 13.3941 11.9672 10.7866 9.7991 8.9649 8.2534 7.6410 42 34.1581 28.2348 23.7014 20.1856 17.4232 15.2245 13.4524 12.0067 10.8134 9.8174 8.9774 8.2619 7.6469 43 34.8100 28.6616 23.9819 20.3708 17.5459 15.3062 13.5070 12.0432 10.8380 9.8340 8.9886 8.2696 7.6522 44 35.4555 29.0800 24.2543 20.5488 17.6628 15.3832 13.5579 12.0771 10.8605 9.8491 8.9988 8.2764 7.6568 45 36.0945 29.4902 24.5187 20.7200 17.7741 15.4558 13.6055 12.1084 10.8812 9.8628 9.0079 8.2825 7.6609 46 36.7272 29.8923 24.7754 20.8847 17.8801 15.5244 13.6500 12.1374 10.9002 9.8753 9.0161 8.2880 7.6645 47 37.3537 30.2866 25.0247 21.0429 17.9810 15.5890 13.6916 12.1643 10.9176 9.8866 9.0235 8.2928 7.6677 48 37.9740 30.6731 25.2667 21.1951 18.0772 15.6500 13.7305 12.1891 10.9336 9.8969 9.0302 8.2972 7.6705 49 38.5881 31.0521 25.5017 21.3415 18.1687 15.7076 13.7668 12.2122 10.9482 9.9063 9.0362 8.3010 7.6730 50 39.1961 31.4236 25.7298 21.4822 18.2559 15.7619 13.8007 12.2335 10.9617 9.9148 9.0417 8.3045 7.6752

30