7: Compounding Frequency

Save this PDF as:

Size: px
Start display at page:

Transcription

1 7.1 Compounding Frequency Nominal and Effective Interest 1 7: Compounding Frequency The factors developed in the preceding chapters all use the interest rate per compounding period as a parameter. This chapter examines the interest rate itself more closely, focusing on what happens as the frequency of compounding changes. For example, the debt owed on \$1 after year 1 at 18% per year compounded annually is \$118., but the debt is \$ at 18% per year compounded monthly. This chapter shows how to convert common financial language into terms suitable for use with factors and how to compute the interest rate on a loan from the borrower's perspective. Then it explains how to analyze cash flows that occur at intervals other than the stated compounding period, and examines cash flows that occur continuously. All of these situations commonly occur in banking, industrial, and governmental analyses. 7.1 Nominal and Effective Interest This section explains common financial terminology, and then it presents formulas that convert this terminology into numbers usable in financial computations. Terminology An interest rate of 18% per year compounded monthly translates to 1½% per month for computational purposes. If \$1 is borrowed today, then the debt after one year is: \$ = 1 (F P, 1½%, 12-) (7-1) The compounding period is one month, so time must be measured accordingly. For example, the last parameter in equation (7-1) is in terms of months (12-) instead of years. The only use of the quoted 18% yearly rate is to compute the monthly rate. An interest rate measures the speed at which debt or capital grows during a stated period of time. For example, if a debt of \$1 would grow to \$ after a year, then its yearly rate of growth is 19.56%: 19.56% = ( ) / 1 (7-2) The rate at which interest actually accumulates over a period is known as the effective interest rate or the yield for that period, so in this case the effective rate or yield is 19.56% per year. Even if the loan were repaid with \$11.5 after 1 month, the effective rate per year is still 19.56%, the percentage by which the debt would increase if unpaid for 12 months. This is similar to someone running a mile in 4 minutes; the person does not have to run for an hour to compute the average speed of 15 miles per hour. The 18% value can be misleading to someone who does not understand financial terminology and compounding. It is known as a nominal rate, meaning in name only. Lenders tend to quote a nominal rate because it is less than the effective rate and hence more attractive. Conversely, institutions providing investment services tend to quote an effective rate or yield since it is greater than the nominal rate and best presents their offerings.

2 7.2 Compounding Frequency Loans and Unknown Interest Rates 2 Another common rate is the annual percentage rate or APR that lenders disclose due to the federal Truth in Lending law. The APR formula is easy for someone without an education in economic analyses to use, but it only approximates the correct effective rate computed using the procedures described below. Formulas Let r be a nominal rate over a long period composed of P short compounding periods, such as 18% per year compounded monthly where the long period (one year) has 12 short compounding periods (months). Then the rate per compounding period is given by i P = r / P. (7-3) Thus 18% per year compounded monthly implies a rate of 18% / 12 or 1½% per month. Compounding at rate i P occurs P times during the long period, so an initial amount X would grow to X (1+ i P ) P by the end of the long period. Expressing this growth as a percentage increase provides the growth rate over the long period, i e = [X (1+ i P ) P X ] / X. (7-4) This is the effective interest rate over the long period, and it simplifies to effective interest rate formula that can be written either as or i e = (1+ i P ) P 1 (7-5) i e = (1+ r /P) P 1. (7-6) If the effective rate is known, then equation (7-5) indicates that i P = (1+ i e ) 1/ P 1, (7-7) so the nominal rate r is P i P. The following examples illustrate computing i P and i e and using them in factors. Nominal rates do not accurately measure the growth of capital or debt and cannot be used in the factors developed thus far in the text. Example 7.1 Effective Rates Table 7.1 shows the effective yearly interest rates corresponding to a nominal rate of 18% per year with different compounding periods. For example, 18% per year compounded monthly corresponds to rates of 1½% per month or % per year. Table 7.1 Compounding Periods Period i P i e Year 18.% =.18 / 1 18.% = (1+.18 / 1) 1 1 Half-Year 9.% =.18 / % = (1+.18 / 2) 2 1 Quarter 4.5% =.18 / % = (1+.18 / 4) 4 1 Month 1.5% =.18 / % = (1+.18 / 12) 12 1 Week.346% =.18 / % = (1+.18 / 52) 52 1 Day.49% =.18 / % = (1+.18 / 365) 365 1

3 7.2 Compounding Frequency Loans and Unknown Interest Rates 3 M i = 1% / month 1, 36 i = 1% / month 1, Figure 7.1 Rates and Cash Flow Diagrams X X 3 i = % / yr 1, Example 7.2 Using Rates in Factors Chablis borrows \$1, at 12% per year compounded monthly. How much must she pay each month over a 3 year period? The nominal rate of 12% is not the rate at which her debt grows, so it cannot be used in the factors. Debt grows at a rate of 12% / 12 or 1% per month, so the monthly payments M in Figure 7.1 are: M = \$33.21 = 1, (A P, 1%, 36-) (7-8) The cash flows occur monthly for this repayment method, so both the interest rate and time must be expressed in terms of months. If she repays in one lump sum X after 3 years, then the problem can be solved in terms of months or years. In terms of months, her payment is given by: X = \$1,43.77 = 1, (F P, 1%, 36-) (7-9) If time is measured in terms of years, then the interest rate must be expressed on a yearly basis. The yearly interest rate is and the payment is % = (1+.12 / 12) 12 1, (7-1) X = \$1,43.77 = 1, (F P, %, 3-) (7-11) Notice that the cash flow diagram always must be expressed in terms of the interest rate used to solve the problem. 7.2 Loans and Unknown Interest Rates This section explains how to compute interest rates on loans and how to choose the best loan. Hundreds of the author's students have done projects that contrasted interest rates supplied by lenders with rates computed by the students, and the rates were rarely the same. There are several reasons for this. Lenders tend to quote nominal rather than effective rates, and they also use the APR formula that only approximates effective rates. More significantly, lenders and borrowers have different viewpoints. Lenders generally seek to compute rates based on their profit, whereas borrowers want to include all cash flows. For example, a lender with a stated rate of 1% per year might issue a check to a borrower for \$1, and require payment of \$1,1 after 1 year. However, suppose that the lender charges \$35 for administrative and risk related items, such as \$1 for loan processing costs and \$25 for a credit check. The borrower actually receives \$965 (1, - 35) at time in exchange for \$1,1 a year later, so the interest rate based on net cash flows is

4 7.2 Compounding Frequency Loans and Unknown Interest Rates % = (1,1 965) / 965. (7-12) This lender implicitly defines interest as solely the payment (\$1) for the use of capital (\$1,). If the administrative (\$1) and risk related (\$25) charges include a profit component for the lender, then this component typically is not reflected in the lender s rate. The borrower s definition recognizes all costs of borrowing, including administrative and risk related charges, in addition to the payment for the use of capital. A prudent borrower examines the offerings of several lenders, seeks out all cash flows, and uses the net cash flow approach. Examples of costs of borrowing include credit checks, administrative costs, loss of cash discounts, additional legal fees, extra insurance, interest losses in non-interest-paying escrow accounts used to pay taxes or insurance, and so forth. The Truth in Lending law attempts to base the APR calculation on a net cash flow basis, but varying interpretations and practices make the APR unreliable, as well as its approximate nature. The following two examples illustrate the procedure for computing interest rates from the borrower s viewpoint and how to select the best loan. Example 7.3 Computing an Unknown Interest Rate Cicero borrows \$1, from Cataline. Credit checks and 33 administrative costs result in Cicero s paying Cataline \$5 at time. The note is \$32 per month for 36 months, plus a monthly \$1 loan insurance charge. What is the interest rate 95 per year? The borrower s viewpoint recognizes all costs of the loan. Figure 7.2 shows that Cicero receives a net amount Figure 7.2 Loan of \$95 (1, - 5) at time, and pays \$33 (32 + 1) per month. Using trial and error to solve the discounted amount formula \$95 = 33 (P A, i m, 36-), (7-13) yields a monthly rate i m of 1.262%. The effective yearly rate is %= ( ) (7-14) Notice that the borrower uses only net cash flows, and the rate from the lender s viewpoint is of no relevance to the borrower, other than satisfying curiosity. In this case, the lender might compute a monthly rate of.786%, the solution to \$1, = 32 (P A, i m, 36-), (7-15) and quote a rate of 9.429% (.786% 12) per year compounded monthly. Example 7.4 Selecting the Best Loan Consider the preceding example, but suppose that Cataline offers to finance the initial loan charges of \$5 at 9.429% per year compounded monthly, increasing the monthly note from \$33 to \$34.6. What is the interest rate of this alternative from Cicero s viewpoint? Cicero now receives a net amount of \$1, at time and pays \$34.6 each month, as shown in Figure 7.3. How Cataline does his computations is not Cicero s concern unless his knowledge helps to negotiate more favorable terms. The monthly rate from Cicero s view , Figure 7.3 Alternative Loan

5 7.2 Compounding Frequency Loans and Unknown Interest Rates 5 point is given by the solution to \$1, = 34.6 (P A, i m, 36-), (7-16) so i m equals 1.239%, and the effective yearly rate is % = ( ) (7-17) Financing the initial charges decreases the interest rate from % to %, but it requires paying interest on an additional \$5 at a rate considerably more than Cicero can earn on his savings. He has an extra \$5 (1, - 95) at time, but \$1.6 less ( ) each month to put into a savings account paying 6% per year compounded monthly. The compound amount of these changes is -\$3.1 = 5 (F P,.5%, 36-) 1.6 (F A,.5%, 36-), (7-18) so Cicero will have \$3.1 less in savings at time 36 if he finances the \$5. If Cicero s objective is to have the most money possible after 36 months, then he should not finance the extra \$5, even though this lowers the interest rate on the loan. Choosing the lowest loan rate is a good approximate procedure or heuristic for selecting loans, but it might not maximize one s final balance. The next chapter pursues the topic of selecting the best alternative in more detail. 7.3 Multi-Period Series An understanding of nominal and effective interest rates contributes to determining unknown interest rates on loans, and it also is helpful for multi-period series in which the intervals between equal cash flows are longer than the compounding periods. Such series require the use of either an effective rate for the longer period or equivalent payments every compounding period, as illustrated by the following example. Example 7.5 Monthly Rate and Quarterly Cash Flows Chablis borrows \$1, at 12% per year compounded monthly. If she makes quarterly payments over a 3 year period, then how much is each payment? Figure 7.4 shows two ways to solve this problem. The method of measuring time in quarters and using the quarterly rate is shown on the left side of the figure. Use the effective interest rate formula with a quarter as the long period and a month as the short period to obtain the effective quarterly rate of 3.31%, 3.31% = (1 +.1) 3 1, (7-19) where 1% is the monthly rate and there are 3 months per quarter. Once the quarterly rate is known, then use capital recovery factor for the 12 quarterly payments: \$1.64 = 1, (A P, 3.31%, 12-). (7-2) The equivalents shown on the right side of the figure provide another solution procedure that will be useful later. A lender seeking to earn 1% per month would want monthly payments of \$33.21: M = \$33.21 = 1, (A P, 1%, 36-). (7-21)

6 7.3 Compounding Frequency Multi-Period Series Q 12 M Q Q 6 Q 33 Q 36 M M M i = 3.31% / qr i = 1% / mo 1, 1, Figure 7.4 Monthly Rate, Quarterly Payments The lender also would accept the compound amounts of these monthly payments at the end of each quarter since either the monthly or their compound amounts produce the same balance. The first compound amount Q 3 occurs at month 3, and its value is \$1.64: Q 3 = \$1.64 = (F A, 1%, 3-). (7-22) The second compound amount Q 6 is at month 6, and its value is also \$1.64: Q 6 = \$1.64 = (F A, 1%, 6-3). (7-23) All compound amounts are the same because the last parameter of the sinking fund factor remains 3 whether it is calculated as 3-, 6-3,, 33-3, or 36-33, so as before. Q = \$1.64 = (F A, 1%, 3), (7-24) Example 7.6 Monthly Rate and Yearly Cash Flows Multi-period series can be difficult to grasp initially, so consider the problem shown in Figure 7.5. If \$1, borrowed at 12% per year compounded monthly is repaid with three yearly notes, then how much is each payment? This is similar to the preceding example, except there are 12 compounding periods between payments instead of 3, so try to anticipate the logic. The monthly rate is 1%, so the effective yearly rate is and the yearly payment is % = (1 +.1) 12 1, (7-25) \$ = 1, (A P, %, 3-). (7-26) Alternatively, the monthly payment at 1% per month is M = \$33.21 = 1, (A P, 1%, 36-), (7-27) Y M Y M Y M Y i = % / yr 1, , i = 1% / mo Figure 7.5 Monthly Rate, Yearly Payments

7 7.3 Compounding Frequency Multi-Period Series 7 and a lender would accept the compound amount Y of 12 such monthly payments each year: Y = \$ = (F A, 1%, 12). (7-28) The two procedures always give the same results if enough digits are used. 7.4 Cash Flows Within Periods Sometimes cash flows occur within the time period for which the interest rate is stated. For example, suppose that the effective interest rate of an account, i e, is advertised as 9.381% per year, but it is known that monthly compounding is done. Solve the effective interest rate formula i e = (1 + i P ) P 1 (7-29) to express the rate for the short period i P in terms of the long period using the rate per period formula: i P = (1 + i e ) 1 / P 1. (7-3) In this case, there are 12 compounding periods per year, so the monthly interest rate is i m = ( ) 1/12 1 (7-31) or.75%. Now a cash flow diagram can be constructed that shows the cash flows each month, with an interest rate of.75% used in the factors. If compounding does not occur within the period, then the timing of the cash flows must be adjusted to either the beginning or end of the period. For example, before computers modernized the banking industry, daily compounding generally was not feasible, and monthly compounding was more common. A deposit made during a month did not begin earning interest until the next month. Conversely, any withdrawals during a month reduced the amount that was compounded for that month. This policy of using adjusted dates for compounding purposes effectively forces all cash flows to occur on a periodic basis. It still is used by institutions that compound daily, but an investor loses only a day s interest instead of a month s interest. Example 7.7 Bonds An investor who wants to earn 1% per year is considering buying a bond with a coupon rate of 8% payable semiannually. The maturity period is 1 years, and the redemption value is \$3,. What is the most that should be paid for the bond? A coupon rate of 8% payable semiannually means that the rate is actually 4% per half-year, so a coupon of \$12 (4% 3,) is received every half-year. The cash flow diagram on the right must be in terms of half-years, since that is the frequency of the cash flows. The investor s effective discount rate is 1% per year, so the semi-annual rate is 3, 12 Figure 7.6 Bond i s = (1 +.1) 1/2 1 (7-32) or 4.881%. The most that the investor should pay is the discounted amount, E = 12(P A, 4.881%, 2-) + 3,(P F, 4.881%, 2-) (7-33) E 1 2 2

8 7.4 Compounding Frequency Cash Flows Within Periods 8 or \$2, Example 7.8 Effect of Not Compounding Cash Flows Within a Period A company deposits \$1, of sales revenues at the end of each month in an account that has an effective interest rate of 9% per year, and it is known that monthly compounding occurs. The amount in the account after 5 years must be determined. One analyst mistakenly ignores the monthly compounding and models the cash flows as shown at the top of Figure 7.7, showing yearly deposits of \$12, with an interest rate of 9% per year. This results in a computed compound amount of \$718,165.27: \$718, = 12, (F A, 9%, 5-) (7-34) Another analyst correctly models the cash flows with monthly deposits of \$1, at.727% per month,.727% = (1 +.9) 1/12 1, (7-35) to obtain a compound amount of: \$747, = 1, (F A,.727%, 6-) (7-36) In the preceding example, the first analyst s simplification produced a result 3.9% smaller than the correct one. Repeating this process with an effective rate of 15% per year produces a result 6.3% too small. Nonetheless, project economics traditionally ignores compounding within a year, a practice that dates back to the use of slide rules. This is justified if approximate results are adequate, but modern computational aides such as calculators and spreadsheets can remove this source of error. This is particularly true in practice where economic calculations require relatively little time compared to obtaining the estimates of cash flows. In the classroom, most of the time is spent on computations, so the simplification of yearly cash flows seems reasonable. 7.5 Continuous Compounding Some financial institutions offer continuous compounding, and continuous reinvestment is the general rule in industry. Table 7.1 shows effective rates as compounding progresses from a yearly to a daily basis, and the change in effective rates diminishes each time. For example, the effective rate increases from 18.% to 18.81% when the compounding changes from yearly to semi-annually, but it only increases from % to % when compounding goes from weekly to daily. This section shows how to compute nominal, effective, and periodic rates as compounding becomes continuous, and then it illustrates the use of these rates in factors. Nominal and Effective Rates Each decrease in the length of the compounding period increases the number of compounding periods each year, until a limiting condition occurs. The formula for effective interest becomes i e = limit (1+ r /P) P 1, (7-37) P E 5 12, a) 9% / yr 1, E b).727% / mo Figure 7.7 Flows within Periods

9 7.5 Compounding Frequency Continuous Compounding 9 where P is the number of compounding periods. The periods for r and i e are the same. For example, if r is a yearly nominal rate, then i e is a yearly effective rate; if r is monthly, then i e is monthly. The limit term equals e r, so the effective rate for continuous compounding is: i e = e r 1. (7-38) For example, a yearly nominal rate of 18% compounded continuously implies a yearly effective rate for use in factors of %: % = e (7-39) This is slightly larger than the % rate for daily compounding shown in Table 7.1. Sometimes the effective rate is known, and the nominal rate must be determined. Solving equation (7-38) for r results in r = ln(1 + i e ), (7-4) where ln is the natural logarithm function. As expected, a yearly effective rate of % corresponds to a yearly nominal rate of 18% compounded continuously: Periodic Rates 18% = ln( ). (7-41) Nominal rates r frequently are quoted on a yearly basis, such as 18% per year compounded continuously. If cash flows occur P times per year, such as monthly where P equals 12, then the rate must be expressed in terms of that period. Equation (7-38) indicates that the yearly effective rate is e r 1, and equation (7-3) provides the rate for the shorter period as i P = (1 + e r 1) 1 / P 1 (7-42) or i P = e r / P 1. (7-43) The monthly rate that corresponds to a nominal yearly rate of 18% compounded continuously is: i m = 1.511% = e.18 / (7-44) This is a true monthly rate that can be used with factors for monthly cash flows. It results in the same effective yearly rate of % ( ) that was computed earlier. Equation (7-43) really is the same as equation (7-38), if the nominal rate is expressed as r / P per short period. For example, a continuous nominal rate of 18% / 12 or 1½% per month corresponds to a monthly rate of i m = 1.511% = e (7-45) In general, continuous nominal rates can be re-expressed in terms of any period, such as 1½% per month, 4½% per quarter, 9% semiannually, 18% per year, 27% per year-and-ahalf, or 36% biennially, without affecting the same effective rate or any computations. This is not the case for discrete compounding where different formulas are used.

10 7.5 Compounding Frequency Continuous Compounding 1 Example 7.9 Rates for Different Periods Table 7.2 converts a nominal rate of 12% per year compounded continuously into rates that can be used in factors for cash flows at different discrete intervals. It does this by expressing the 12% per year as 1% per month, 3% per quarter, and so forth. Note the rates in the table also are valid for an effective rate of % where compounding is continuous, since equation (7-4) indicates that ln( ) equals 12%. Factors Periodic rates such as those computed above accurately measure the growth of capital or debt, so they can be used in all of the single payment, uniform series, and gradient factors developed thus far. For example, if interest is 12% per year compounded continuously, then the discounted amount of \$1, occurring 8 months hence is given by: \$ = 1,(P F, 1.5%, 8-) (7-46) An easier approach is to develop special factors for continuous compounding. If r is the nominal continuous rate per period (e.g., per month, per quarter, etc.), then the periodic rate is e r 1. Substituting e r 1 for i in each factor s formula yields the continuous compounding factors. Single payment factors are used more commonly than the others: and Period Table 7.2 Rates per Period Rate per Period Monthly i m = 1.5% = e.1 1 Quarterly i q = 3.455% = e.3 1 Semiannually i s = % = e.6 1 Yearly i y = % = e.12 1 Biennially i b = % = e.24 1 (P F, r, m) = e r m (7-47) (F P, r, m) = e r m (7-48) Continuous compounding allows cash flows to occur at any time, so m does not have to be an integer. However, it is necessary for m and r to be dimensionally consistent, so one cannot be in terms of months with the other measured in years. Example 7.1 Continuous Compounding Single Payment Factors Consider the cash flows in Figure 7.8 at months 1.5, 3, 7, and 9.2. The interest rate is 12% per year compounded continuously. What is equivalent at time 5? The factors given above easily can be evaluated using a calculator using a monthly nominal rate of 1%, so E 5 = 11 e.1 (5-1.5) + 13 e.1 (5-3) (7-49) + 17 e.1 (7-5) + 18 e.1 (9.2-5), 11 E Figure 7.8 Continuous Compounding Factors

11 7.5 Compounding Frequency Continuous Compounding 11 or \$58.58 will produce the same balance as the original cash flows. 7.6 Continuous Cash Flows Several situations can be modeled using continuous cash flows, such as transactions involving automatic teller machines or on-line credit card purchases of gasoline. This model also can be applied to industrial operations and maintenance expenses that occur fairly uniformly during the year. This section shows how to reduce a continuous cash flow to a single payment that can be manipulated as necessary using the continuous single payment factors. Suppose that funds flow at a rate of f (t), such as \$1, per f (t ) year. Then the amount of funds that flow during some small interval E b at time t is approximately f (t) t, as shown in the figure on the right. For example, if t should be 1 day or 1/365 of a year, then the cash a t b flow for t would be \$1,(1/365). If f (t) is a constant, then f (t) t Figure 7.9 Funds is an exact value, otherwise it is approximate. The compound Flow Equivalent amount at time b is f (t) t e r (b-t). The sum of the compound amounts for all intervals t between a and b is the equivalent of the entire funds flow at time b. Calculus provides this sum as E b b a r( b t) f ( t) e dt, (7-5) where a, b, f (t), and r are all dimensionally consistent. For example, they must all be expressed in terms of years or all in terms of months. The most commonly occurring case is when f (t) is some constant K, such as \$1, per year, where the expression for the equivalent simplifies to: E b e K r( b a) Example 7.11 Constant Funds Flow Function All transactions at a new service station require inserting a credit card into a pump, so that payment automatically is credited to an account E \$5, / mo E 6 paying 6% per year compounded continuously Revenues are expected to be \$5, per Figure 7.1 Constant Funds month from month 3 through month 6. How Flow Function much would have to be received at month to produce the same balance? How much will be in the account at month 12? The first step in solving this problem is to reduce the funds flow to a single equivalent at month 6 using equation (7-51). Dimensional consistency requires expressing the nominal interest rate as ½% per month, and then e 1 E 6 5, (7-52).5 or \$151, Thereafter, single payment factors can be used to compute r 1.5(6 3) t E 12 (7-51)

12 7.6 Compounding Frequency Continuous Cash Flows 12 or \$146,664.6 and or \$155, Example 7.12 Discrete Approximations E = 151,13.65e.5(6-) (7-53) E 12 = 151,13.65e.5(12-6) (7-54) Cash flows that are actually continuous frequently are treated as if they occurred at the end of a period. For example, consider a continuous cash flow of \$365, per year. Suppose that the effective interest rate is 1% per year and that compounding is continuous during the year. Then its actual compound amount after a year can be computed first by determining the nominal rate per year, or 9.531%, and then using equation (7-51) r = ln(1.1) (7-55).9531(1 ) e 1 CA 365,.9531 (7-56) to obtain \$382, Table 7.3 shows the compound amounts caused by assuming Table 7.3 Discrete Approximations end-of-period flows. For example, the Period CA % Error compound amount associated with daily flows assumes \$1, per day at a daily Actual \$382,96.14.% interest rate given by equation (7-43) as Day 382, i d = e.9531/365 1 (7-57) Week 382, or.2612%. These \$1, flows compound Month 381, to 1,(F A,.2612%, 365-) or Quarter 378, \$382,91.15 after a year, a.13% error. Half-Year 373, Similarly, a 52-week and 12-month year Year 365, are used to compute the compound amounts for those periods. The end-of-year assumption typically used in the classroom introduces a 4.69% error. 7.7 Summary This chapter first explains how to convert a nominal rate such as 18% per year compounded monthly to the actual monthly rate of 1½% per month. This monthly rate can be used in factors, or it can be used to compute an effective rate or yield for a longer period, such as a year, using the effective interest rate formula in equation (7-5). Lenders tend to quote nominal rates rather than effective rates. Moreover, these nominal rates frequently are based only on cash flows that the lender claims a profit. For example, the quoted rate might not include costs that a borrower must pay for a credit check, administrative charges, or insurance required as a condition of the loan. A prudent borrower always draws the cash flow diagram from his or her own perspective, and then computes the rate per compounding period. If the period is less than a year, such as a month, then it might be desirable to determine the effective rate per year.

13 7.7 Compounding Frequency Summary 13 The next topic is multi-period series, such as a yearly series when compounding occurs on a monthly basis. There are two approaches to solving problems of this type. The first is to compute the effective rate for the time interval between cash flows. This typically results in a rate that is not in the tables, so the series factors must be computed with a calculator. The second method is to replace each original series flow with its equivalent series having an interval equal to the compounding period. Sometimes cash flows occur more frequently than the quoted interest rate. If the quoted rate is nominal, then the rate for a shorter compounding period equals the quoted rate divided by the number of shorter periods within the nominal period. If the quoted rate is effective, then use the rate per period formula in equation (7-3). If cash flows occur more frequently than the stated compounding frequency, then they are not credited to the account until the beginning of the next period. Continuous compounding allows cash flows to earn interest at any time, not just at the beginning or end of compounding periods. If cash flows should occur at regular intervals, then equation (7-43) converts continuous rate into an effective rate per period that can be used in previously developed factors. Single payments occur frequently enough to warrant developing formulas for the continuous compounding single payment factors. The compound amount of continuous funds flows can be determined by integrating the funds flow function multiplied by the single payment compound amount factor over the region of the flow. The most common case where this occurs is when the funds flow function is a constant, and equation (7-51) provides a convenient formula for the compound amount at the end of the flow. The continuous compounding single payment factors allow equivalents to be computed at other times. Questions Section 1: Nominal and Effective Interest 1.1 An account pays 12% per year compounded monthly. What interest rate should be used in factors for cash flows that occur on a monthly, quarterly, semi-annual, annual, and biennial basis? (1%, 3.31%, %, %, %) 1.2 A bank advertises that its yield on certificates of deposits is 7.8% per year. If compounding is monthly, then what is the yearly nominal rate? (7.5343%) Section 2: Loans and Unknown Interest Rates 2.1 Horace is considering a chariot loan that Catullus quotes at a rate of 9% per year compounded monthly. The credit check and loan origination fee cost \$2. If Horace borrows \$14, with a 36 month repayment period, then his monthly principal and interest payment will be \$ Catullus also requires a credit life insurance policy costing \$2.25 monthly that will pay off the loan if Horace reaches his expiration date prior to paying off the loan, plus extra collision insurance beyond what he would have purchased. This extra insurance costs \$15 per year, payable at the beginning of each year. There are also costs such as recording the title and so forth that amount to \$5; these costs would have to be paid even if the car were not financed. From Horace s viewpoint, what is the effective yearly interest rate? (13.966%)

14 7: Compounding Frequency Questions Suppose that the lender in question 2.1 offers to finance the \$35 for credit check, origination fee, and initial extra insurance payment in exchange for increasing the monthly principal and interest payment by \$ What would the interest rate be on this loan? Should Horace accept this offer of additional financing if he can put his money into an account paying 6% per year compounded monthly? (13.36%, reduce balance by \$18.97 if finance \$35) Section 3: Multi-Period Series 3.1 A loan of \$1, is to be repaid with quarterly payments over 5 years. The interest rate is 8% per year compounded monthly. What are the payments? (\$61.24) 3.2 Suppose that the interest rate is 9% per year, compounded monthly. Deposits of \$1, are made at years 3, 4, and 5, and then equal withdrawals are made at years 1, 11, and 12 resulting in a final balance of \$. a) What is the effective interest per year? (9.3869%) b) Use the effective interest rate to compute the withdrawals. (\$1,873.2) c) What is the rate per month? (.75%) d) Use monthly equivalents to compute the withdrawals. (\$1,873.2) e) Use monthly equivalents to compute the withdrawals if the interest rate changes immediately after the last deposit to 12% per year compounded monthly. (\$2,234.5) 3.3 Withdrawals of \$1, are planned at months 3, 42, 54, and 66. The interest rate is 12% per year compounded monthly. Use monthly equivalents of the withdrawals to determine how much must be deposited today. (\$2,53.2) Section 4: Cash Flows Within Periods 4.1 An investor wants to earn a yield of 9% from a \$1, bond with a coupon rate of 6% payable semiannually. The bond s life is 1 years, and it was issued 4 years ago. The eighth payment will be made immediately after the purchase. What is the maximum that the investor should pay for the bond? (\$9,13.48) 4.2 A company deposits \$1, of sales revenues at the end of each month in an account that has an effective interest rate of 12% per year, and it is known that monthly compounding occurs. Determine the amount in the account after 5 years and the percent error introduced by assuming a single cash flow of \$12, at the end of the year, where Percent Error = (Approximate Value True Value) / True Value. (\$83,412.71, % or 5.112% low) 4.3 A savings program pays 6% per year compounded monthly. However, it credits deposits during the month as if they occurred at the end of the month, and it treats withdrawals during the month as if they occurred at the beginning of the month. Joe McSave makes 6 mid-month deposits of \$1, followed by 6 mid-month withdrawals of \$4. How much is in the account at the end of 12 months? (\$381.77) Section 5: Continuous Compounding 5.1 An account pays 1% per year compounded continuously. a) What is its nominal monthly rate? (.83333%) b) What is its effective monthly rate? (.83682%) c) What is its effective yearly rate based on the nominal yearly rate? (1.5179%) d) Use the effective yearly rate to compute the effective monthly rate. It should equal the previously computed value. (.83682%)

15 7: Compounding Frequency Questions 15 e) If deposits of \$1 are made at the end of months 1, 2,, 12, then how much is in the account at the end of month 12? (\$1,256.8) 5.2 If a certificate of deposit s yearly effective rate is 8% and compounding is continuous, then what is its nominal yearly rate? (7.6961%) 5.3 Interest is 9% per year compounded continuously, and cash flows of \$1 are made at years 6, 7.25, 12.5, and 14. What single cash flow at time 1 is equivalent? (\$421.3) Section 6: Continuous Cash Flows 6.1 Funds flow at a constant rate of \$12, per year, and interest is 9% per year compounded continuously. Consider the funds that flow from month 3.5 to month 9.. a) What is their discounted amount at month 1? (\$52,88.21) b) What is their compound amount at month 19.5? (\$6,75.73) 6.2 Other types of flow rates also can occur in addition to the constant funds flow. For example, sometimes revenues decline exponentially over time. This leads to a funds flow function f (t) of the form Ve wt. Using this function in equation (7-5) results in the following compound amount: E b e V r( b a) wa e w r wb (7-58) As an example, suppose that the flow rate of an oil well at year t is 5,e.1t barrels per year. If oil sells for \$2 per barrel, then the funds flow rate function is \$1,e.1t per year. What is the compound amount at month 18 of a flow from months 6 to 18 (years.5 to 1.5) if the interest rate is 8% per year compounded continuously? (\$94,33.64) 6.3 Compute the percent errors for Table 7.3 if the effective interest rate is 15%, but compounding is continuous? (.%,.191%,.1343%,.5812%, %, %, %)

PRACTICE EXAMINATION NO. 5 May 2005 Course FM Examination. 1. Which of the following expressions does NOT represent a definition for a n

PRACTICE EXAMINATION NO. 5 May 2005 Course FM Examination 1. Which of the following expressions does NOT represent a definition for a n? A. v n ( 1 + i)n 1 i B. 1 vn i C. v + v 2 + + v n 1 vn D. v 1 v

APPENDIX. Interest Concepts of Future and Present Value. Concept of Interest TIME VALUE OF MONEY BASIC INTEREST CONCEPTS

CHAPTER 8 Current Monetary Balances 395 APPENDIX Interest Concepts of Future and Present Value TIME VALUE OF MONEY In general business terms, interest is defined as the cost of using money over time. Economists

Chapter 3 Understanding Money Management. Nominal and Effective Interest Rates Equivalence Calculations Changing Interest Rates Debt Management

Chapter 3 Understanding Money Management Nominal and Effective Interest Rates Equivalence Calculations Changing Interest Rates Debt Management 1 Understanding Money Management Financial institutions often

The Basics of Interest Theory

Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest

5.1 Simple and Compound Interest

5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?

Introduction to Bond Valuation. Types of Bonds

Introduction to Bond Valuation (Text reference: Chapter 5 (Sections 5.1-5.3, Appendix)) Topics types of bonds valuation of bonds yield to maturity term structure of interest rates more about forward rates

3. Time value of money. We will review some tools for discounting cash flows.

1 3. Time value of money We will review some tools for discounting cash flows. Simple interest 2 With simple interest, the amount earned each period is always the same: i = rp o where i = interest earned

Grade 7 & 8 Math Circles. Finance

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 7 & 8 Math Circles October 22/23, 2013 Finance A key point in finance is the time value of money, a concept which states that a dollar

Chapter 4 Nominal and Effective Interest Rates

Chapter 4 Nominal and Effective Interest Rates Chapter 4 Nominal and Effective Interest Rates INEN 303 Sergiy Butenko Industrial & Systems Engineering Texas A&M University Nominal and Effective Interest

3. Time value of money

1 Simple interest 2 3. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned

Time Value of Money 1

Time Value of Money 1 This topic introduces you to the analysis of trade-offs over time. Financial decisions involve costs and benefits that are spread over time. Financial decision makers in households

Compound Interest Chapter 8

8-2 Compound Interest Chapter 8 8-3 Learning Objectives After completing this chapter, you will be able to: > Calculate maturity value, future value, and present value in compound interest applications,

With compound interest you earn an additional \$128.89 (\$1628.89 - \$1500).

Compound Interest Interest is the amount you receive for lending money (making an investment) or the fee you pay for borrowing money. Compound interest is interest that is calculated using both the principle

The Time Value of Money (contd.)

The Time Value of Money (contd.) February 11, 2004 Time Value Equivalence Factors (Discrete compounding, discrete payments) Factor Name Factor Notation Formula Cash Flow Diagram Future worth factor (compound

Introduction to Real Estate Investment Appraisal

Introduction to Real Estate Investment Appraisal Maths of Finance Present and Future Values Pat McAllister INVESTMENT APPRAISAL: INTEREST Interest is a reward or rent paid to a lender or investor who has

Section 8.3 Notes- Compound Interest

Section 8.3 Notes- Compound The Difference between Simple and Compound : Simple is paid on your investment or principal and NOT on any interest added Compound paid on BOTH on the principal and on all interest

Solutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material. i = 0.75 1 for six months.

Solutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material 1. a) Let P be the recommended retail price of the toy. Then the retailer may purchase the toy at

\$496. 80. Example If you can earn 6% interest, what lump sum must be deposited now so that its value will be \$3500 after 9 months?

Simple Interest, Compound Interest, and Effective Yield Simple Interest The formula that gives the amount of simple interest (also known as add-on interest) owed on a Principal P (also known as present

Calculating interest rates

Calculating interest rates A reading prepared by Pamela Peterson Drake O U T L I N E 1. Introduction 2. Annual percentage rate 3. Effective annual rate 1. Introduction The basis of the time value of money

Ch 3 Understanding money management

Ch 3 Understanding money management 1. nominal & effective interest rates 2. equivalence calculations using effective interest rates 3. debt management If payments occur more frequently than annual, how

Mathematics. Rosella Castellano. Rome, University of Tor Vergata

and Loans Mathematics Rome, University of Tor Vergata and Loans Future Value for Simple Interest Present Value for Simple Interest You deposit E. 1,000, called the principal or present value, into a savings

Compound Interest Formula

Mathematics of Finance Interest is the rental fee charged by a lender to a business or individual for the use of money. charged is determined by Principle, rate and time Interest Formula I = Prt \$100 At

NOTES E. Borrower Receives: Loan Value LV MATURITY START DATE. Lender Fixed Fixed Fixed Receives: Payment FP Payment FP Payment FP

NOTES E DEBT INSTRUMENTS Debt instrument is defined as a particular type of security that requires the borrower to pay the lender certain fixed dollar amounts at regular intervals until a specified time

CHAPTER 1. Compound Interest

CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.

Personal Financial Literacy

Personal Financial Literacy 6 Unit Overview Many Americans both teenagers and adults do not make responsible financial decisions. Learning to be responsible with money means looking at what you earn compared

University of Virginia - math1140: Financial Mathematics Fall 2011 Exam 1 7:00-9:00 pm, 26 Sep 2011 Honor Policy. For this exam, you must work alone. No resources may be used during the quiz. The only

Financial Mathematics

Financial Mathematics 1 Introduction In this section we will examine a number of techniques that relate to the world of financial mathematics. Calculations that revolve around interest calculations and

SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS Unit Five Moses Mwale e-mail: moses.mwale@ictar.ac.zm BBA 120 Business Mathematics Contents Unit 5: Mathematics

)n = (1 + i(m) 3. According to the question, choose the appropriate reference time point, denoted by t 0.

General Annuity. Discrete Annuity Dot Product Strategy is an algebraic evaluation of the present value of annuity, which is defined as a sequence of n periodical payments, where n is either finite or infinite.

Lecture 4: Amortization and Bond

Lecture 4: Amortization and Bond Goals: Study Amortization with non-level interests and non-level payments. Generate formulas for amortization. Study Sinking-fund method of loan repayment. Introduce bond.

SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS

SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS This page indicates changes made to Study Note FM-09-05. April 28, 2014: Question and solutions 61 were added. January 14, 2014:

Time Value of Money. MATH 100 Survey of Mathematical Ideas. J. Robert Buchanan. Department of Mathematics. Fall 2014

Time Value of Money MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014 Terminology Money borrowed or saved increases in value over time. principal: the amount

Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis

Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................

Module 1: Corporate Finance and the Role of Venture Capital Financing Time Value of Money 1.0 INTEREST THE COST OF MONEY 1.01 Introduction to Interest 1.02 Time Value of Money Taxonomy 1.03 Cash Flow Diagrams

Chapter 28 Time Value of Money

Chapter 28 Time Value of Money Lump sum cash flows 1. For example, how much would I get if I deposit \$100 in a bank account for 5 years at an annual interest rate of 10%? Let s try using our calculator:

Time-Value-of-Money and Amortization Worksheets

2 Time-Value-of-Money and Amortization Worksheets The Time-Value-of-Money and Amortization worksheets are useful in applications where the cash flows are equal, evenly spaced, and either all inflows or

Compound Interest. Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate:

Compound Interest Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate: Table 1 Development of Nominal Payments and the Terminal Value, S.

Simple Interest. and Simple Discount

CHAPTER 1 Simple Interest and Simple Discount Learning Objectives Money is invested or borrowed in thousands of transactions every day. When an investment is cashed in or when borrowed money is repaid,

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM FM SAMPLE QUESTIONS

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Copyright 2005 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions

ENGINEERING ECONOMICS AND FINANCE

CHAPTER Risk Analysis in Engineering and Economics ENGINEERING ECONOMICS AND FINANCE A. J. Clark School of Engineering Department of Civil and Environmental Engineering 6a CHAPMAN HALL/CRC Risk Analysis

Future Value or Accumulated Amount: F = P + I = P + P rt = P (1 + rt)

F.1 Simple Interest If P dollars (called the principal or present value) earns interest at a simple interest rate of r per year (as a decimal) for t years, then: Interest: I = P rt Examples: Future Value

ISE 2014 Chapter 3 Section 3.11 Deferred Annuities

ISE 2014 Chapter 3 Section 3.11 Deferred Annuities If we are looking for a present (future) equivalent sum at time other than one period prior to the first cash flow in the series (coincident with the

CHAPTER 2 TIME VALUE OF MONEY

CHAPTER 2 TIME VALUE OF MONEY 2.1 Concepts of Engineering Economics Analysis Engineering Economy: is a collection of mathematical techniques which simplify economic comparisons. Time Value of Money: means

Statistical Models for Forecasting and Planning

Part 5 Statistical Models for Forecasting and Planning Chapter 16 Financial Calculations: Interest, Annuities and NPV chapter 16 Financial Calculations: Interest, Annuities and NPV Outcomes Financial information

Calculations for Time Value of Money

KEATMX01_p001-008.qxd 11/4/05 4:47 PM Page 1 Calculations for Time Value of Money In this appendix, a brief explanation of the computation of the time value of money is given for readers not familiar with

Notes for Lecture 3 (February 14)

INTEREST RATES: The analysis of interest rates over time is complicated because rates are different for different maturities. Interest rate for borrowing money for the next 5 years is ambiguous, because

CE303 Introduction to Construction Engineering

CE303 Introduction to Construction Engineering Engineering Economics Engineering Economics Study of the desirability of making an investment Very little, if any, true economics (micro or macro) in this

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount

Compound Interest and Continuous Growth

Warm Up 1 SECTION 6.3 Compound Interest and Continuous Growth 1 Use the compound interest formula to calculate the future value of an investment 2 Construct and use continuous growth models 3 Use exponential

MATHEMATICS OF FINANCE AND INVESTMENT

MATHEMATICS OF FINANCE AND INVESTMENT G. I. FALIN Department of Probability Theory Faculty of Mechanics & Mathematics Moscow State Lomonosov University Moscow 119992 g.falin@mail.ru 2 G.I.Falin. Mathematics

Basic financial arithmetic

2 Basic financial arithmetic Simple interest Compound interest Nominal and effective rates Continuous discounting Conversions and comparisons Exercise Summary File: MFME2_02.xls 13 This chapter deals

Chapter Two. THE TIME VALUE OF MONEY Conventions & Definitions

Chapter Two THE TIME VALUE OF MONEY Conventions & Definitions Introduction Now, we are going to learn one of the most important topics in finance, that is, the time value of money. Note that almost every

Chapter. Discounted Cash Flow Valuation. CORPRATE FINANCE FUNDAMENTALS by Ross, Westerfield & Jordan CIG.

Chapter 6 Discounted Cash Flow Valuation CORPRATE FINANCE FUNDAMENTALS by Ross, Westerfield & Jordan CIG. Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 1. The simple interest per year is: \$5,000.08 = \$400 So after 10 years you will have: \$400 10 = \$4,000 in interest. The total balance will be

Chapter 4 Discounted Cash Flow Valuation

University of Science and Technology Beijing Dongling School of Economics and management Chapter 4 Discounted Cash Flow Valuation Sep. 2012 Dr. Xiao Ming USTB 1 Key Concepts and Skills Be able to compute

Savings Accounts, Term Deposits, and Guaranteed Investment Certificates

280 Chapter 8 Simple Interest and Applications 13. \$1800 due two months ago and \$700 due in three months are to be repaid by a payment of \$1500 today and the balance payment in four months. Find the balance

MGF 1107 Spring 11 Ref: 606977 Review for Exam 2. Write as a percent. 1) 3.1 1) Write as a decimal. 4) 60% 4) 5) 0.085% 5)

MGF 1107 Spring 11 Ref: 606977 Review for Exam 2 Mr. Guillen Exam 2 will be on 03/02/11 and covers the following sections: 8.1, 8.2, 8.3, 8.4, 8.5, 8.6. Write as a percent. 1) 3.1 1) 2) 1 8 2) 3) 7 4 3)

Section 5.2 Compound Interest

Section 5.2 Compound Interest With annual simple interest, you earn interest each year on your original investment. With annual compound interest, however, you earn interest both on your original investment

SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Interest Theory

SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Interest Theory This page indicates changes made to Study Note FM-09-05. January 14, 2014: Questions and solutions 58 60 were

Present Value Concepts

Present Value Concepts Present value concepts are widely used by accountants in the preparation of financial statements. In fact, under International Financial Reporting Standards (IFRS), these concepts

ICASL - Business School Programme Quantitative Techniques for Business (Module 3) Financial Mathematics TUTORIAL 2A This chapter deals with problems related to investing money or capital in a business

Finding the Payment \$20,000 = C[1 1 / 1.0066667 48 ] /.0066667 C = \$488.26

Quick Quiz: Part 2 You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? You want to receive \$5,000 per month in retirement.

III INTEREST FORMULAS AND EQUIVALENCE. willing to pay for the use of money. The question arises why people may want to pay for use of others

III 27 INTEREST FORMULAS AND EQUIVALENCE 3.0 INTEREST Interest is money earned (paid) for use of money. This definition of interest implies that people are willing to pay for the use of money. The question

Annuities, Sinking Funds, and Amortization Math Analysis and Discrete Math Sections 5.3 and 5.4

Annuities, Sinking Funds, and Amortization Math Analysis and Discrete Math Sections 5.3 and 5.4 I. Warm-Up Problem Previously, we have computed the future value of an investment when a fixed amount of

1 Pricing of Financial Instruments

ESTM 60202: Financial Mathematics ESTEEM Fall 2009 Alex Himonas 01 Lecture Notes 1 September 30, 2009 1 Pricing of Financial Instruments These lecture notes will introduce and explain the use of financial

The following is an article from a Marlboro, Massachusetts newspaper.

319 CHAPTER 4 Personal Finance The following is an article from a Marlboro, Massachusetts newspaper. NEWSPAPER ARTICLE 4.1: LET S TEACH FINANCIAL LITERACY STEPHEN LEDUC WED JAN 16, 2008 Boston - Last week

Chapter 6. Discounted Cash Flow Valuation. Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Answer 6.1

Chapter 6 Key Concepts and Skills Be able to compute: the future value of multiple cash flows the present value of multiple cash flows the future and present value of annuities Discounted Cash Flow Valuation

2. What is your best estimate of what the price would be if the riskless interest rate was 9% (compounded semi-annually)? (1.04)

Lecture 4 1 Bond valuation Exercise 1. A Treasury bond has a coupon rate of 9%, a face value of \$1000 and matures 10 years from today. For a treasury bond the interest on the bond is paid in semi-annual

The Institute of Chartered Accountants of India

CHAPTER 4 SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY APPLICATIONS SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY- APPLICATIONS LEARNING OBJECTIVES After studying this chapter students will be able

CHAPTER 5. Interest Rates. Chapter Synopsis

CHAPTER 5 Interest Rates Chapter Synopsis 5.1 Interest Rate Quotes and Adjustments Interest rates can compound more than once per year, such as monthly or semiannually. An annual percentage rate (APR)

Time Value of Money. Background

Time Value of Money (Text reference: Chapter 4) Topics Background One period case - single cash flow Multi-period case - single cash flow Multi-period case - compounding periods Multi-period case - multiple

Rate of Return Analysis

Basics Along with the PW and AW criteria, the third primary measure of worth is rate of return Mostly preferred over PW and AW criteria. Ex: an alternative s worth can be reported as: 15% rate of return

CHAPTER 17 ENGINEERING COST ANALYSIS

CHAPTER 17 ENGINEERING COST ANALYSIS Charles V. Higbee Geo-Heat Center Klamath Falls, OR 97601 17.1 INTRODUCTION In the early 1970s, life cycle costing (LCC) was adopted by the federal government. LCC

LO.a: Interpret interest rates as required rates of return, discount rates, or opportunity costs.

LO.a: Interpret interest rates as required rates of return, discount rates, or opportunity costs. 1. The minimum rate of return that an investor must receive in order to invest in a project is most likely

Saving and Investing

Presentation Slides \$ Lesson Twelve Saving and Investing 04/09 pay yourself first (a little can add up) example 1: Save this each week At % Interest In 10 years you ll have \$7.00 5% \$4,720 \$14.00 5% \$9,400

Exercise 6 Find the annual interest rate if the amount after 6 years is 3 times bigger than the initial investment (3 cases).

Exercise 1 At what rate of simple interest will \$500 accumulate to \$615 in 2.5 years? In how many years will \$500 accumulate to \$630 at 7.8% simple interest? (9,2%,3 1 3 years) Exercise 2 It is known that

Chapter 1. Theory of Interest 1. The measurement of interest 1.1 Introduction

Chapter 1. Theory of Interest 1. The measurement of interest 1.1 Introduction Interest may be defined as the compensation that a borrower of capital pays to lender of capital for its use. Thus, interest

, plus the present value of the \$1,000 received in 15 years, which is 1, 000(1 + i) 30. Hence the present value of the bond is = 1000 ;

2 Bond Prices A bond is a security which offers semi-annual* interest payments, at a rate r, for a fixed period of time, followed by a return of capital Suppose you purchase a \$,000 utility bond, freshly

Overview of Lecture 5 (part of Lecture 4 in Reader book)

Overview of Lecture 5 (part of Lecture 4 in Reader book) Bond price listings and Yield to Maturity Treasury Bills Treasury Notes and Bonds Inflation, Real and Nominal Interest Rates M. Spiegel and R. Stanton,

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and

FIN Chapter 5. Time Value of Money. Liuren Wu

FIN 3000 Chapter 5 Time Value of Money Liuren Wu Overview 1. Using Time Lines 2. Compounding and Future Value 3. Discounting and Present Value 4. Making Interest Rates Comparable 2 Learning Objectives

Financial Mathematics for Actuaries. Chapter 1 Interest Accumulation and Time Value of Money

Financial Mathematics for Actuaries Chapter 1 Interest Accumulation and Time Value of Money 1 Learning Objectives 1. Basic principles in calculation of interest accumulation 2. Simple and compound interest

CALCULATOR TUTORIAL. Because most students that use Understanding Healthcare Financial Management will be conducting time

CALCULATOR TUTORIAL INTRODUCTION Because most students that use Understanding Healthcare Financial Management will be conducting time value analyses on spreadsheets, most of the text discussion focuses

5. Time value of money

1 Simple interest 2 5. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned

CHAPTER 3. Exponential and Logarithmic Functions

CHAPTER 3 Exponential and Logarithmic Functions Section 3.1 (e-book 5.1-part 1) Exponential Functions and Their Graphs Definition 1: Let. An exponential function with base a is a function defined as Example

Review Solutions FV = 4000*(1+.08/4) 5 = \$4416.32

Review Solutions 1. Planning to use the money to finish your last year in school, you deposit \$4,000 into a savings account with a quoted annual interest rate (APR) of 8% and quarterly compounding. Fifteen

Study Questions for Actuarial Exam 2/FM By: Aaron Hardiek June 2010

P a g e 1 Study Questions for Actuarial Exam 2/FM By: Aaron Hardiek June 2010 P a g e 2 Background The purpose of my senior project is to prepare myself, as well as other students who may read my senior

11.1 What You Will Learn

11.1 What You Will Learn Percent, Fractions, and Decimal Numbers (skip, assume that you known) Percent Change Percent Markup and Markdown 11.1-1 Percent Change Formula: Percent change = amount in latest

Overview of Lecture 4

Overview of Lecture 4 Examples of Quoted vs. True Interest Rates Banks Auto Loan Forward rates, spot rates and bond prices How do things change when interest rates vary over different periods? Present

Top 50 Banking Interview Questions

Top 50 Banking Interview Questions 1) What is bank? What are the types of banks? A bank is a financial institution licensed as a receiver of cash deposits. There are two types of banks, commercial banks

Annuity is defined as a sequence of n periodical payments, where n can be either finite or infinite.

1 Annuity Annuity is defined as a sequence of n periodical payments, where n can be either finite or infinite. In order to evaluate an annuity or compare annuities, we need to calculate their present value

Time Value of Money. Work book Section I True, False type questions. State whether the following statements are true (T) or False (F)

Time Value of Money Work book Section I True, False type questions State whether the following statements are true (T) or False (F) 1.1 Money has time value because you forgo something certain today for

ST334 ACTUARIAL METHODS

ST334 ACTUARIAL METHODS version 214/3 These notes are for ST334 Actuarial Methods. The course covers Actuarial CT1 and some related financial topics. Actuarial CT1 which is called Financial Mathematics

Fin 3312 Sample Exam 1 Questions

Fin 3312 Sample Exam 1 Questions Here are some representative type questions. This review is intended to give you an idea of the types of questions that may appear on the exam, and how the questions might

Time Value of Money CAP P2 P3. Appendix. Learning Objectives. Conceptual. Procedural

Appendix B Time Value of Learning Objectives CAP Conceptual C1 Describe the earning of interest and the concepts of present and future values. (p. B-1) Procedural P1 P2 P3 P4 Apply present value concepts

Chapter 5 Time Value of Money

1. Future Value of a Lump Sum 2. Present Value of a Lump Sum 3. Future Value of Cash Flow Streams 4. Present Value of Cash Flow Streams 5. Perpetuities 6. Uneven Series of Cash Flows 7. Other Compounding