Excel 2002 Formulas phần 5 ppsx

EXAMPLE 20 What are the payments on a loan of $200,000 over 10 years, at 0.5% interest per month (with payments in arrears)? This example is illustrated in Figure 11-4. Figure 11-4: Calculating a loan payment Function required: PMT(rate, nper, pv, fv, type) The following formula returns $2,220.41: =PMT(0.5%,120,200000,0,0) This result can be verified by using the PV function to calculate the loan amount. The following formula returns $200,000: =PV(0.5%,120,-2220.41,0,0) In this example, the loan is fully repaid after 10 years, and the fv argument is zero. Also note that the payments are to be monthly, and the monthly loan rate has been quoted. Therefore, the 10-year term is converted to months. EXAMPLE 21 I can afford payments of $2,500 per month, and can borrow at 0.45% (per month) over 20 years. How much can I afford to borrow on a fully redeemable mortgage? Function required: PV(rate, nper, pmt, fv, type) This formula returns $366,433.74: =PV(0.45%,240,-2500,0,0) Chapter 11: Introducing Financial Formulas 309 4800-x Ch11.F 8/27/01 11:56 AM Page 309 Note that, with mortgages, we always assume payments are in arrears and that the type argument is 0. Also note that the rate of interest (and the payments) are monthly. Therefore, the term of 20 years must be converted to months. You can check the answer by using the calculated answer to determine the rate on a mortgage of $366,433.74 over 240 months. The following formula returns 0.45%: =RATE(240,-2500,366433.74,0,0) EXAMPLE 22 I currently owe $150,000 on a mortgage, and make payments of $1,900 per month. The current interest rate is 0.45% per month. How long will it take to repay the loan? Function required: NPER(rate, pmt, pv, fv, type) The following formula returns 97.76: =NPER(0.45%,-1900,150000,0,0) Because interest and payments are monthly, the formula returns the amortiza- tion period in months. This answer, although correct in mathematical terms, has a practical implication. Payments are actually made on exact monthly anniversaries. This calculation implies that the loan somehow gets repaid 0.76 of the way through the 98th month. In reality, you have a choice: make an additional payment at the end of 97 months, or make a reduced level payment after 98 months. These options can be calculated using the FV function. To calculate the additional payment at the end of 97 months, calculate the amount due using this formula (which returns –$1,429.85): =FV(0.45%,97,-1900,150000,0) Therefore, the final payment after 97 months is –$3,329.85 (that is, the normal payment of –$1,900 plus –$1,429.85). To calculate the reduced payment after 98 months, use this formula (which returns +$463.72): =FV(0.45%,98,-1900,150000,0) Therefore, the final payment after 98 months is –$1,436.28 (that is, the normal payment of –$1,900 plus $463.72). A relatively frequent problem arises where the payment is less than the amount of the interest portion on the outstanding balance. In this example, the outstanding loan is $150,000, and interest in the first month is $675 ($150,000 * 0.45%). If the payment is less than this amount, the outstanding 310 Part III: Financial Formulas 4800-x Ch11.F 8/27/01 11:56 AM Page 310 balance will continue to increase, and the loan will extend to infinity (rather than seem to last for infinity). If this happens, the NPER function returns the error message #NUM!. EXAMPLE 23 A consumer credit agreement provides that I borrow $1,000 and pay $100 per month in advance for 12 months. What is the rate of interest? Function required: RATE(nper, pmt, pv, fv, type, guess) The following formula returns 3.503153%: =RATE(12,-100,1000,0,1) Before you start to think how generous this agreement is, remember that pay- ments are per month. Therefore, the result is the monthly effective rate! The annual effective equivalent rate is 51.16%, calculated as follows: =((1+0.03503153)^12)-1 The annual rate, based on the nominal compounded monthly basis, returns 42.05%, calculated as follows: =3.503153 * 12 There is a large difference between the annual effective rate and the equiva- lent nominal rate compounded monthly.The size of the difference increases with the level of the rates used. EXAMPLE 24 I borrow $300,000 on a balloon mortgage over 15 years, with monthly payments on $100,000. The balance of $200,000 is due at the end of the term. The rate of inter- est is 0.4% per month, and payments are made monthly in arrears. What will the payments be? A common type of mortgage (used to increase the amount that can be borrowed) is the so-called “balloon” mortgage. The loan is divided into two elements: 1) the “payment” element, where payments fully redeem part of the loan by the end of the term, and 2) the “balloon” element. During the loan term, interest only (no princi- pal) is paid on the balloon element. The principal balance is paid as a lump sum at the end of the loan. Chapter 11: Introducing Financial Formulas 311 4800-x Ch11.F 8/27/01 11:56 AM Page 311 The ability to use an fv argument in the PV, PMT, RATE, and NPER functions make it relatively easy to perform balloon mortgage calculations. Function required: PMT(rate, nper, pv, fv, type) The following formula returns –$1,580.41: =PMT(0.4%,180,300000,-200000,0) Note that the total mortgage of $300,000 is used for the pv argument. This calculation can be checked using the calculated payment to determine the PV. This formula returns $299,999.43 (the rounding error is caused by using a rounded payment amount): =PV(0.4%,180,-1580.41,-200000,0) The payments on a balloon basis can be compared with payments on a tradi- tional mortgage. This formula returns $202,509.64 (traditional mortgage): =PV(0.4%,180,-1580.41,0,0) And payments for the $300,000 traditional mortgage are –$2,341.24, calculated with this formula: =PMT(0.4%,180,300000,0,0) The previous amortization calculation examples can be modified for balloon mortgages by providing an fv argument in the PV, PMT, NPER, and RATE functions. You can also calculate the balloon mortgage element itself with the FV function. This is a calculation that requires a careful interpretation of the sign of the result. If the FV function returns a positive value, that means that the original mortgage has been overpaid and this amount is now due to the borrower. If it returns a negative amount, this is the amount of the balloon element. A balloon element will exist in cases where the amount of the payments do not fully pay the loan during the mort- gage term at the quoted interest rate. Typically, these calculations are made in two stages. First, calculate the payment on the normal amortization loan (usually in accordance with lender rules). Second, calculate how much “balloon” element an additional payment will allow. Example 25 provides the details. EXAMPLE 25 If the bank insists on an amortization of $200,000 of a loan, how much extra can I borrow on the balloon mortgage basis if I can afford payments of $3,000 per month? The term of the loan is 10 years, and the current rate is 0.4% per month. 312 Part III: Financial Formulas 4800-x Ch11.F 8/27/01 11:56 AM Page 312 Function required: PMT(rate, nper, pv, fv, type) The first step is to calculate the payment for a $200,000 normal amortization loan. The following formula returns –$2,101.81: =PMT(0.4%,120,200000,0,0) If payments of $3,000 are affordable, the additional amount of $898.19 can be paid as interest on the balloon element (that is, $3,000 – $2,101.81). The balloon element can now be calculated because the amount of interest is known. This for- mula, which represents the balloon element, returns $224,546.88: =898.19 / 0.4% The calculation can be checked by calculating the payment based on a total mortgage of $424,546.88 with a balloon element of $224,546.88. The following formula returns –$3,000: =PMT(0.4%,120,424546.88,-224546.88,0) Converting Interest Rates The previous examples have been conveniently expressed to allow easy matching of the interest rate with the payment frequency and total term. Often, however, interpreting a financial problem will be more difficult. There are two situations in which interest rate conversions must be made: ◆ When you must do calculations involving a frequency of payments or a number of time periods, and the rate that you are required to use does not match the frequency of payments or time period. ◆ When you have done calculations involving a frequency of payments or a number of time periods, and you need to express the resulting interest rate in terms of a rate per year or some other period of time. To create accurate formulas, you will need to understand the principle of equiv- alence of interest rates. Stated simply, any given interest rate for one period of time is equivalent to another interest rate for a different period of time. Methods of Quoting Interest Rates There are three commonly used methods of quoting interest rates: ◆ Nominal rate: The interest is quoted on an annual basis, along with a compounding frequency per year. For example, the commonly quoted APR of, say, 6% compounded monthly, where 0.5% is charged per month. Chapter 11: Introducing Financial Formulas 313 4800-x Ch11.F 8/27/01 11:56 AM Page 313 ◆ Annual effective rate: A rate of interest in which the given rate represents the percentage earned in one year. For example, with a 10% annual effec- tive rate, $1,000 earns $100 interest at the end of a year. ◆ Periodic effective rate: A rate of interest in which the given rate represents the percentage earned during a period of less than a year. For example, with a rate of 3% per half year, $300 earns $9 after six months. An interest rate quoted using any of these three methods can be converted to any of the other three methods. For example, consider an interest rate of 1% per month on $100. In the first month, the investment earns $1 in interest. If the inter- est credited is not withdrawn, it will be added to the principal, and the subsequent interest will be based on the new balance. A 1% monthly interest rate is equivalent to a 12.6825% per annum interest rate (the effective rate). This is calculated by using the following formula: =(1+0.01)^12 – 1 Another example of a nominal rate is an interest rate quoted as 6% per annum, compounded quarterly. This means that 1.5% (that is, 6% / 4) is paid or received every three months. Most banks and financial institutions quote interest on a nominal basis com- pounded monthly. However, when reporting returns from investments or when comparing interest rates, it is common to quote annual effective returns, which makes it easier to compare rates. For example, we know that 12% per annum com- pounded monthly is more than 12% per annum compounded quarterly — but we don’t know (without an intermediate conversion calculation) how much more it is. Converting Interest Rates Using the Financial Functions Add-in As you will see, 10 different conversions may be required in converting among Nominal, Annual Effective, and Periodic Effective systems. The companion CD-ROM contains an add-in (named Financial Functions), written by Norman Harker.This add-in provides custom functions (written in VBA) to calculate interest rate conversions. You’ll also find a workbook that demonstrates the use of these functions.In addition, these functions are used in many of the examples in this and subsequent chapters. For your conve- nience, the VBA functions are defined in the example workbooks. Therefore, you do not need to install the add-in to work with the example workbooks. 314 Part III: Financial Formulas 4800-x Ch11.F 8/27/01 11:56 AM Page 314 When using the Financial Functions add-in, you can either enter the function manually, or use Excel’s Insert Function dialog box (the functions are located in the Financial category). Table 11-1 lists the 10 interest rate conversion functions con- tained in the Financial Functions add-in. The table also shows (where applicable) the equivalent Excel formula. TABLE 11-1 CUSTOM VBA INTEREST RATE CONVERSION FUNCTIONS Add-in Function Description Equivalent Excel Formula Effx_Nomx Converts an Effective rate (none) (Effx,Freqx) for a period of less than a year to the equivalent Nominal rate for that frequency. Effx_AnnEff Converts an Effective rate =EFFECT(Effx* (Effx,Freqx) for a frequency of less than Freqx,Freqx) a year to an equivalent Annual Effective rate. Effx_Nomy(Effx, Converts an Effective rate =NOMINAL(EFFECT(Effx* Freqx,Freqy) for a frequency of less than Freqx,Freqx),Freqy) a year to an equivalent Nominal rate for a different frequency. Effx_Effy(Effx, Converts an Effective rate =NOMINAL(EFFECT(Effx for a frequency of less than *Freqx,Freqx,Freqy) a year to an equivalent Freqx),Freqy)/Freqy Effective rate for a different frequency, which is also less than a year. Nomx_Effx Converts a Nominal rate to (none) (Nomx,Effx) the equivalent Effective rate for the frequency of the Nominal rate. Nomx_AnnEff Converts a Nominal rate to =EFFECT(Nomx,Freqx) (Nomx,Freqx) the equivalent Annual Effective rate. Continued Chapter 11: Introducing Financial Formulas 315 4800-x Ch11.F 8/27/01 11:56 AM Page 315 TABLE 11-1 CUSTOM VBA INTEREST RATE CONVERSION FUNCTIONS (Continued) Add-in Function Description Equivalent Excel Formula Nomx_Nomy(Nomx, Converts a Nominal rate for =NOMINAL(EFFECT Freqx,Freqy) a frequency to an equivalent (Nomx,Freqx),Freqy) Nominal rate (for a different frequency). Nomx_Effy(Nomx, Converts a Nominal rate to =NOMINAL(EFFECT Freqx,Freqy) an equivalent Effective rate (Nomx,Freqx) for a frequency of less than ,Freqy)/Freqy a year, which is not the frequency of the given Nominal rate. AnnEff_Effx Converts an Annual Effective =NOMINAL(AnnEff,Freqx) (AnnEff,Freqx) rate to an equivalent Effective /Freqx rate for a frequency of less than a year. AnnEff_Nomx Converts an Annual Effective =NOMINAL(AnnEff,Freqx) (AnnEff,Freqx) rate to an equivalent Nominal rate. The function names and arguments may appear confusing at first, but you will soon get the hang of them. The name of each function is made up of three parts: ◆ The interest rate you have (Effx, AnnEff, or Nomx). Note that the com- pounding frequency of the effective and nominal rates are denoted by x. ◆ The linking symbol, which is an underscore character (_). ◆ The interest rate you want (Effx, Effy, AnnEff, Nomx, or Nomy). Again, compounding frequencies are denoted by x (if it is the same as the fre- quency of the rate you have), or y (if it is different). The ordering of arguments is also easy to master: ◆ The first argument is always the interest rate you have. ◆ The second argument is always the Freqx, which is the frequency of the Effx or Nomx rate. Note that every conversion function uses a Freqx argument, and it is always the second argument. 316 Part III: Financial Formulas 4800-x Ch11.F 8/27/01 11:56 AM Page 316 ◆ If there is a second known frequency other than x or annual, there is a third argument, Freqy. Effective Cost of Loans Lending institutions typically advertise their “headline” rates to make them appear as low as possible. A savvy borrower is able to interpret these rates to determine how much the loan is really costing. The only safe and constant comparison is to look at the effective cost in terms of the annual effective interest rate, or some other common rate such as the annual nominal rate compounded monthly. This section presents four examples that demonstrate how to calculate the effec- tive cost of loans. All of the examples in this section are available on the companion CD-ROM. These examples use the custom VBA interest rate conversion functions. Impact of Fees and Charges upon Effective Interest In addition to the interest on a mortgage, banks often charge “points,” or set-up fees, and account service fees. These fees add to the effective cost of the loan. But by how much? EXAMPLE 26 A bank quotes a mortgage rate of 7% nominal compounded monthly, and you are interested in borrowing $150,000 over 10 years with monthly payments. The bank charges an up-front loan arrangement fee of 2% of the loan, plus an account ser- vice fee of $25 per month. What is the annual effective cost of the loan? Figure 11-5 shows a worksheet that’s set up to solve this problem. The known information is entered into the Base Data section of the worksheet. Table 11-2 lists the key formulas that perform the calculations. For clarity, the formulas are shown using actual values rather than cell references. Chapter 11: Introducing Financial Formulas 317 4800-x Ch11.F 8/27/01 11:56 AM Page 317 Figure 11-5: This worksheet calculates the effective cost of a loan. TABLE 11-2 FORMULAS USED IN FIGURE 11-5 Cell Calculation Formula (Using Actual Values) B16 Set-up fee =$150,000 * 2% B17 Effective borrowing =$150,000 – $3,000 B18 Loan term periods =10 * 12 B19 Loan rate period =Nomx_Effx(7%,12) B20 Loan payment =PMT(0.583333%,120,150000,0,0) B21 Loan payment + fee =–$1,741.63–$25 B22 Effective cost of =RATE(120, the loan –1766.63,147000,0,0) Cell B19 uses a custom VBA function. 318 Part III: Financial Formulas 4800-x Ch11.F 8/27/01 11:56 AM Page 318 [...]... =CUMPRINC(Nomx_Effx (5. 6%,12),10*12, 250 000,1,12,0) We can check these answers using the PMT function to calculate the aggregate of the payments The following formula returns $32,706.74, which is the aggregate of the preceding results: =PMT(Nomx_Effx (5. 6%,12),10*12, 250 000,0,0)*12 These formulas all use the Nomx_Effx custom VBA function 323 4800-x Ch11.F 324 8/27/01 11 :56 AM Page 324 Part III: Financial Formulas. .. can be done either by using –12 ,50 0 as the outgoing, or by reversing the sign of the result by using –FV (as in the example) If the equivalent amount is to be calculated in advance, we would use the same principles and apply the PV function 4800-x Ch11.F 8/27/01 11 :56 AM Page 3 25 Chapter 11: Introducing Financial Formulas Limitations of Excel s Financial Functions Excel s primary financial functions... secured on a property that I am building, and the bank is prepared to lend, subject to payments not exceeding 75% of the estimated income of $9 ,50 0 per month How much can I borrow? The following formula uses the custom AnNEff_Effx function, and returns $55 0,422.02: =PV(AnnEff_Effx(8%,12,10*12,- 950 0* 75% ,0,0)*(1+AnnEff_Effx(8%,12))^12 Valuing a Series of Regular Payments We can extend the basic principle of... following formula returns $200,344.00: =PV(AnnEff_Effx(10%,12),48 ,50 00,0,1) This formula returns $139 ,55 9.07: =PV(AnnEff_Effx(10%,12),36, 650 0,0,1)*(1+AnnEff_Effx(10%,12))^-48 This formula returns $638,331.47: =PV(AnnEff_Effx(10%,12),36, 850 0,1300000,1)*(1+AnnEff_Effx(10%,12))^(48+36) And the total of the three elements checks at $978,224 .54 Subject to exceptions involving just one or two changes in the... validation to allow the user to select the type of flow (1, 2, 4, 12, 13, 26, 52 , 3 65, 366) That choice determines the appropriate interest conversion calculation, and also affects the labels in row 5, which contain formulas that reference the text in cell D3 339 4800-x Ch12.F 340 8/27/01 11 :56 AM Page 340 Part III: Financial Formulas Figure 12-11: This worksheet allows the user to select the time period... advance He could have purchased an identical system for $2 ,50 0 cash or on normal credit terms What is the effective cost of this loan? Again, the Effx_AnnEff VBA function provides the simplest solution This formula returns 51 .16%: =Effx_AnnEff(RATE(12,-(3000/12), 250 0,0,1),12) 319 4800-x Ch11.F 320 8/27/01 11 :56 AM Page 320 Part III: Financial Formulas Such calculations are often more difficult when the... EXAMPLE 35 What is the present value of a property yielding an income of $5, 000 per month for four years, rising to $6 ,50 0 per month for the next three years, and rising to $8 ,50 0 per month for the final three years? After 10 years, the property will be worth an estimated $1,300,000 A discount rate of 10% per annum may be assumed and all payments are in advance The following formula returns –$978,224 .54 :... are in advance The following formula returns –$978,224 .54 : =PV(AnnEff_Effx(10%,12),48 ,50 00,0,1) + PV(AnnEff_Effx(10%,12),36, 650 0,0,1)* (1+AnnEff_Effx(10%,12))^-48 + PV(AnnEff_Effx(10%,12),36, 850 0,1300000,1)* (1+AnnEff_Effx(10%,12))^-(48+36) 4800-x Ch11.F 8/27/01 11 :56 AM Page 327 Chapter 11: Introducing Financial Formulas Note how the final value of $1,300,000 has been nested in the final PV function... right to the future income stream, the sign would have to be reversed The following formula returns $978,224 .54 : =PV(AnnEff_Effx(10%,12),48 ,50 00,PV(AnnEff_Effx(10%,12),36, 650 0,PV(AnnEff_Effx(10%,12),36, 850 0,1300000,1),1),1) Of these two approaches, the first formula (using the basic discounting formulas) looks easier as a method; it looks easier to build using the megaformula technique or to break up... add-in is installed EXAMPLE 31 A consumer is borrowing $ 250 ,000 on a mortgage, repayable over 10 years at 5. 6% nominal compounded monthly with payments monthly in arrears What will the payments of interest and principal be in the first year of the loan? The following formula, for principal payments, returns $13 ,51 2.31: =CUMIPMT(Nomx_Effx (5. 6%,12),10*12, 250 000,1,12,0) The following formula returns $19,194.42 . is $ 150 ,000, and interest in the first month is $6 75 ($ 150 ,000 * 0. 45% ). If the payment is less than this amount, the outstanding 310 Part III: Financial Formulas 4800-x Ch11.F 8/27/01 11 :56 AM. equivalent rate is 51 .16%, calculated as follows: =((1+0.0 350 3 153 )^12)-1 The annual rate, based on the nominal compounded monthly basis, returns 42. 05% , calculated as follows: =3 .50 3 153 * 12 There. construction by using formulas nested in or applied to the basic amortization formulas. Chapter 11: Introducing Financial Formulas 3 25 4800-x Ch11.F 8/27/01 11 :56 AM Page 3 25 Deferred Start to