Excel 2002 Formulas phần 5 ppsx

The table also shows where applicablethe equivalent Excel formula.TABLE 11-1 CUSTOM VBA INTEREST RATE CONVERSION FUNCTIONS Add-in Function Description Equivalent Excel Formula Effx,Freq

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:

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

Note that, with mortgages, we always assume payments are in arrears and thatthe type argument is 0 Also note that the rate of interest (and the payments) aremonthly 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 returns0.45%:

=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 tion period in months This answer, although correct in mathematical terms, has apractical implication Payments are actually made on exact monthly anniversaries.This calculation implies that the loan somehow gets repaid 0.76 of the way throughthe 98th month In reality, you have a choice: make an additional payment at theend of 97 months, or make a reduced level payment after 98 months These optionscan be calculated using the FV function

amortiza-To calculate the additional payment at the end of 97 months, calculate theamount due using this formula (which returns –$1,429.85):

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%:

equiva-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 est is 0.4% per month, and payments are made monthly in arrears What will the payments be?

inter-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 theterm, 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 atthe end of the loan

Chapter 11: Introducing Financial Formulas 311

The ability to use an fv argument in the PV, PMT, RATE, and NPER functionsmake 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 arounded payment amount):

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.

Function required: PMT(rate, nper, pv, fv, type)

The first step is to calculate the payment for a $200,000 normal amortizationloan 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 bepaid as interest on the balloon element (that is, $3,000 – $2,101.81) The balloonelement 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 totalmortgage of $424,546.88 with a balloon element of $224,546.88 The followingformula 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 inwhich interest rate conversions must be made:

◆ When you must do calculations involving a frequency of payments or anumber of time periods, and the rate that you are required to use does notmatch the frequency of payments or time period

◆ When you have done calculations involving a frequency of payments or anumber 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 alence of interest rates Stated simply, any given interest rate for one period of time

equiv-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 quotedAPR of, say, 6% compounded monthly, where 0.5% is charged per month

Chapter 11: Introducing Financial Formulas 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 tive rate, $1,000 earns $100 interest at the end of a year

effec-◆ 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 toany of the other three methods For example, consider an interest rate of 1% permonth 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 subsequentinterest 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 byusing 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 receivedevery three months

Most banks and financial institutions quote interest on a nominal basis pounded monthly However, when reporting returns from investments or whencomparing interest rates, it is common to quote annual effective returns, whichmakes 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

com-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 amongNominal, 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 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.

conve-When using the Financial Functions add-in, you can either enter the functionmanually, or use Excel’s Insert Function dialog box (the functions are located in theFinancial 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,Freqx) for a period of less than a

year to the equivalentNominal rate for thatfrequency

Effx_AnnEff Converts an Effective rate =EFFECT(Effx*

(Effx,Freqx) for a frequency of less than Freqx,Freqx)

a year to an equivalentAnnual 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 equivalentNominal 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) 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

TABLE11-1 CUSTOM VBA INTEREST RATE CONVERSION FUNCTIONS (Continued)

Add-in Function Description Equivalent Excel

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 willsoon 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 quency of the rate you have), or y (if it is different).

fre-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 theEffx or Nomx rate Note that every conversion function uses a Freqxargument, and it is always the second argument

◆ 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 determinehow much the loan is really costing The only safe and constant comparison is tolook at the effective cost in terms of the annual effective interest rate, or some othercommon rate such as the annual nominal rate compounded monthly

This section presents four examples that demonstrate how to calculate the tive cost of loans

effec-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-upfees, 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 knowninformation is entered into the Base Data section of the worksheet Table 11-2 liststhe key formulas that perform the calculations For clarity, the formulas are shownusing actual values rather than cell references

Chapter 11: Introducing Financial Formulas 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)

Cell B19 uses a custom VBA function.

The payments are based on the loan amount of $150,000, but the effective cost

is based upon the fact that, after deducting the set-up fee, the borrower receivesonly $147,000 Similarly, actual payments are higher by the amount of the accountservice fee

The impact of these costs varies according to the term: The shorter the term, thegreater the impact If a mortgage is not capable of being transferred to a new housewhen the borrower moves, the calculation should be based on the likely time thatthe mortgage will last — usually about seven years

“Flat” Rate Loans

Many consumer credit agreements use a loan agreement in which a percentage ofthe loan is added to the loan, and payments are based on the aggregate of the loanamount plus the flat interest divided by the number of payments You can useExcel’s RATE function to calculate the effective costs of such loans

in the loan agreement

Interest-Free Loans

Another interesting calculation is the effective cost of a so-called “interest-free”

loan offer In making these calculations, you need to know the price for which youcould get the product elsewhere (without the interest-free package)

EXAMPLE 28

A consumer buys a hi-fi system at a list price of $3,000 on “interest-free” terms over 12 months, with the payments in advance He could have purchased an iden- tical system for $2,500 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 mula returns 51.16%:

for-=Effx_AnnEff(RATE(12,-(3000/12),2500,0,1),12)

Chapter 11: Introducing Financial Formulas 319

Such calculations are often more difficult when the equivalent cash price is jective (for example, the used car market).

sub-You can perform similar calculations for other types of agreement, such as “Pay25% down today, no more to pay for 12 months.” Again, the key is to establish theequivalent cash price, and then compare the calculations with that price, ratherthan a price that is inflated by the retailer who’s offering the credit

Most states and countries have consumer credit legislation that governs the tation of interest rates In many localities, the only major regulation of interest-freetype agreements is that the retailer may not offer the same product at a cash pricedifferent from that quoted in the interest-free agreement

quo-“Annual Payments / 12” Loan Costs

A practice that is rooted in the precalculator days is to calculate payments on an

“annual in arrears” basis, and to charge the borrower 1/12 of that amount eachmonth That calculation was facilitated by preprepared tables of monthly paymentsper $1,000 of loan The practice prevails (especially in UK Building Societies) partlybecause it produces a lower advertised rate than Nominal or Effective rate regimes

EXAMPLE 29

A bank offers a mortgage of $100,000 at a rate of 7% over 10 years, where ments per month are based on 1/12 of the annually calculated payment being paid monthly in arrears What is the annual effective cost?

pay-The following formula (which uses the Effx_AnnEff VBA function) returns.7522% (the per annum effective rate):

=Effx_AnnEff(RATE(10*12,PMT(7%,10,100000,0,0)/12,100000,0,0),12)

Calculating the Interest and Principal Components

This section discusses four Excel functions that enable you to:

◆ Calculate the interest or principal components of a particular payment(the IPMT and PPMT functions)

◆ Calculate cumulate interest or principal components between any twotime periods

The examples in this section are available on the companion CD-ROM.

Using the IPMT and PPMT Functions

You may need to know (or simply be curious about) how much of a particular ment constitutes interest, and how much of the payment goes toward the principal

pay-This information might be useful in determining tax effects on interest payments Ifyou’ve studied any of the loan amortization examples, you know that the interestelement is not constant over the life of a loan Rather, the interest componentdecreases, while the principal component increases

If you’ve created an amortization schedule, these functions are not larly useful, because you can simply refer to the schedule The IPMT and PPMT functions are most useful when you need to determine the interest/principal breakdown of a particular payment.

particu-The syntax for these two functions is as follows (bold arguments are required):

IPMT(rate,per,nper,pv,fv,type)

PPMT(rate,per,nper,pv,fv,type)

As with all amortization functions, the rate, per, and nper must match in terms

of the time period If the loan term is measured in months, the rate argument must

be the effective rate per month, and the per argument (that is, the period of interest)must be a particular month

EXAMPLE 30

A consumer obtains a three-year car loan (monthly payments) for $20,000 at an annual rate of 8% What are the interest and principal portions for the final loan payment?

Figure 11-6 shows the solution, set up in a worksheet

Chapter 11: Introducing Financial Formulas 321

Figure 11-6: This worksheet calculates the interest and principal components for any periods of a loan.

Function required: IPMT(rate,per,nper,pv,fv,type)

This formula calculates the interest portion of the final payment, and returns–$4.15:

prin-You can check the calculations by using the PMT function (which returns the total payment, interest plus principal) The following formula returns –$626.73, which is the loan payment amount (and the sum of the two previ- ous formulas):

=PMT(8%/12,36,-20000,0)

Chapter 11: Introducing Financial Formulas 323

Using the CUMIPMT and CUMPRINC Functions

The IPMT and PPMT functions can be useful But, more often, you will need toknow the interest or principal component for a group of consecutive periods In thiscase, the CUMIPMT and CUMPRINC functions are of greater service These func-tions are useful for creating annualized amortization schedules, and for establish-ing qualifying interest for tax return purposes

The syntax for these functions is shown here (all arguments are required):

CUMIPMT(rate,nper,pv,start_period,end_period,type) CUMPRINC(rate,nper,pv,start_period,end_period,type)

These functions are available only when the Analysis ToolPak 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 $13,512.31:

=CUMIPMT(Nomx_Effx(5.6%,12),10*12,250000,1,12,0)

The following formula returns $19,194.42 (total interest payments):

=CUMPRINC(Nomx_Effx(5.6%,12),10*12,250000,1,12,0)

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*12,250000,0,0)*12

These formulas all use the Nomx_Effx custom VBA function.

Matching Different Interest and Payment Frequencies

Previous examples involved nominal interest compounding frequencies that matchthe frequency of payments Thus, for example, we might have a quoted nominalrate compounded monthly with payments that are also monthly As usual, the realworld isn’t always as cooperative

EXAMPLE 32

A bank quotes a nominal rate compounded monthly of 6.3%, but allows payments weekly at the equivalent interest rate If I borrow $300,000 over 10 years, what will the weekly payments be?

The easy way to resolve such problems is to use the custom Nomx_Effy interestconversion function This formula returns $777.51:

=PMT(Nomx_Effy(6.3%,12,52),10*52,300000,0,0)

EXAMPLE 33

We have set up annual accounts, but need to handle a monthly outgoing of

$12,500 Rather than annualize by multiplying by 12, what is the equivalent annual amount using a deposit rate of 7% per annum nominal compounded monthly? The monthly payment is in arrears, and the equivalent amount is to be calculated at the end of each year.

First, calculate the monthly effective rate (using a custom VBA function) Thefollowing formula returns 0.58333%:

If the equivalent amount is to be calculated in advance, we would use the sameprinciples and apply the PV function

Limitations of Excel’s Financial Functions

Excel’s primary financial functions (PV, FV, PMT, RATE, NPER, CUMIPMT, andCUMPRINC) are very useful, but they have two common limitations:

◆ They can handle only one level of interest rate

◆ They can handle only one level of payment

For example, the NPER function cannot handle the variations in payments thatarise with credit card calculations In such calculations, the monthly payment isbased upon a reducing outstanding balance, and may also be subject to a minimumamount rule

The common solution to the problem of varying payments is to create a cashflow schedule and use other financial functions that can handle multiple paymentsand rates Examples of the process appear in the next two chapters Briefly, thefunctions involved are:

◆ FVSCHEDULE, which handles accumulation of a Present Value at differentrates and which, when used in a formula, can calculate the present value

of a future amount at different rates

◆ IRR, which handles the calculation of a single rate from regular cashflows

◆ NPV, which handles the calculation of the sum of the present values ofregular cash flows and which by formula can handle the sum of accumu-lated values of regular cash flows

◆ MIRR, which is a specialist IRR aimed at avoiding the multiple IRR lem by applying different rates to negative and positive regular cashflows

prob-◆ XIRR, which handles the calculation of a single rate from irregular cashflows

◆ XNPV, which handles the calculation of the sum of the present values ofirregular cash flows and which, in a formula, can handle the sum of accu-mulated values of irregular cash flows

In a situation that involves only one or two variations, it may be possible toavoid cash flow construction by using formulas nested in or applied to the basicamortization formulas

Chapter 11: Introducing Financial Formulas 325

Deferred Start to a Series of Regular Payments

In some cases, a series of cash flows may have a deferred start We can calculate the

PV of a regular series of cash flows with a deferred start by using a formula likethis:

The following formula uses the custom AnNEff_Effx function, and returns

$550,422.02:

12

=PV(AnnEff_Effx(8%,12,10*12,-9500*75%,0,0)*(1+AnnEff_Effx(8%,12))^-Valuing a Series of Regular Payments

We can extend the basic principle of discounting successive, but different, levels ofpayment by chaining the PV functions For example, if PV1, PV2, and PV3 repre-sent different present values of series of payments for time periods NPER1, NPER2,and NPER3, the discounted value of all series of payments can be found by:

PV1 + PV2(1+I)^-NPER1 + PV2(1+I)^-(NPER1+NPER2)

EXAMPLE 35

What is the present value of a property yielding an income of $5,000 per month for four years, rising to $6,500 per month for the next three years, and rising to $8,500 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:

=PV(AnnEff_Effx(10%,12),48,5000,0,1) + PV(AnnEff_Effx(10%,12),36,6500,0,1)*

(1+AnnEff_Effx(10%,12))^-48 + PV(AnnEff_Effx(10%,12),36,8500,1300000,1)*

(1+AnnEff_Effx(10%,12))^-(48+36)

Note how the final value of $1,300,000 has been nested in the final PV function.

The same answer could be achieved by “nesting” the successive Present Valueinside the preceding function as future values But remembering that as the PV atthat time represents a right to the future income stream, the sign would have to bereversed The following formula returns $978,224.54:

PV(AnnEff_Effx(10%,12),36,6500,- PV(AnnEff_Effx(10%,12),36,8500,1300000,1),1),1)

=PV(AnnEff_Effx(10%,12),48,5000,-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 into three cells that are then added together

The following formula returns $200,344.00:

=PV(AnnEff_Effx(10%,12),36,8500,1300000,1)*(1+AnnEff_Effx(10%,12))^-And the total of the three elements checks at $978,224.54

Subject to exceptions involving just one or two changes in the series of ments, the solution will be to set up a cash flow schedule This will be covered afterthe next chapter because we first have to outline the basic tools of NPV and IRR

pay-Summary

This chapter introduced the financial functions and provided the basic concepts oftime value of money and equivalent interest rates The chapter presented a series ofexamples that used the key financial functions for accumulations, discounting, andloan amortization

The next chapter presents examples that use Excel for depreciation calculations,and introduces the techniques of calculating net present values (NPV) and internalrates of return (IRR)

Chapter 11: Introducing Financial Formulas 327

Chapter 12

Discounting and Depreciation Financial Functions

IN THIS CHAPTER

◆ Using the NPV and IRR functions

◆ Understanding the various approaches for cash flows

◆ Using cross-checking to verify results

◆ Dealing with multiple internal rates of return

◆ Understanding the limitations of IRRs and NPVs

◆ Extending NPV analysis using more than one rate

◆ Using the NPV function to calculate accumulated values

◆ Using the depreciation functions

T HE NPV (N ET P RESENT V ALUE )and IRR (Internal Rate of Return) functions are haps the most commonly used of the financial analysis tools This chapter providesmany examples of using these functions for various types of financial analysis

per-Using the NPV Function

The NPV function returns the sum of any series of regular cash flows, discounted tothe present day using a single discount rate The syntax for Excel’s NPV function isshown here (arguments in bold are required):

NPV(rate,value1,value2, )

Cash inflows are represented as positive values, and cash outflows are negativevalues The NPV function is subject to the same restrictions that apply to financialfunctions such as PV, PMT, FV, NPER, and RATE The only exception is that the

If the discounted negative flows exceed the discounted positive flows, the tion will return a negative amount Similarly, if discounted positive flows exceeddiscounted negative flows, the NPV function will return a positive amount.

func-If the NPV is positive, this indicates that at period zero, the investor could payout up to this additional amount and still achieve the discount rate If the NPV isnegative, then the investor does not get the required discount rate That rate is

often called a hurdle rate The implication of a negative NPV is that the investor is

paying out too much The “right price” requires the addition of the shortfall to theTime 0 cash flow

The discount rate used must be a single effective rate for the period used for thecash flows Therefore, if flows are set out monthly, you must use the monthly effec-tive rate

Definition of NPV

Excel’s NPV function assumes that the first cash flow is received at the end of the

first period It is important to understand that this differs from the definition used

by most financial calculators, and it is also at odds with the definition used byinstitutions such as the Appraisal Institute of America (AAI) For example, the AAIdefines NPV as the difference between the present value of positive cash flows andthe present value of negative cash flows

If you use Excel’s NPV function without making an adjustment, the result willnot adhere to this definition

Therefore, when using Excel’s NPV function, you will need to take into accountthe time Point 0 cash flow For this reason, the procedure to adopt when calculatingNPV using Excel is as follows:

◆ Treat the number of periods as points in time rather than the time periodbetween points

◆ Always include a Point 0, even if cash flows do not arise until the end ofperiod 1 (Point 1)

◆ Use a formula like the one shown here to include the Point 0 cash flow:

=NPV(Rate,Range)*(1+Rate)

If you use this procedure, your calculations will adhere to the accepted tions of NPV, and the results will coincide with those made on your trusty financialcalculator By the way, it’s not that Microsoft got it wrong The online help clearlystates that the first cash flow in the range is assumed to be received at the end ofthe first period If you use the previous formula and always have a Time 0 period(even if it is $0), you will always get the correct answer

defini-NPV Function Examples

This section contains a number of examples that demonstrate the NPV function

All of the examples in this section are available on the companion CD-ROM.

EXAMPLE 1

Figure 12-1 shows a worksheet set up to calculate the net present value for a series

of cash flows in the range B6:B13

Figure 12-1: This worksheet uses the NPV function.

The NPV calculation in cell B15 uses the following formula This formula returns–$33,629.14:

=NPV(B3,B6:B13)*(1+B3)

The worksheet in Figure 12-1 also shows a method of cross-checking the NPVcalculation Column E contains a duplicate of the original cash flow, with oneexception The Point 0 cash flow is equal to the original Point 0 cash flow, minusthe calculated NPV In this example, the Point 0 cash flow is –$166,370.86 Thecross-check formula in cell E15, shown here, returns $0.00:

at the same rate must be 0 If it is 0, this means that the required discount ratewas met

Chapter 12: Discounting and Depreciation Financial Functions 331

The present values are calculated in column D, by multiplying each cash flow byits corresponding present value factor The formula in cell D7 is:

=C7*B7

Column D contains all the present values calculated, and the sum of that column

is the sum of the present values By definition, the sum of the present values (cellD16) should equal the NPV

EXAMPLE 3

This example (see Figure 12-3) calculates the net present value of a cash flow with

an initial (Time 0) positive cash flow

Figure 12-3: This worksheet calculates the net present value for a cash flow that has an initial flow.

The net present value calculation is in cell B15, which contains the followingformula:

If we do not know the value, we put 0 in the capital column at period 0, and theNPV represents the value using the required discount rate If we know the quotingprice, we can put that in as a negative at period 0, and the NPV then represents howmuch more or less we should pay to get the required discount rate

Chapter 12: Discounting and Depreciation Financial Functions 333

Figure 12-4: This worksheet demonstrates cash flows with a terminal value.

The NPV calculation in cell D15 is:

EXAMPLE 6

This example is a simplistic valuation model that uses initial and terminal flows(see Figure 12-6) It represents a typical investment example in which the aim is todetermine if, and by how much, an asking price exceeds a criterion rate of return

Figure 12-6: This worksheet demonstrates cash flows with terminal values.

The following formula indicates that, at $280,000 asking price, the discountedpositive cash at the criterion rate of return is $148,026.29:

The simplest solution is to use the AnnEff_Effx function (which is also used insome of the examples in Chapter 11) This is a custom VBA function that makes itvery easy to convert an interest rate to the monthly effective basis required by amonthly cash flow

The AnnEff_Effx function is defined in the example workbook on the CD-ROM The interest rate conversion functions are also available in the Financial Functions add-in (also on the CD-ROM).

Chapter 12: Discounting and Depreciation Financial Functions 335

Figure 12-7 shows a rental of $12,000 paid quarterly in advance It also shows

an initial price of $700,000 and a sale (after three years) for $900,000 Note thatbecause rent is paid in advance, the purchaser gets a cash adjustment to the price.However, at the end of three years (12 quarters), the same rule applies, and the rentpayable for the next quarter is received by the new owner If we discount at 7% perannum effective, this shows an NPV of $166,099.72

Often, rental flows are annualized This might sound a bit peculiar However,before the advent of calculators and computers, this was the approach adopted byappraisers who used precalculated tables of annual constants that they applied tothe aggregate annual rent Figure 12-8 shows the same data, but this time we haveadopted the approach of assuming that the rent of $48,000 per annum is paidannually in arrears Still discounting at 7% per annum effective, we get an NPV of

$160,635.26

Figure 12-7: Calculating the NPV using quarterly cash flows

Figure 12-8: Calculating the NPV by annualizing quarterly cash flows

Using the NPV Function to Calculate Accumulated Amounts

This section presents two examples that use the NPV function to calculate futurevalues or accumulations These examples take advantage of the fact that:

Figure 12-9: Calculating FV using the NPV function

The result is verified in column D, which calculates a running balance of theinterest The results of the future value calculation matches the cumulative interest

Interest is calculated using the interest rate multiplied by the previous month’s ance The running balance is the sum of the previous balance, interest, and the cur-rent month’s cash flow

bal-It is important to properly sign the cash flows Then, if the running balance forthe previous month is negative, the interest will be negative Signing the flowsproperly and using addition is preferable to using the signs in the formulas forinterest and balance

Chapter 12: Discounting and Depreciation Financial Functions 337

EXAMPLE 9

Chapter 11 covers the use of the PMT function to calculate payments equivalent to

a given present value Similarly, we can use the NPV function, nested in a PMTfunction, to calculate an equivalent single-level payment to a series of changingpayments

This is a typical problem where we require a time-weighted average single ment to replace a series of varying payments An example is an agreement in which

pay-a schedule of rising rentpay-al ppay-ayments is replpay-aced by pay-a single ppay-ayment pay-amount In theexample shown in Figure 12-10, the following formula (in cell C27) returns

$10,923.24, which is the payment amount that would substitute for the varyingpayment amounts in column B:

=PMT(C7,C6,-B25,0,C8)

The example in this section gives the user flexibility in choice of rate type andfrequency of the income flow Data validation is used to allow the user to selecteither Effective or Nominal in cell C3 This type of calculation is frequently used tocalculate alternatives of fixed and stepped rentals

Figure 12-10: Calculating equivalent payments with NPV

Using the IRR Function

Excel’s IRR function returns the discount rate that makes the net present value of

an investment zero In other words, the IRR function is a special-case NPV, and wewill use that feature in designing an automatic cross-check

The syntax of the IRR function is:

IRR(range,guess)

The range argument must contain values Empty cells are not treated as zero If the range contains empty cells or text, the IRR function does not return an error Rather, it will return an incorrect result Thus, if range B1:B40 contains text in cells B11:B20, the IRR will calculate on the basis of 30 con- secutive cash flows This is especially dangerous if the text is misleading: a blank,“-”,“nil”,“zero”, or (worst) “O” (the uppercase “o”).

In most cases, the IRR can only be calculated by iteration The guess argument,

if supplied, acts as a “seed” for the iteration process It has been found that a guess

of –0.9 will always produce an answer Other guesses, such as 0, usually (but notalways) produce an answer

An essential requirement of the IRR function is that there must be both negativeand positive income flows: To get a return, there must be an outlay and there must

be a payback There is no essential requirement for the outlay to come first For aloan analysis using IRR, the loan amount will be positive (and come first) and therepayments that follow will be negative

The IRR is a very powerful tool, and its uses extend beyond simply calculatingthe return from an investment This function can be used in any situation in which

we need to calculate a time- and money-weighted average return

EXAMPLE 10

This example sets up a basic matrix for IRR calculations (see Figure 12-11) Thisexample demonstrates the perennial problem of a cash flow frequency returning anIRR for that frequency Thus, if cash flows are monthly, the function will return themonthly IRR The example uses data validation to allow the user to select the type

of flow (1, 2, 4, 12, 13, 26, 52, 365, 366) That choice determines the appropriateinterest conversion calculation, and also affects the labels in row 5, which containformulas that reference the text in cell D3

Chapter 12: Discounting and Depreciation Financial Functions 339

Figure 12-11: This worksheet allows the user to select the time period for the cash flows.

The following formula, in cell D22, is a validity check:

=NPV(D20,D6:D18)*(1+D20)

The IRR is the rate at which the discounting of the cash flow produces an NPV ofzero The formula in cell D22 uses the IRR in an NPV function applied to the samecash flow The NPV discounting at the IRR (per quarter) is $0.00 — so the calculationchecks

EXAMPLE 11

You may have a need to calculate an average growth rate, or average rate of return.Because of compounding, a simple arithmetic average does not yield the correctanswer Even worse, if the flows are different, an arithmetic average will not takethese variations into account

A solution uses the IRR function to calculate a geometric average rate of return.

This is simply a calculation that determines the single percentage per period thatexactly replaces the varying ones

Example 11 (see Figure 12-12) shows the IRR function being used to calculate ageometric average return based upon index data (in column B) The calculations ofthe growth rate for each year are in column C For example, the formula in cellC5 is:

=(B5-B4)/B4

The remaining columns show the geometric average growth rate between ent periods The formulas in Row 10 use the IRR function to calculate the internalrate of return For example, the formula in cell F10, which returns 5.241%, is:

Figure 12-13 shows a worksheet that uses the present value IRR check This check

is based on the definition of IRR: The sum of positive and negative discountedflows is 0

The net present value is calculated in cell B16:

=NPV(D3,B6:B14)*(1+D3)

The internal rate of return in calculated in cell B17:

=IRR(B6:B14,-0.9)

In column C, formulas calculate the present value They use the IRR (calculated

in cell B17) as the discount rate, and use the period number (in column A) for theexponent For example, the formula in cell C6 is:

=B6*(1+$B$17)^-A6

The sum of the values in column C is 0

Chapter 12: Discounting and Depreciation Financial Functions 341

The formulas in column D use the discount rate (in cell D3) to calculate the sent values For example, the formula in cell D6 is:

pre-=B6*(1+$D$3)^-A6

The sum of the values in column D is equal to the net present value

For serious applications of NPV and IRR functions, it is an excellent idea to usethis type of cross-checking

Figure 12-13: Checking IRR and NPV using sum

EXAMPLE 13

Figure 12-14 shows an example that has two IRR calculations, each of which uses

a different “seed” value for the guess argument As you can see, the formula duces different results

pro-Figure 12-14: A worksheet that demonstrates multiple IRRs

The IRR formula in cell B21 (which returns a result of 13.88%) is:

=IRR(B7:B16,B3)

The IRR formula in cell B22 (which returns a result of 7.04%) is:

=IRR(B7:B16,B4)

So which rate is correct? Unfortunately, both are correct Figure 12-14 shows the

interest and running balance calculations for both of these IRR calculations Bothshow that the investor can pay and receive either rate of interest, and can secure a(definitional) final balance of $0 Interestingly, the total interest received ($1,875) isalso the same

But there’s a flaw This example illustrates a “worst-case scenario” of the cal fallacy of many IRR calculations NPV and IRR analyses make two assumptions:

practi-◆ That we can actually get the assumed (for NPV) or calculated (for IRR)interest on the outstanding balance

◆ That interest does not vary according to whether the running balance ispositive or negative

The first assumption may or may not be correct It’s possible that balances could

be reinvested (but in forward projections in times of changing interest rates, thismight not be the case) But the real problem is with the second assumption Bankssimply do not charge the same rate for borrowing that they pay for deposits

Chapter 12: Discounting and Depreciation Financial Functions 343

Figure 12-15: Multiple internal rate of return

The MIRR function works by separating out negative and positive flows, anddiscounting them at the appropriate rate — the finance rate (for negative flows) andthe deposit rate (for positive flows)

We can replicate the MIRR algorithm by setting up a revised flow, which pares the two NPVs (refer to Figure 12-15, columns C:E) The negative flow NPV isplaced at Period 0, and the positive flow is expressed as its equivalent future value(by accumulating it at the deposit rate) at the end of the investment term The IRR

com-of the revised flow is the same as the MIRR com-of the original (source) flow

This example reveals that the methodology is suspect In separating out negativeand positive flows, the MIRR implies that interest is charged on flows Banks, ofcourse, charge interest on balances An attempt at resolving the problem is shown

in the next example

EXAMPLE 15

The MIRR function uses two rates: one for negative flows, and one for positive

flows In reality, interest rates are charged on balances and not on flows The

exam-ple in this section applies different rates on negative and positive balances Theinterest calculation uses an IF function to determine which rate to use

When analyzing a project in which interest is paid and received, the end balancemust be 0 If it is greater than 0, then we have actually received more than thestated deposit rate If it is less than 0, then we still owe money and the finance ratehas been underestimated This example assumes a fixed finance rate and calculatesthe deposit rate needed to secure a 0 final balance.

In the Risk Rate Equivalent IRR method, the finance rate is fixed by the user Theinterest received on positive balances is initially “seeded” by the user Interest onnegative balances is charged at the finance rate Interest on positive balances is atthe seed rate If the seed rate is the exact return, the final balance will be 0 Excel’sTools→ Goal Seek command can be used to determine the exact rate by iteratingthe interest rate on positive balances to derive a final balance of 0 This is themethod used in the example in Figure 12-16

Figure 12-16: Accumulating balance approach for multiple IRRs

The revised flow, derived from changes to the running balance, should have anIRR approaching zero The Risk Rate Equivalent IRR may be compared with a com-parator rate such as the Risk Free Rate of Return (traditionally 90-day Treasurybills)

But what does this all mean? It means that if I pay 9% on negative balances, thisproject gives me 8.579% rate on positive balances The name “Risk Rate EquivalentIRR” refers to the fact that it determines how the project compares with the return

on money invested in a bank or 90-day Treasury bills

There is no requirement that the finance rate be fixed A bank might do tions in the same way, but fix the deposit rate and allow “Goal Seek” to calculatethe equivalent lending rate

calcula-Chapter 12: Discounting and Depreciation Financial Functions 345

Using the FVSCHEDULE Function

The FVSCHEDULE function calculates the future value of an initial amount, afterapplying a series of varying rates over time Its syntax is:

FVSCHEDULE(principal,schedule)

The FVSCHEDULE function is available only when the Analysis ToolPak

add-in is add-installed.

EXAMPLE 16

This example, shown in Figure 12-17, uses the FVSCHEDULE function to calculate

an accumulated amount, together with other formulas that use the base data tocalculate an index and the geometric average growth rate

This worksheet contains details of an index of share prices between 1997 and

2001, with 1997 being assigned an index of 100 This example can answer a

ques-tion such as: If we bought $1,000 of shares in 1997, what would they be worth in

2001, and what has been the average compound growth rate?

The share value, in cell B13, is $1,296.81 This is the equivalent of 6.714%compounded on the initial investment of $1,000

Figure 12-17: Using the FVSCHEDULE function

The Accumulated Amount (cell B13) is calculated with the following formula:

=FVSCHEDULE(B3,B7:B10)

Note that the FVSCHEDULE function does not follow the sign convention It

returns a future value with the same sign as the present value Also, be aware that the growth rates must be the periodic effective rates for the time peri- ods In the example, the time period is in years, so the growth rates are in annual terms.

The formula in cell B14 calculates the geometric average growth rate:

=RATE(4,0,-B3,B17,0)

Note that the formula uses a negative sign for the third argument (present value)

You can also calculate the geometric average rate of return by using a single mula (cell B15):

for-=RATE(4,0,-B3, FVSCHEDULE(B3,B7:B10),0)

This example also demonstrates a convenient way to calculate an index based

on a schedule of growth rates (column C) This topic is covered in detail in the nextchapter

Depreciation Calculations

This section covers depreciation, a critical element for many investment mance analyses Excel offers five functions to calculate depreciation of an assetover time Depreciating an asset places a value on the asset at a point in time, based

perfor-on the original value and its useful life The functiperfor-on that you choose depends perfor-onthe type of depreciation method that you use

Table 12-1 summarizes Excel’s depreciation functions and the arguments used

by each For complete details, consult Excel’s online help system

Chapter 12: Discounting and Depreciation Financial Functions 347

TABLE 12-1 EXCEL’S DEPRECIATION FUNCTIONS

SLN Straight-line The asset depreciates by the Cost, Salvage, Life

same amount each year of its life

DB Declining balance Computes depreciation at Cost, Salvage, Life,

a fixed rate Period, [Month]

DDB Double-declining balance Computes Cost, Salvage, Life,

depreciation at an accelerated rate Period, Month, [Factor]Depreciation is highest in the first period and

decreases in successive periods

SYD Sum of the year’s digits Allocates a large Cost, Salvage, Life, Period

depreciation in the earlier years of an asset’s life

VDB Variable-declining balance Computes the Cost, Salvage, Life, Start

depreciation of an asset for any period Period, End Period, (including partial periods) using the [Factor], [No Switch]double-declining balance method or some

other method you specify

*Arguments in brackets are optional.

The arguments for the depreciation functions are described as follows:

◆ Cost: Original cost of the asset.

◆ Salvage: Salvage cost of the asset after it has fully depreciated.

◆ Life: Number of periods over which the asset will depreciate.

◆ Period: Period in the Life for which the calculation is being made.

◆ Month: Number of months in the first year; if omitted, Excel uses 12.

◆ Factor: Rate at which the balance declines; if omitted, it is assumed to be

2 (that is, double-declining)

◆ Rate: Interest rate per period If you make payments monthly, for

exam-ple, you must divide the annual interest rate by 12

◆ No-switch: True or False Specifies whether to switch to straight-line

depreciation when depreciation is greater than the declining balancecalculation