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

Each example also contains a cross-check to ensure the accuracy of the

calculation.

EXAMPLE 12

What is the present value of the right to receive $25,000 in five years, discounting

at 6.5% per annum?

Figure 11-3 shows this example, set up on a worksheet.

Figure 11-3: Calculating a present value

Function required: PV(rate, nper, pmt, fv, type)

This formula returns –$18,247.02:

=PV(6.5%,5,0,25000,0)

Note the logic of the signs. If we have a right to receive, the fv argument is pos-

itive — we must pay out in the present to receive this positive right in the future.

With no payment, the type argument is irrelevant.

The accuracy of the computation can be assured by cross-checking the answer

with another function. In this case, we might check whether $18,247.02 will accu-

mulate to $25,000 in five years at 6.5%. The following cross-check formula does

indeed return $25,000:

=FV(6.5%,5,0,-18247.02,0)

304

Part III: Financial Formulas

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EXAMPLE 13

A property yields a rental of $25,000 for the next 25 years. If I discount at 8%,

how much should I pay? Assume a zero value after 25 years and that rent is paid

annually in arrears.

Function required: PV(rate, nper, pmt, fv, type)

The following formula returns –$266,869.40:

=PV(8%,25,25000,0,0)

This result can be checked using the RATE function. This formula returns 8.00%:

=RATE(25,25000,-266869.40,0,0)

Typically, real estate payments are made in advance. In such a case, Example 13

would be modified by making the Type argument 1.

EXAMPLE 14

Assume that the Example 13 rent of $25,000 is received in perpetuity. If we dis-

count at 8%, how much should we pay?

This is an example of a discounting problem that Excel can’t solve using its

functions. The problem is that we can’t use “perpetuity” as the nper argument. The

solution is to use a very long time period, such as 1,000 years. The result is cer-

tainly accurate enough for most purposes.

Function required: PV(rate, nper, pmt, fv, type)

The following formula returns –$312,500.00:

=PV(8%,1000,25000,0,0)

Another option is to use a formula to calculate the present value:

PV = PMT/RATE

For this example, the following formula returns $312,500.00:

=25000/0.08

Note that the sign is different because the formula has not adopted the strict sign

convention.

If rent is paid in advance, we merely adapt this “cheating” approach by using 1

for the Type argument. The following formula returns $337,500.00:

=PV(8%,1000,25000,0,1)

Chapter 11: Introducing Financial Formulas

305

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The formula approach is varied, and the general formula for valuing income in

advance is as follows:

PV = PMT*(1+RATE)/RATE

For this example, the following formula returns $337,500.00:

=25000*(1+.08)/.08

Other examples can be expressed in discounting terms, but were covered earlier

in the “Simple Accumulation Problems” section.

EXAMPLE 15

A property currently worth $2,000,000 is subject to a lease at a peppercorn rent for

five years. A purchaser has paid $1,750,000 for it. Assuming no future growth in

value, what was the discount rate?

A peppercorn rent is a nominal rent that is intended to demonstrate that a

property is leasehold and not freehold.

Function required: RATE(nper, pmt, pv, fv, type, guess)

The following formula returns 2.706609%:

=RATE(5,0,-1750000,2000000,0)

The payment today represents a negative present value. The value in five years

is a (positive) right to receive.

To check the answer, use this formula (which returns $2,000,000.03):

=FV(2.706609%,5,-1750000,0)

The rounding error is caused by hard-coding the rate to only six decimal places.

Normally, the argument would be a cell reference, not a hard-coded value.

EXAMPLE 16

A leasehold interest in a property was recently sold for $230,000. The lease had

four years to run, and rent was payable at $6,000 per month in advance without

rent review or escalation. If we accept a yield of 0.75%, what profit rent is shown

by the transaction? Profit rent is the rental value minus the rent paid.

306

Part III: Financial Formulas

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

The following formula returns $5,680.95:

=PMT(0.75%,48,-230000,0,1)

Adding the rent paid ($6,000) produces a rental value of $11,680.95.

Complex Discounting Problems

Complex discounting problems involve the use of all three monetary amounts: pre-

sent value, payment, and future value. The examples of complex discounting in this

section are essentially re-expressions of the complex accumulation problems.

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

EXAMPLE 17

If I discount at 0.75% per month, how much should I pay for a property yielding

$25,000 per month in advance (which I estimate will be worth $5,000,000 in five

years)?

Function required: PV(rate, nper, pmt, fv, type)

The following formula returns –$4,406,865.34:

=PV(0.75%,60,25000,5000000,1)

This example uses a rate per month, and payments are monthly. Therefore, the

nper argument has been converted to months.

We can check this calculation by using the RATE function. The following for-

mula returns 0.75%:

=RATE(60,25000,-4406865.34,5000000,1)

EXAMPLE 18

I paid $1,200,000 for a property that yields a rent of $12,000 per month in

advance. If I sell it in five years for $1,500,000, what yield will I receive?

Function required: RATE(nper, pmt, pv, fv, type, guess)

The following formula returns 1.29136%:

=RATE(60,12000,-1200000,1500000,1)

Chapter 11: Introducing Financial Formulas

307

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This result can be verified by using the PV function. The following formula

returns –$1,200,000.00:

=PV(1.29136%,60,12000,1500000,1)

It’s important to understand that the rent is quoted monthly in advance, but the

term is five years. This discrepancy is resolved by converting the years to months.

Therefore, the formula returns a monthly rate of interest.

Note that the rent is not converted to an annual rent.This is because a rent

of $12,000 per month in advance is not the same as a rent of $144,000 per

annum in advance. To achieve the equivalent annual amount we would

need to know the rate of discount — which is the one piece of information

we are trying to calculate.

EXAMPLE 19

A property has been purchased for $1,600,000. It yields a rent of $10,000 per

month in advance. If I am to secure a yield of 1% per month, what must the prop-

erty be worth in five years when I plan to sell it?

Function required: FV(rate, nper, pmt, pv, type)

This formula returns $2,081,851.05:

=FV(1%,60,10000,-1600000,1)

This result can be verified using the following formula (which returns

–$1,600,000):

=PV(1%,60,10000,2081851.05,1)

Amortization Problems

Amortization is the term given to the process of paying back loans. This chapter, in

fact, has already covered most of the calculations required, but the problems were

expressed in terms of accumulation.

All the basic examples in this section are available on the companion

CD-ROM.

308

Part III: Financial Formulas

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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

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

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

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

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:

 Nominalrate: 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

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