Finance Functions

The financial functions are a powerful tool in making business or personal

financial decisions. To use the financial functions, select Finance from the

Mode menu, or use the keyboard shortcut Ctrl-F, or use shift then the Fin button.. The financial functions allow you to perform all kinds

of financial calculations including the value of investments, the cost of

financing a loan, the discounted value of a future payment or cash flow, and the

equivalent rate of return of a series of payments. There are also buttons to

allow you to quickly add tax, compute a discount or add a mark-up. The meaning

of some of the functions, and the input values needed, are easily forgotten. If

you allow the cursor to hover over a button, or in the case of pen devices if

you do a "tap and hold" with the stylus, a tool tip will appear which gives the

full name of the function and a hint at the order of arguments.

In finance mode the number of decimal places is automatically set to two,

irrespective of the settings in the Option/Display dialog. The display settings

can be modified to change the decimal separator and digit group (thousands)

separator if required. You can select "," or "." for the digit group character

or set it to "no" to disable digit grouping. The scientific and engineering

decimal display settings also have no effect.

The example below assume that you are using Algebraic logic. For RPN logic

you will need to alter the keystrokes slightly. So for example if the key

sequence given is 100 func 200 = , for RPN

you would use 100 Ent 200
func.

Whilst we have taken every care to ensure that these calculator functions and

the descriptions are accurate, we must remind you that you use them at your own

risk. If you are in any doubt, you are urged to seek professional financial

advice before making major financial decisions.

Using the percent key

The percent key is available in most modes and works in exactly the same way.

In this section we show how you might use it for financial calculations. To get

the percent key use the shift and = buttons. This displays the result of an arithmetic operation as a percentage.

Examples:

Calculate the tax at 17.5% on � 25.00:

250 x 17.5 shift % Result: � 43.75

Calculate 12% mark-up on $250

250 + 12 shift % Result: $ 280

Give 5% discount on goods costing $125

125 - 5 shift % Result: $ 118.75

Using the tax, before tax b/t, discount d/c and mark-up m/u functions.

The percentage calculations described above can be performed more

conveniently using the special keys available in finance mode. The tax, discount

(d/c) and mark-up (m/u) functions all do this automatically, without needing a

percentage to be entered. The percentage rates you regularly use will typically

not change frequently and these can be programmed in to the calculator. Any time

you need to use an ad-hoc percentage in a calculation you can revert to using

the percentage key.

First, here is how to change the percentages. Click on the rate button to bring up the financial rates dialog box. We will look at this

dialog further in the following sections. It shows you the rates used by various

functions. To change the tax, discount or mark-up rates, select the appropriate

edit box and enter the new percentage (on Pocket PC devices the input window

should come up automatically). So, supposing that the sales tax (or value added

tax) rate is 17.5%, we would enter this value for the tax rate. Similarly, if we

regularly offer customers a 20% discount, we would enter that value as the

discount percentage. (Do not confuse the discount percentage with the discount rate, a separate percentage rate which is used in depreciation

calculations). If we use regularly use a mark-up of 40% on goods, we would enter

this value also. Click on the OK button when you are finished with the rates

dialog.

Once the rates have been set you can use the Tax, d/c and m/u buttons to

quickly perform percentage calculations. The tax and mark-up buttons compute the

result of adding the tax or mark-up percentage. The discount button computes the

amount after subtracting the discount percentage. The before tax (b/t) button

can be used to calculate the tax-exclusive amount from a given tax-inclusive

total.

Example: calculate the 17.5% VAT inclusive price of an item costing �

99 excluding VAT.

First set the tax rate to 17.5%.
Enter the amount - 99.0
Click on the tax button: result: 116.32

Example: calculate the VAT exclusive price of an item with a retail

VAT inclusive price of � 99.

If needed, check that the tax rate is set to 17.5%.
Enter the amount - 99.0
Click on the shift button then the b/t (before tax)

button: result: �

84.26

Setting Interest and Discount Rates Rate

To choose the various percentage rates used in financial calculations, use

the Rate button which brings up the Financial Rates dialog box. The

dialog box can also be used to convert between nominal and effective rates for

different interest payment schedules.

The bank interest rate is applied to the loan calculations (Loan, Savg, TM, TMa, PV, PVa, FV and FVa) and the discount rate to the Net Present Value calculations (NPV and VFl ). Typically these will be assigned different values which prevail at the time

of the calculation and to the relevant field of commerce. The rates also vary

between financial institutions, and depending on whether you are dealing with a

loan or savings, and between different financial products from the same

institution.

By default it is assumed that the period in these calculations is annual. In

this case there is no difference between the nominal interest rate and the

effective interest rate. If a different period is required then the Period:

drop-down list box can be used to select biannual, quarterly, monthly, weekly or

daily interest payments. The nominal or effective annual rate can be used to

specify the interest rate by entering a value in the input box - if the nominal

rate is entered the effective rate is automatically calculated and displayed in

the adjacent edit box and vice versa. The effective rate is the annual rate of

interest taking into account the compounding of the individual payments over the

year, and is sometimes referred to as the Annual Effective Rate (AER) or Annual

Percentage Rate (APR - this sometimes is used to include additional related

finance charges). Because of compounding this rate will be slightly higher than

the nominal annual rate. Banks may quote both rates, and lenders are often

legally obliged to display an APR as well as the lower nominal rate which they

tend to give prominence.

Changing the period automatically updates the effective rate. If you change

the nominal rate the effective rate is updated automatically, so you can use

this to work out the effective rate for a given nominal rate and period. You can

also work back to the nominal rate for a given effective rate by changing the

value in the effective rate box, which automatically updates the nominal rate.

This can be useful if a lender or savings institution quotes one of these rates

but not the other.

If the payment is made at the end of each period, the arrears radio

button should be selected. If the payment is made at the beginning of each

period, select the advance radio button. Don't forget that the financial

functions expect a number of periods, so for example if the period is monthly

then five years is 5 x 12 = 60 periods.

The Financial Rates dialog box also allows you to specify the tax, markup and

discount rates used by the Tax and m/u and d/c functions.

Example: A lender quotes an annual rate of 6% for a loan, but interest

is payable monthly. What is the effective annual rate?

The lender will charge interest at 6% divided by 12 months, or 0.5% per

month. The total interest paid per year will be 6% of the capital, but because

of the time value of money it is more expensive than paying this amount at the

end of the period. To get the effective interest rate, open the Rates dialog and

enter 6 for the nominal bank rate and select "monthly" from the period drop-down

box. The effective rate is automatically updated to show 6.16778%. If the lender

had quoted "6% APR" you could work back to the nominal rate by entering this in

the effective rate box, and the nominal rate would show 5.84106%.

Time Value of Money Calculations

The following group of functions are concerned with the time value of money,

and the present or future value of a fixed amount or a series of regular

payments. This includes loan calculations, mortgages and annuities. The three

functions TM, PV and FV are used to calculate the term, present and future

values of fixed amounts. They each have a corresponding annuity function which

is used to compute the term, present or future values of a series of regular

payments. The word "annuity" is used in this context in a general sense meaning

any recurring regular payment. The payment is not necessarily an annual one, and

not necessarily a regular income purchased for a fixed price as in pensions

planning.

These functions all use the bank rate to compute time-dependent values. The

period defaults to one year but can also be set to be biannual, quarterly,

monthly, weekly or daily. The bank rate and period are set in the Rate dialog box. This dialog can also be used to define whether payments are made or

received at the beginning or end of each period.

Savg Savings Payment

To compute the periodic payment needed to accumulate a given amount, enter

the final amount (future value), click the Savg button, then enter the term

(number of payments). Press =
to get the amount of each payment.

Example: You decide to start saving for a holiday in 12 months time.

How much should you save each month in order to have �1000 available at the end

(assuming 10% bank rate)? What total amount must you put aside?

Check that the bank rate is set at 10% and that the period is monthly,

payment in advance.

1000 Savg 12 = Result: 78.92

X 12 = Result: 947.10

The total amount saved is � 947.10. The interest earned makes the amount up

to � 1000.

Loan Loan Payment

To compute the required periodic payments to repay a given loan, enter the

amount of the loan (present value), click the Loan button, then enter the term (number of payments). Press =
to get the amount of each payment.

Usually the term of a mortgage is defined in years, but the repayment periods

and corresponding calculations are carried out monthly.

Example: What is the monthly payment to pay off a loan of $30,000

dollars over 25 years at an interest rate of 10% ?

First click on Rate to check that the interest rate is set to 10% (and change it if needed),

and set the period of the payments to monthly. You can note that the effective

interest rate changes to about 10.47%. You should also set the payments to

"arrears", assuming you will be paying the mortgage at the end of each month.

The period is now in months, which over 25 years is 25x12 months (i.e. 300

months).

3 0 0 0 0 Loan ( 2 5 X 12 ) =

Result: $ 272.61 (the monthly payment).

Int Calculate Interest Payments

To calculate the interest payment, per period, on a loan or saving, enter the

loan or investment amount, then shift and then Int .

Example: Your business borrows $500,000. What is the monthly interest

payment?

Check your rates (e.g. use 10%, monthly payment).

500000 Int Result: 4166.67 - the monthly interest payment.

TM Term of Investment

To compute the term required for an investment to increase to a given value,

enter the principal, followed by TM followed by the desired compounded sum at the term. The result is the number

of periods (generally years) required for the investment to reach this value. If

you wish to use a different period (monthly for example) you need to change the

value of the interest rate accordingly, by pressing Rate .

Example: How many years of annual compound interest are required for

an initial investment of $100 to reach the value of $200 (assuming 10% bank

interest rate)?

First make sure the bank rate is set to 10% and the period is set to annual

1 0 0 TM 2 0 0 =

Result: 7.27 (i.e. 8 years to exceed $200).

TMa Term of an Annuity

To compute the term required for a series of regular payments to increase to

a given value, enter the payment, followed by TMa followed by the

desired compounded sum at the term.

Example: You plan to buy a boat costing $20000. You can afford to save

$300 per month. How long will it take to save for the boat (assuming 7% bank

interest over the period, and monthly compounding, payment in advance)?

First check the rate and period are set correctly in the Rate dialog.

300 TMa 20000 =

Result: 56.20

The number of periods is 56.2, or about 4 years and 9 months.

PV Present Value of an Investment

This function computes the present value of a future amount. This can be used

to find out how much you would need to invest now to be worth some specified

amount in the future. You can also use it to work out the present value of some

future amount. This can be useful if you need to compare alternative investments

which yield amounts at different times in the future.

Example: Which is worth more now; � 500 in two years time, or � 1000

in ten years time?

To answer this we need to make a guess of the interest rate over the next ten

years or so. Let's assume this to be 5% and, for argument's sake, compounded

annually. We can then work out the present value of the two future amounts so

that we can make a comparison.

500 PV 2 = Result: 453.51

1000 PV 10 = Result: 613.91

This suggests that the � 1000 is worth waiting for. In practice you might

take into account any commercial risk of default on the payment over the longer

period.

Example: You plan to give a four year old child $1000 on their

eighteenth birthday. What amount do you need to invest today (assuming 7% bank

interest over the period, and monthly compounding)?

First check the rate and period are set correctly in the Rate dialog.

1000 PV ( 14 X 12 ) =

Result: 376.38

One problem with such long term calculations is that the bank rate will very

probably vary a great deal over a long period. In this case it is necessary to

use a guess of the likely average rate over time based on historical interest

rates. It would also be wise to choose a fairly conservative value for the

expected interest rate.

PVa Value of Annuity

To calculate the present value of an annuity at maturity, enter the amount of

the payment, then FVa , followed by the term (number of periods). Despite the literal meaning of the

word annuity, it is possible to use a period other than annual in the

calculation, in which case you need to change the value of the interest rate

accordingly, by pressing Rate .

Example: You decide to finance the purchase of a car costing $15,000

over a period of five years, but can only afford a monthly payment of $300. If

the finance company offers an APR of 10% compounded monthly, what down payment

would be required?

Make sure rate is 10%, period is monthly, payment in arrears.

300 PVa ( 5 X 12 ) =

Result: $ 14119.61

- 15000 =

Result: $ -880.39

A down payment of $880 is required.

FV Future Value of an Investment

To compute the value of a single amount invested for a given number of years

at the current interest rate, enter the principal, click FV and the

number of investment periods. Usually the number of periods is the number of

years. If you wish to use a different period (monthly for example) you need to

change the value of the interest rate accordingly, by pressing Rate .

Example: What is the value of $100 invested for five years at compound

interest (assuming a 10% annual interest rate)?

First check that the Rate dialog is set to the correct bank rate (10%) and

period (annual).

1 0 0 FV 5 =

Result: $161.05

You can also compute the value of the same investment if interest is computed

monthly by setting the period to monthly in the Rate dialog and

adjusting the number of periods accordingly:

1 0 0 FV ( 5 X 12 ) =

Result: $ 164.53

FVa Future Value

of Annuity

To calculate the future value of an annuity at maturity, enter the amount of

the annual payment, then FVa , followed by the term (number of periods).

Despite the literal meaning of the word annuity, it is possible to use a

period other than annual in the calculation, in which case you need to change

the value of the interest rate accordingly, by pressing Rate .

Example: What is the value at maturity of a 30 year annuity with an

annual payment of $200 (assuming an interest rate of 10%)?

Make sure rate is 10%, period is annual, payment in advance.

2 0 0 FVa 3 0 =

Result: $ 36188.68

Cash Flow Functions

Cash flow functions are designed to compute the present value of an irregular

series of payments. These functions use the discount rate in the Rate dialog box. In order to use the IRR and VFl functions you will need to set the

calculator to array mode. To do this, select the Option/Matrix menu and set the

grid to the desired size, and check the box labelled "Show matrix in all modes".

If the array is two-dimensional, the values corresponding to successive periods

are ordered right to left in rows, and then row by row from top to bottom.

IRR Internal Rate of Return

The internal rate of return is the discount rate at which the present value

of a series of payments would be zero. The practice is to compare this with the

bank rate to decide whether the investment is preferable to simply investing the

capital in a bank. In real situations there is usually an element of risk and

uncertainty in the expected future payments and this should be taken into

account when making investment decisions.

To compute the internal rate of return on an irregular series of payments,

first select the Option/Matrix menu and set the grid to the desired size, and

check the box labelled "Show matrix in all modes". Then enter the values of the

payments into the array (if the array is two-dimensional, remember that the

elements are ordered from left to right and then from top to bottom). The

investments should be entered as negative numbers. Make sure all cells are

selected and then click on the IRR button to get the IRR value.

It is possible for a set of cash flows to exist for which there is no IRR, or

for the value to be too large or negative, in which case an overflow error is

displayed.

Example: You plan to invest $1000 and expect to receive nothing in the

first year, $100 in the second year, $200 in the third year, and then $300 in

the fourth, fifth and sixth years. What is the internal rate of return?

Enter the values -1000, 100, 200, 300, 300, 300 into the array (make sure all

cells are selected when you finish). Then click on the IRR button.

The array is filled with the internal rate of return, which is computed as 5.55 .

NPV Calculate Net Present Value of a Future Cash Flow

To calculate the present value of an amount to be paid at some time in the

future, enter the value of the amount, followed by NPV and then the number of periods before the payment will be made. The length of

the period is determined by the settings in the Rate dialog box, and

defaults to one year. If you wish to use a different period (monthly for

example) you need to change the value of the period accordingly.

The NPV function is exactly the same as the PV function

except that it uses the discount rate instead of the bank rate.

You can enter a negative number of periods, in which case you get the present

value of a payment which was paid at some time in the past. If you require the

net present value of a series of periodic cash flows, use PVa taking

care to set the bank rate to the required discount rate. Alternatively use the VFl function with equal amounts.

Example: What is the net present value of $100 to be paid in five

years time (assuming 5% discount rate).

First check that the discount rate is set to 5%; period to annual.

1 0 0 NPV 5 =

Result: $ 78.35

VFl Net Present Value of Cash Flows

To calculate the net present value of a series

of uneven cash flows, first select the Option/Matrix menu and set the grid to

the desired size, and check the box labelled "Show matrix in all modes". Next

enter the cash flow for each period into the array. Make sure that all cells are

selected and press the VFl button. The array is now filled with the

net present value for each corresponding period. The value corresponding to the

last entry is the present value of the whole cash flow.

Typically the period is annual. If you wish to use a different period

(monthly for example) you need to change the value of the discount rate

accordingly, by pressing Rate .

Example: A project requires an initial capital expenditure of

$1,000,000. After five years the capital equipment is to be written off. The

expected annual revenue stream at the end of each year, less running costs, is:

year 1 - $100,000; year 2 - $200,000, year 3 - $300,000, year 4 - $300,000, year

5 - $300,000. The net revenues exceed the initial capital cost, but is the

investment a good one, assuming a discounting rate of 5% per annum?

Input the values, with a value of zero for year 1:

0.00
100000.00
200000.00
300000.00
300000.00
300000.00

VFl

Result:

0.00
95238.10
276643.99
535795.27
782606.01
1017663.86

The result shows that, taking into account the time value of money, the

revenue flows have a net present value of $1017663.86, so that the project is

just profitable (but probably not worth the risk!).

Depreciation Functions

The depreciation functions compute a depreciation factor for a given asset

life (in periods) and number of periods. For example, the depreciation factor

for an asset with a life of 10 years after 5 years with straight line

depreciation would be computed as 1 0 SLD 5. The resulting factor, 0.5, can be applied to the

value of the asset less any salvage value at the end of the period.

SLD Straight Line Depreciation

To calculate the fraction of the value of an asset which is depreciated after a

given time using straight-line depreciation, enter the initial cost less any

salvage value, then X , followed by the useful asset life, then SLD , followed by the number of periods after which the depreciation is to be

calculated, followed by = . The result is the total

(cumulative) depreciation charge.

Usually the number of periods is the number of years. If you wish to use a

different period (monthly for example) you need to change the value of the

discount rate accordingly, by pressing Rate .

Example: What is the depreciation after five years on a capital asset

costing $10,000 with a ten year life and a salvage value of $1000 at the end of

its life?

1 0 SLD 5 =

Result: 0.50 (the depreciation factor)

X ( 1 0 0 0 0 - 1 0 0 0 ) =

Result: 4500.00

The depreciation is $4500, therefore the value of the asset after 5 years is

$10000 - $4500 = $5500.00

Using RPN logic, you would enter:

10000 Ent 1000 - 10 Ent 5 SLD X

DDB Double declining Balance

Depreciation

The double-declining balance method of depreciation is an accelerated

depreciation method which provides more rapid depreciation charges in the early

part of the lifetime of the asset. This method is often preferred when

calculating depreciation charges for tax purposes, for example. To calculate the

fraction of the value of an asset which is depreciated after a given time using

the double-declining balance method of depreciation, enter the initial cost

(ignoring the residual value), then X , followed by the useful asset life, then DDB , followed by the number of periods after which the depreciation is to be

calculated, followed by = . The result is the total (cumulative)

depreciation charge.

Usually the number of periods is the number of years. If you wish to use a

different period (monthly for example) you need to change the value of the

discount rate accordingly, by pressing Rate .

Example: What is the depreciation after five years on a capital asset

costing $10,000 with a ten year life and a salvage value of $1000 at the end of

its life?

( 1 0 DDB 5 ) X 1 0 0 0 0 =

Result: 6723.20

The depreciation is $6723, therefore the value of the asset after 5 years is

$10000 - $6723 = $3277

Applying the depreciation to the whole of the asset value, rather than the value

less the residual value, results in a faster rate of depreciation (which is

usually advantageous). This does mean that the depreciation charge may bring the

depreciated value below the residual value near the end of the life of the

asset. When this happens the usual practice is to reduce the depreciation charge

to a value which leaves the residual value and allow the asset to remain at this

value until it is sold or disposed of.

SYD Sum-of-Years-Digits Depreciation

The Sum-of-Years-Digits is another accelerated depreciation method. To

calculate the fraction of the value of an asset which is depreciated after a

given time using sum-of-years-digits, enter the initial cost less any salvage

value, then X

, followed by the useful asset life, then SYD

, followed by the number of periods after which the depreciation is to be

calculated, followed by = . The result is the total (cumulative)

depreciation charge.

Usually the number of periods is the number of years. If you wish to use a

different period (monthly for example) you need to change the value of the

discount rate accordingly, by pressing Rate .

Example: What is the depreciation after five years on a capital asset

costing $10,000 with a ten year life and a salvage value of $1000 at the end of

its life?

1 0 SYD 5 =

X ( 1 0 0 0 0 - 1 0 0 0 ) =

Result: 6545.45

The depreciation is $6545, therefore the value of the asset after 5 years is

$10000 - $6545 = $3455

Currency Conversion

You can use the conversion feature to perform conversions between some of the

major currencies and the former currencies of the European Union. Select

"Currency" as the property and then select the To and From currencies from the

drop-down lists. Apart from the former currencies of countries now using the

Euro (which had a fixed exchange rate) the currency conversions fluctuate and so

the conversions will not be up-to-date. The date at which the conversion was set

is indicated for each currency. You can get updated currency files from time to

time at our web site (http://www.calculator.org/),

or you can edit the currency conversions yourself as needed.

For more information on using the conversion utility see the relevant section.

More examples

Example: You have $100,000 in a savings account, earning 6% annual interest, credited

monthly. In addition, $500 is deposited every month. How long will the funds

last if $1,500 is withdrawn every month?

That's a net withdrawal of $1000/month (the withdrawal and deposit amounts can

be offset). Interest is 0.5%/month (or you could correct for compounding if

appropriate). None of the built-in functions yield a term for an annuity type

investment, but we can use the formula:

Term = - log(1 - PV.rate/pmt) / log(1 + rate)


where PV (present value) is 100000 and pmt (payment) is 1000 so

Term = - log(1 - 100000 x 0.005 / 1000) / log(1 + 0.005)

= 138.975 months, or about 11 1/2 years.

You can check this value using the Mortgage function by computing the mortgage

repayment on a $100000 loan over 139 months, which gives a payment of about

$1000. You can also check using the Annuity function that a $1000 payment over

139 months yields about $200000, if the rate is set to 0.005%. You can then

discount this to the present value (using NPV with a term of 139) to get back to

about $100000.


Example: In the above example, how much can be withdrawn each month to

make equal monthly withdrawals for 10 years?

This can be done using the Mortgage function, entering the principal amount and

the term (i.e. $100000 and 120 months). The result should be $1110.21 per month.

If you are paying in $500 per month also, you can simply offset that with the

withdrawal.

Example: You have a $50,000 mortgage payable over 30 years. How does the

balance change over this period.

First select the Option/Matrix menu and set the grid to 12 x 30, and check the

box labelled "Show matrix in all modes". Switch to Matrix mode and click on the shift and ind buttons to fill the array with indices then - 359 = +/-
to reverse the order. Use Min to save the values for later use. Switch back to Finance mode.

Set the rate to 5%, the period to monthly, and select "Arrears". Calculate the

monthly payment:

50000 Loan ( 30 X 12 ) = Result:
268.41 (in all cells)

 </p>  
      
 <p>

Now calculate the present values:

shift PVa MR =

The result will be the balance outstanding after each period, with

    months along the x-axis and year number on the y-axis. The balance falls 
    
    slowly at first and decreases rapidly near the end to zero when the 
    
    mortgage is paid off.</p>
   
 <p>

Comparison of the Various Financial Functions

For time value of money calculations, the variables are the interest rate,

the period, whether the payment is at the beginning or end of the period

(advance or arrears), the periodic payment (PMT), the present value (PV), the

future value (FV) and the term. Normally the interest rate and period are fixed

(in the case of this software by the Rates dialog). Therefore if three of PMT,

PV, FV and TERM are known the other can be calculated. In addition in most

calculations one of the four is zero leaving two unknowns.

The table below shows the various functions available with the unknown

quantities indicated by an "x", a zero showing the fixed zero value, and "ans"

to indicate the value computed by the function.

If you need to compute any of the various quantities by hand, the

    underlying arithmetic functions are as follows (r = rate):</p>
 <p>TM(PV, FV) = log(FV/PV)/log(1+r)</p>
 <p>TMa(PMT, FV) = log(1 + FV.r/PMT) / log(1 + r)</p>
 <p>PV(FV, term) = FV.(1 + r)<sup> -term</sup></p>
 <p>PVa(PMT, term) = PMT.(1 - (1+r)<sup>-n</sup>) / r or PMT.(1 + r).(1 - (1 
    
    + r)<sup>-n</sup>) / r (advance)</p>
 <p>FV(PV, term) = PV.(1 + r)<sup> term</sup></p>
 <p>FVa(PMT, term) = PMT.((1 + r)<sup> term</sup> - 1) / r or PMT.(1 + 
    
    r).((1 + r)<sup> term</sup> - 1) / r (advance)</p>
 <p>Loan(PV, term) = PV.r / (1 - (1 + r) <sup>-term</sup>) or PV.r / (1 - (1 
    
    + r) <sup>-term</sup>) / (1 + r) (advance)</p>
 <p>Savg(FV, term) = FV.r / ((1 + r) <sup>term</sup> - 1) or FV.r / ((1 + r) <sup>
    term</sup> - 1) / (1 + r) (advance)</p>
 <p>NPV(FV, term) = FV.(1 + r<sub>d </sub>)<sup> 
 -term</sup></p>
 <p>V<sub>flow</sub> (FV<sub>n</sub>) =  <font face="Arial"><font size="5">&#931;</font> 
    FV<sub>n</sub>.(1 + r<sub>d</sub>)
 <sup>-n</sup></font></p>
 <p><font face="Arial">Int(PV) = PV.r</font></p>
 <p>