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RPN Financial 12 manual · Chapter 7 of 16

Time value of money

Loans, mortgages, savings, leases and annuities with n, i, PV, PMT and FV: how the five keys work, the sign convention, solving for each unknown, BEGIN mode and part periods.

7.1How the five financial keys work#

Money today is worth more than the same money later, because today's money can earn interest. The five financial keys connect the pieces of a loan or savings plan. Know any four and the calculator finds the fifth.

KeyStands forNotes
nNumber of periodsMonths for a monthly loan, years for a yearly one
iInterest rate per periodIn percent. A 6% annual rate on monthly payments is 0.5 per month
PVPresent valueThe amount at the start: the loan, the first deposit
PMTPaymentThe regular amount each period
FVFuture valueThe amount at the end: a balloon, a savings goal
0=PV+PMT⋅1−(1+i)−ni+FV (1+i)−n0=PV+PMT\cdot\frac{1-(1+i)^{-n}}{i}+FV\,(1+i)^{-n}
The equation behind the keys (payments at the end of each period). The calculator solves it for whichever value you leave out.
n1i2PV3PMT4FV5CHS6789÷yˣ1/x%TΔ%%EEX456×R/SSSTR↓x⇄yCLxENTER123−ONfgSTORCL0.Σ++
  1. n periods (g: n × 12)
  2. i rate % (g: i ÷ 12)
  3. PV present value
  4. PMT payment
  5. FV future value
  6. CHS sign
The five financial keys, with the 12× and 12÷ shortcuts.

7.2Store a value or calculate it#

Each financial key has two jobs. Key in a number and press the key: the number is stored. Press the key straight after another financial key, without typing a number, and it calculates its value from the other four. The calculated value is shown and stored too.

  1. n stored
  2. i stored
  3. PV stored
  4. FV stored
  5. PMT calculates
Solving for the payment: store four values, then press PMT.

7.3The sign convention: money in and money out#

Money you receive is positive. Money you pay out is negative. A loan you take out is money in (positive PV) and your payments are money out (negative PMT). A deposit into savings is money out (negative) and the balance you get back is money in (positive).

01234+200,000.00PV−1,199.10PMT−1,199.10PMT−1,199.10PMT−1,199.10PMT
A loan from the borrower's side (first periods shown): the loan comes in, the payments go out.

If you get an answer with the wrong sign, check the signs you keyed in. Use CHS to change a sign.

7.4Monthly payments: 12× and 12÷#

Loans are usually quoted with years and an annual rate, but paid monthly. g12×n (12×) multiplies X by 12 and stores it as n. g12÷i (12÷) divides X by 12 and stores it as i. So 30 years and 6% become 360 months and 0.5 percent a month.

Example 7.1

Convert 30 years to months

  1. Key in 30 and press g, 12×.

    30g12×n
Result of the example "Convert 30 years to months" The display shows: 0.00 / 0.00 / 0.00 / 360.00.
You should see360.00
Example 7.2

Convert 6% a year to a monthly rate

  1. Key in 6 and press g, 12÷.

    6g12÷i
Result of the example "Convert 6% a year to a monthly rate" The display shows: 0.00 / 0.00 / 0.00 / 0.50.
You should see0.50

7.5Find the payment on a loan#

To find the monthly payment on a 200,000 loan over 30 years at 6% a year: n = 360, i = 6 ÷ 12, PV = 200,000 and FV = 0 (the loan is fully repaid). Then press PMT.

The loan is set up: the last value stored (FV = 0) is on the display. The display shows: 360.00 / 0.50 / 200,000.00 / 0.00.
The loan is set up: the last value stored (FV = 0) is on the display.
  1. 360 n
  2. 6 g12÷i
  3. 200000 PV
  4. 0 FV
  5. PMT
Mortgage payment.
Example 7.3

Monthly payment on a 30-year mortgage

  1. Key in 360 and press n. Key in 6, press g 12÷. Key in 200000 and press PV. Key in 0 and press FV.

    360n6g12÷i200000PV0FV

  2. Press PMT with no number typed.

    PMT
    After step 2: Press PMT with no number typed. The display shows: 0.50 / 200,000.00 / 0.00 / -1,199.10.
Result of the example "Monthly payment on a 30-year mortgage" The display shows: 0.50 / 200,000.00 / 0.00 / -1,199.10.
You should seeThe payment is −1,199.10 a month (money out).
Example 7.4

Same loan using years

  1. Key in 30 and press g 12×. Then 6 and g 12÷. Then the loan and FV.

    30g12×n6g12÷i200000PV0FVPMT
Result of the example "Same loan using years" The display shows: 0.50 / 200,000.00 / 0.00 / -1,199.10.
You should see-1,199.10
Example 7.5

Car loan: 20,000 over 60 months at 4.5%

  1. Key in 20000 PV, 60 n, 0 FV and 4.5 then g 12÷, then PMT.

    20000PV60n0FV4.5g12÷iPMT
Result of the example "Car loan: 20,000 over 60 months at 4.5%" The display shows: 60.00 / 0.00 / 0.38 / -372.86.
You should see-372.86

7.6A loan with a balloon payment#

If part of the loan is repaid in one lump sum at the end (a balloon), put it in FV as a negative number. The payments are then smaller.

Example 7.6

Lease-style loan: 25,000 over 36 months at 6%, with 10,000 left at the end

  1. Key in 36 n, 6 g 12÷, 25000 PV, 10000 CHS FV, then PMT.

    36n6g12÷i25000PV10000CHSFVPMT
Result of the example "Lease-style loan: 25,000 over 36 months at 6%, with 10,000 left at the end" The display shows: 0.50 / 25,000.00 / -10,000.00 / -506.33.
You should seeThe payment is −506.33 a month. It is lower than for a loan with no balloon.

7.7Find how much you can borrow#

Turn the problem round. If you can afford 1,199.10 a month, how large a loan is that? Enter n, i, PMT (negative) and FV, then press PV.

Example 7.7

Loan you can afford

  1. Key in 360 n, 6 g 12÷, 1199.10 CHS PMT, 0 FV, then PV.

    360n6g12÷i1199.10CHSPMT0FVPV
Result of the example "Loan you can afford" The display shows: 0.50 / -1,199.10 / 0.00 / 199,999.82.
You should seeAbout 200,000.

7.8Find what savings will grow to#

Put a single deposit in PV (negative, because you pay it out), the rate and the periods, set PMT to 0, and press FV.

012345678910−100.00PV+162.89FV
One deposit of 100 grows at 5% for 10 years.
FV=PV (1+i)nFV=PV\,(1+i)^{n}
Compound growth of a single deposit (sign aside).
Example 7.8

100 at 5% for 10 years

  1. Key in 10 n, 5 i, 100 CHS PV, 0 PMT, then FV.

    10n5i100CHSPV0PMTFV
Result of the example "100 at 5% for 10 years" The display shows: 5.00 / -100.00 / 0.00 / 162.89.
You should see162.89

7.9Save a regular amount each period#

For regular deposits, put the deposit in PMT (negative), leave PV at 0 and press FV.

0123456789101112−100−100−100−100+1,268.25FV
Deposits of 100 at the end of each month for 12 months.
Example 7.9

100 a month for 12 months at 1% a month

  1. Key in 12 n, 1 i, 100 CHS PMT, 0 PV, then FV.

    12n1i100CHSPMT0PVFV
Result of the example "100 a month for 12 months at 1% a month" The display shows: 1.00 / -100.00 / 0.00 / 1,268.25.
You should see1,268.25

7.10Payments at the beginning of the period#

Press gBEG7 (BEG) when payments are made at the start of each period. The same deposits then earn one extra period of interest. Press gEND8 (END) to go back.

Example 7.10

Deposits at the start of each month

  1. Press g BEG, then the same deposits as before.

    gBEG712n1i100CHSPMT0PVFV
Result of the example "Deposits at the start of each month" The display shows: 1.00 / -100.00 / 0.00 / 1,280.93.
You should seeMore than 1,268.25, because each deposit earns one more month of interest.
Example 7.11

Lease payment: payments at the start

  1. BEG, 36 n, 0.5 i, 25000 PV, 10000 CHS FV, then PMT.

    gBEG736n0.5i25000PV10000CHSFVPMT
Result of the example "Lease payment: payments at the start" The display shows: 0.50 / 25,000.00 / -10,000.00 / -503.81.
You should see-503.81

7.11Find how long it takes: n#

Key in the rate, the amount now (negative), the amount you want and PMT of 0, then press n. If the answer is not a whole number, n is rounded up to the next whole period.

Example 7.12

How long to double 100 at 5%?

  1. Key in 5 i, 100 CHS PV, 200 FV, 0 PMT, then n.

    5i100CHSPV200FV0PMTn
Result of the example "How long to double 100 at 5%?" The display shows: -100.00 / 200.00 / 0.00 / 15.00.
You should seeIt takes 15 periods (14.2 rounded up).
Example 7.13

Months to clear a debt

  1. Debt 10,000 at 1% a month, paying 500 a month. Key in 1 i, 10000 PV, 500 CHS PMT, 0 FV, then n.

    1i10000PV500CHSPMT0FVn
Result of the example "Months to clear a debt" The display shows: 10,000.00 / -500.00 / 0.00 / 23.00.
You should see23.00

7.12Find the interest rate: i#

Enter n, PV, PMT and FV and press i. The answer is the rate per period. Multiply by 12 (key in 12 then ×) for the annual rate when periods are months.

Example 7.14

Rate on a car loan

  1. 20000 PV, 60 n, 400 CHS PMT, 0 FV, then i.

    60n20000PV400CHSPMT0FVi
Result of the example "Rate on a car loan" The display shows: 20,000.00 / -400.00 / 0.00 / 0.62.
You should see0.62 percent a month.
Example 7.15

The same rate as an annual rate

  1. Press 12 and × to multiply by 12.

    60n20000PV400CHSPMT0FVi12×
Result of the example "The same rate as an annual rate" The display shows: -400.00 / -400.00 / 0.00 / 7.42.
You should see7.42

7.13Find the present value#

To find what a future amount is worth today, enter n, i and FV and press PV. To find what a stream of payments is worth today, enter n, i and PMT and press PV.

Example 7.16

Value today of 200 in 15 years at 5%

  1. 15 n, 5 i, 200 FV, 0 PMT, then PV.

    15n5i200FV0PMTPV
Result of the example "Value today of 200 in 15 years at 5%" The display shows: 5.00 / 200.00 / 0.00 / -96.20.
You should seeYou would need to pay in 96.20 today (shown as money out).

7.14Zero interest#

With an interest rate of 0 the payment is simply the loan divided by the number of periods.

Example 7.17

Interest-free loan

  1. 360 n, 0 i, 36000 PV, 0 FV, then PMT.

    360n0i36000PV0FVPMT
Result of the example "Interest-free loan" The display shows: 0.00 / 36,000.00 / 0.00 / -100.00.
You should see-100.00

7.15Part periods: n with a fraction#

When n is not a whole number, the extra part period at the start earns simple interest by default. Press STO EEX to turn on the C indicator and use compound interest for the part period instead. The two answers differ slightly.

Example 7.18

Simple interest for the part period

  1. 12.5 n, 1 i, 100 CHS PMT, 0 FV, then PV.

    12.5n1i100CHSPMT0FVPV
Result of the example "Simple interest for the part period" The display shows: 1.00 / -100.00 / 0.00 / 1,119.91.
You should see1,119.91
Example 7.19

Compound interest for the part period

  1. Press STO EEX first (C is lit), then the same keys.

    STOEEX12.5n1i100CHSPMT0FVPV
Result of the example "Compound interest for the part period" The display shows: 1.00 / -100.00 / 0.00 / 1,119.92.
You should see1,119.92

7.16Simple interest#

For interest that is not compounded, enter n as the number of days, i as the annual rate and PV as the principal (negative). Then press fINTi (INT). X shows the interest on a 360-day year. The principal is in Y, and the interest on a 365-day year is in Z.

I=P×r100×days360I=P\times\frac{r}{100}\times\frac{\text{days}}{360}
Example 7.20

Interest on 450 for 60 days at 7%

  1. 60 n, 7 i, 450 CHS PV, then f INT.

    60n7i450CHSPVfINTi
Result of the example "Interest on 450 for 60 days at 7%" The display shows: 7.00 / 5.18 / 450.00 / 5.25.
You should see5.25
Example 7.21

Add the principal

  1. Press + to add the principal in Y.

    60n7i450CHSPVfINTi+
Result of the example "Add the principal" The display shows: 7.00 / 7.00 / 5.18 / 455.25.
You should see455.25
Example 7.22

The 365-day interest

  1. Press R↓ then x⇄y to bring the 365-day figure into X.

    60n7i450CHSPVfINTiR↓x⇄y
Result of the example "The 365-day interest" The display shows: 5.25 / 7.00 / 450.00 / 5.18.
You should see5.18

7.17Start a new problem#

The financial registers keep their values until you change them. Press fFINx⇄y (CLEAR FIN) before a new problem so that no old value sneaks into the answer. It clears n, i, PV, PMT and FV but not the stack or the other registers.

Example 7.23

Clear the financial registers

  1. Store a loan, press f, FIN, then recall PV.

    360n200000PVfFINx⇄yRCLPV
Result of the example "Clear the financial registers" The display shows: 0.00 / 360.00 / 200,000.00 / 0.00.
You should see0.00

See also Time value of money: the five money keysCalculate a monthly loan payment

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