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A Master Time Value of Money Formula
Floyd Vest
For Financial Functions on a calculator or computer, Master Time Value of Money
(TVM) Formulas are usually used for the Compound Interest Formula and for Annuities.
(See Formula 7 below. See the Appendix in the TI83 (p. A-55) or TI84 manuals. The
manuals are almost identical for finance. You can download parts of a manual at
www.ti.com/calc.) The following is a derivation of the TVM Formula for Future Value
( FV ) with examples and exercises.
You Try It #
Check the Appendix, Financial Functions in the TI83/84 manual and find the Time Value
of Money Formula for FV. Compare it to Formula 7 in this article.
Compound Interest Formula. First we need to derive the Compound Interest Formula,
which is
(1) FV = PV(1 + i )
N
, where FV is the future value, PV is the present value , i is the
interest rate per compounding period, and N is the number of compounding periods. We
are using the terminology in the TI83 and TI84 manuals.
Example 1
For an example we calculate the FV for PV = $100 invested for ten years at the annual
nominal rate of 4%, compounded quarterly. The rate per compounding period is
i =
= (annual nominal rate) ÷ (number of compounding periods per year). N is the
number of compounding periods, which is 10 × 4 = 40 quarterly periods. So
FV = 100 1 +
40
= $148.89. The investment of $100 grew to $148.89 in ten years.
You Try It #
For Example 1 above, what is the Future Value if 4% is compounded daily? Money
Market Funds usually compound daily.
For the derivation, consider that compound interest earns interest on interest as recorded
in the following table:
Beginning PV Interest FV at end of period
Period 1 PV i ( PV ) PV (1 + i )
Period 2 PV (1 + i ) i [ PV (1 + i )] PV (1 + i ) × PV (1 + i ) = PV (1 + i )
2
Period 3 PV (1 + i )
2
i [ PV (1 + i )
2
] PV (1 + i )
3
Period N PV (1 + i )
N – 1
i [ PV (1 + i )
N – 1
] PV (1 + i )
N
We conclude, from the last line of the table, the Compound Interest Formula for FV to be
FV = PV(1 + i )
N
as described above.
Future Value of an Ordinary Annuity. To continue the derivation of the TVM Formula,
we need the formula for the Future Value of an Ordinary Annuity of N payments ( PMT ),
each drawing compound interest at the rate i per period as illustrated on the following
timeline.
0 1 2 3... N – 1 N
|________|________|_______|_____________________________|_______|____
PMT PMT PMT PMT PMT
By the Compound Interest Formula, the FV of the Ordinary Annuity is the accumulation
of PMT s and the interest on each:
FV = PMT (1 + i )
N – 1
N – 2
N – 3
+... + PMT (1 + i )
1
+ PMT.
Multiplying through by (1 + i ) we get
(1 + i ) FV = PMT (1 + i )
N
N – 1
N – 2
2
Next we subtract to get
(1 + i ) FV – FV = PMT (1 + i )
N
- PMT , with intermediate terms subtracting out. Solving
for FV we have
(2) FV = PMT
N
i
i
as the formula for the Future Value of an Ordinary Annuity
where PMT is the payments over N periods, each drawing compound interest at the rate i
per period. For an Ordinary Annuity, the PMT s occur at the end of periods.
Algebraic manipulation is conducted on Formula 5 to get the TVM Formula for FV to be
(6) FV = (1 )
N i i
PMT G PMT G
i PV
i i
− × ×
. The algebra is left as an exercise.
Cash Flow Sign Conventions. One property of the final Master TVM Formula is that
sign changes are made to accommodate Cash Flow Sign Conventions. For example
consider a Cash Flow timeline with PV and PMT s negative and FV positive.
0 1 2 3... N – 1 N
|________|________|_______|_____________________________|_______|
FV
PV PMT PMT PMT PMT PMT
In an investment of PV and PMT s, money going out is considered negative, and indicated
under the timeline, and FV coming in at the end is positive, and indicated with an upward
arrow on the timeline.
Master TVM Formula. To accommodate this sign convention, we will in Formula 6
change - PMT to PMT and PV to - PV to get Formula 7.
(7) FV = (1 )
i N i
PMT G PMT G
i PV
i i
× ×
, where we have an annuity with
payments PMT , interest rate per period of i , N periods, and a PV at time zero.
i
G is
described above.
This Master TVM Formula for FV is in the manuals for the TI83 and TI84. Financial
functions on other calculators may have different sign conventions. (See the TIBA35,
TIBAII, and HP19B.)
Example 3
We will do an example for Formula 7. Considering the above timeline, we invest
PV = - $100,000, PMT of - $10,000 at the end of each year for N = 25 years in a retirement
program averaging 6% per year. Notice the sign conventions. What is the FV of the
savings program? Substituting we have
i
G = 1 since we have end of period PMT s.
FV =
25
. FV = $977,832.19.
You Try It #
For Example 3 above, investments earn 6% compounded monthly and the family has
$200,000 in investments and are saving $500 at the end of each month for the next 25
years. How much is in their retirement fund?
To get the TVM Formulas for PV and PMT from Formula 7, you simply use algebra to
solve. To solve for N , you can use logarithms. These solutions are left as exercises. To
solve for i for an Annuity requires an iterative program. See the references for some
iterative calculator programs. To get the Compound Interest Formula from Formula 7,
simply let PMT = 0, and remember the sign changes. For the Compound Interest
Formula, you can solve algebraically for i.
The concept of discounting and PVOA. From the Compound Interest Formula,
FV = PV (1 + i )
N
we get PV = FV (1 + I )
- N . This is referred to as discounting FV to get
PV. For an Ordinary Annuity, we can discount each of the PMT s to get a PV of an
Ordinary Annuity (PVOA) as the sum, PVOA =
1
PMT (1 i )
−
2
PMT (1 i )
−
3
PMT (1 i )
−
( 1)
N
PMT i
− −
N
PMT i
−
It left as an exercise to show that the PVOA of an Ordinary Annuity is
(8) PVOA = PMT
N
i
i
−
, and to derive a formula for the Present Value of an
Annuity Due (PVAD).
The concept of discounting is important because it comes up in loans, mortgages, long
term financial planning, inflation, bond pricing, discounting Cash Flows to Initial Equity
or Net Present Value in business planning, Yield to Maturity (YTM), Internal Rate of
Return (IRR), other applications. Note here that the term PV has taken on two meanings.
Examples for Exercises
The following is an example of the format of some of the application problems in the
Exercises requesting formulas, answers, commentary, timelines, knowns, unknowns,
variables, formulas, Financial Functions, code, and summary.
Instructions: Solve the following problems with formulas, a scientific calculator pad, list
and label knowns, unknowns, draw a time line. Use any financial formulas that are
convenient.
Then, solve with Financial Functions on a calculator or computer. See the calculator
manual for the required code. The problems are taken from Chapter 14 of the TI83 and
TI84 manuals. Write code and commentary. Identify the output, and put on units.
Summarize the answer to the problem.
Problem: Consider a problem on p. 14 - 3 of the TI83 manual. A car costs $9000. You
can afford monthly payments of $250 a month for four years. What (APR) annual
percentage rate will make this possible?
Exercises
For some of the following exercises, solve with mathematics of finance formulas and
with financial functions on a calculator. Use any mathematics of finance formula that is
convenient. For all problems, show all your work, label all inputs, show formulas, label
all answers, and summarize. For the basic mathematics of finance formulas and their
derivations, see Luttman or Kasting in Unit 1 of this course.
- At what annual interest rate, compounded monthly, will a deposit of $
accumulate to $2000 in seven years? See p. 14-3 of the TI83 Manual. Consider
Formula 9 below.
(9) FV = PV 1 +
r
k
N
, where r is the annual nominal rate compounded k times per year.
N is the number of compounding periods. I % is a percent. r =
I
- For a mortgage of $100,000 at 18% per year for 30 years, what is the monthly
payment? See p. 14-4 in the TI 83 Manual.
- For a mortgage of $100,000 at 8.5% per year for 30 years, what is the monthly
payment? See p. 14-6 of the TI 83 Manual.
- For a 30-year, 11% mortgage with $1000 a month payments, what is the amount of the
mortgage? See p. 14 - 7 of the TI 83 Manual.
- On p. 14- 3 of the TI 83 Manual, one problem is I % = 6%, PV = 9000, PMT = 350, FV
= 0, P / Y = 3, what is N? Do this problem.
- Set up the Cash Flow equation for the second timeline on p. 14-8 and verify the
answers. The npv (Net Present Value) at 6% = $2920.65 means the sum of the cash
flows discounted at 6% gives $2920.65. The (Internal Rate of Return) irr = 27.88%
makes the npv equal to zero. The Internal Rate of Return is the rate that discounts the
remaining cash flows to the initial equity CF0. To do this calculation with a scientific
calculator pad, store the 1 + i in STO and use RCL.
By algebra, derive Formula 6 from Formula 5.
Derive from Formula 3, the TVM Formula in the Appendix of the TI 83 Manual for N ,
which is N =
ln
PMT! FV " i
PMT + PV " i
ln( 1 + i )
for
i
G = 1.
- Derive from Formula 3, the TVM Formula in the Appendix of the TI 83 Manual for
PV with
i
G = 1, which is PV =
N
PMT PMT
FV
i i i
− × −
. Indicate the required
sign changes to satisfy the TVM sign conventions for cash flows.
- Derive from Formula 3, the TVM Formula in the Appendix of the TI 83 Manual for
PMT with
i
G = 1, which is PMT = - i
N
PV FV
PV
i
and indicate the required sign
changes to satisfy the TVM sign conventions for cash flows. You may want to use the
identity
N
N N
PV i PV
PV
i i
- What do you change in Formula 7 to get the Compound Interest Formula? Make
changes and show the formula.
- Derive the Formula for the Sum of an Ordinary Annuity from Formula 7. What
strange results do you get and what sign change is required?
- Derive a formula for the Present Value of an Annuity Due.
- Write a paragraph with cash flow timelines discussing some of the entries and sign
changes required in the TVM Solver and required by the TVM formulas.
- Consider a fund of PV dollars that provides annual withdrawals of PMT s for N years.
Draw a cash flow timeline with sign conventions and give the signs for PV and PMT s.
Give a formula for this fund and withdrawals.
- (a) Consider a loan of PV dollars that is repaid with PMT s. Draw a cash flow
timeline with sign conventions and give the sign changes for the variables. Give a
formula for this loan and payments.
(b) Notice that financial variables have multiple meanings. It is this multivalency and
abstractness that gives the power and generality to mathematics. In some publications,
even more generic financial terms such as Rent ( R ), Principle ( P) , and Sum ( S ) are used.
Sometimes applied problems in finance require formulas that are not among the common
ones. Then you need to derive your own formula. For an example of such a derivation,
derive a formula for the Sum
n
S of the first n terms of a geometric sequence with first
term a , common ratio r , second term ar and n th term ar
n – 1
. Use the technique used to
derive Formula 2 above.
- A simplification of a formula for i in the Appendix to the TI83 Manual is
i =
ln(1 )
i
e
. Prove this formula. Use the definition of ln(1 + i ).
- Some problems require two timelines. A person makes a debt of $2000 that must be
paid back in ten years at 5% interest compounded semiannually. To accumulate money
Side Bar Notes
Load Funds and No-load Funds. From USA Today , May 27, 2011: “Let’s look at the
American Funds Growth Fund of America. The A shares … have gained 4.17% a year
for the last decade. … To put this in perspective: A $10,000 investment in the fund
would have gained $5,046 in 10 years.”
“If you paid the fund’s maximum 5.75% sales charge, however, your total return was
3.55%, …. Your gain has now shrunk to $4,174. You owe taxes on the distributions.
… your after tax return would be 3.21%.”
“And if you had sold the fund after 10 years and paid taxes on you gains, you’d be left
with a 2.96% average annual gain.”
“So you invested $10,000 in a taxable account, paid the sales charge, paid taxes on
distributions and gains. Your $10,000 is now $13,387. At that rate, you’d double your
money in about 24 years.”
“But you can reduce your cost. One easy way of course, is to buy a no-load fund. You
pay no commission.” If you had the money invested in a Roth IRA or 401k, it would be
after-tax money and there would be no income taxes. See IRS Publication 590.
For a no-load fund, see www.vanguard.com for the Vanguard Wellington Fund, which is
the nation’s oldest balanced fund. 6.20% in the last 10 years. Expense ratio 0.34%.
Some salesmen of load funds will tell you that you will pay indirectly a commission on a
no-load fund. How is that with a 0.34% expense ratio on the no-load fund, and a 0.69%
expense ratio on the load fund.
Problem: See if your can reproduce the above calculations. You can do most of them.
References
Most of the following references are cited in the annotated bibliography for this course,
and provide derivations, exercises, and calculator programs for practice with the financial
formulas, calculator Financial Functions, and graphs.
Vest, Floyd “Time, Money, & Polynomials,” HiMAP Pull-Out, Consortium 37.
Vest, Floyd, “What Can a Financial Calculator Do for a Mathematics Teacher or
Student?” Journal of Computers in Mathematics and Science Teaching , pp. 67- 78 , Fall
Vest, Floyd, “Computerized Business Calculus Using Calculators,” Journal of Computers
in Mathematics and Science Teaching , Summer,1991.
Vest, Floyd, “Tim and Tom’s Financial Adventure,” HiMAP Pull-Out, Consortium 39.
Vest, Floyd, “Making Money With Algebra,” HiMAP Pull-Out, Consortium 33.
Answers to Exercises
- To solve the problem with a formula, use the Compound Interest Formula 9 given
above: Substituting in Formula 9, we have 2000 = 1250 1 +
r
84
since there are
7 × 12 = 84 compounding periods. Using algebra and a scientific calculator pad to solve
for r , we have r = 0 .06733 = 6.733% as the annual nominal rate.
For the financial calculator code and explanation for the TVM Solver, see page 14-3 of
the TI 83 Manual, or see the TI 84 Manual.
- We will use Formula 8 above. Substituting we get
100 , 000 = PMT
! 360
Solving and using a scientific calculator pad gives PMT = $1507.09 as the monthly
payment. We could have used Formula 3 above with FV = 0 and PMT would have come
out negative.
For a financial calculator code, see p. 14-4 of the TI 83 Manual, or see the TI 84 Manual,
for the TVM Solver.
- For this problem, the TI83 Manual presents the tvm_Pmt financial function, which is
menu item 2 under 2
nd
Finance.
Code and comments: Put the knowns in the TVM Solver. Put 0 in PMT.
Code: 2
nd
Finance ∨ (to tvm_Pmt ) Enter Enter (On the home screen you
read - 768.91. The monthly payment is $768.91.)
- Substituting into Formula 6 and calculating with a scientific calculator pad gives
PV = $105,006.35, as the amount of the mortgage.
The TI 83 Manual presents the tvm_PV [(N,I%,PMT,FV,P/Y,C/Y)] function for this
problem. On the home screen, you store with the STO> key, 360 in N, 11 in I%,
(-)1000 in PMT, 0 in FV, 12 in P/Y and calculate PV.
Code and comments: (On the home screen.) 360 → 2
nd
Finace > Enter
(To Sto 360 in N.) Alpha : 11 → 2
nd
Finace > ∨ (To I%) Enter
Alpha : (-)1000 → 2
nd
Finace > ∨ (To PMT.) Enter Alpha : 12
nd
Finance > ∨ (To P/Y.) Enter Enter 2
nd
Finace ∨
(To tvm_PV.) Enter Enter (You read 105006.35.) The amount of the mortgage
is $105,006.36. (It is more convenient to enter values directly into the TVM Solver
when possible.) The above code didn’t store values to the TVM Solver.
- Substituting into Formula 8 above and using logs to solve, you get N = 36.
payments. The TVM Solver gives the same answer.
- To get the Compound Interest Formula FV = PV (1 + i )
N
from Formula 7, set
PMT = 0,
i
G = 1 and make a sign change.
1 2. To get the Formula for the Sum of an Ordinary Annuity from Formula 7, set PV = 0
and
i
G = 1, and make a sign change to get Formula 2.
- To get a Formula for the Present Value of an Annuity Due, use Formula 8 and since
PMT s are at the beginnings of periods, use PVAD =
n
i
PMT i
i
−
- For the fund, PV is negative and PMT is money coming in and is positive.
- For the loan, PV is money coming in and is positive, PMT s are money going out and
are negative.
- They should save $272.96 at the end of each year for ten years.
