TIME VALUE OF MONEY CALCULATIONS
I.
Introduction
Throughout this class, we will be discussing the “time value of money”. Intuitively, we already know that a dollar received today is worth more than a dollar received one year from today. Which would you rather have?
When we use finance rules to quantify the relationship of money received at different points in time, we are studying the “time value of money”.
We will discuss five key factors associated with making time value calculations:
Present
Value, Future Value, Annuity Payment amounts, the interest rate, and, the
number of periods.
Periodic payments (such as a house note, or car note) are the Annuity Payments. We will also have to deal with different time periods (yearly, monthly). For instance, your car note and your house note are due monthly, but, the interest rate is often quoted as an “annual rate”.
You can use Excel to easily solve these problems.
II.
Practical Example of Time Value of Money
If you’re in your 40’s, you’re probably thinking about planning for retirement.
If you’re in your 20’s, you’re probably not. And that’s unfortunate. Because, as the below example demonstrates, the time value of money works wonderful magic if you begin planning for retirement early. Here’s that example:
A. Retirement Planning – assume 10% return on your investments. (This assumption is realistic, since the stock market’s historical return = 10%. Of course, past performance is not a guarantee of future performance.)
Assume there are two people who are both the same age today (20 years old). One of these people has taken a finance class about the time value of money, and begins saving for retirement now. The other one was watching Monday Night Football, and, missed the lecture on the time value of money. Which will have more saved for retirement when they retire at 65 years of age? Here’s what happened:
At 20 years of age, Person A put up $4,000 in an IRA for the next two years when he/she filed their tax return. Then Person A just let the money grow for the next 43 years. You retire at the age of 65.
Age 20 (at the end of the year) $4,000
Age 21 (at the end of the year) $4,000
(a total of two payments)
After two years, this person has $ 8,400.00
Then 43 years later, at the age of 65,
Person A has
$506,016.58 <
--- amount you’ll have at age 65
At 38 years of age, Person B wakes up to the reality that he/she must plan for retirement. So they put up $4,000 in an IRA every year for the next 27 years. Person B also retires at the age of 65.
Age 38 $4,000 (a total of 27 payments)
39 $4,000
40 $4,000…
64 $4,000
Person B, who put up money every year for
27 years, will have at age 65:
$484,399.77
That’s less than the Person A, who put up
money twice and stopped. That’s the time
value of money.
A few points to consider:
1. It is important for both A and B that the money was put into an IRA. In an IRA, the money grows tax free. There are two types of IRA’s: the Regular IRA and the Roth IRA. You should consider both before deciding on your plan. See how taxes keep coming up in personal finance?
2. How much in retirement savings would person A have if he/she continued the $4,000 contributions every year (instead of stopping after two years)? $2,875,619
III. Compound Interest vs. Simple Interest
With Time Value of Money – There are two ways to calculate interest:
Compound interest in multiple periods – “interest on interest and principle”
vs.
Simple interest – “interest only on principle”
Compound interest is better for the person receiving the interest because you receive interest on both the interest and principle. This compounding is KEY to the magic of time value of money.
ALWAYS
assume Compound Interest in this class, unless Simple Interest is stated.
Example:
SIMPLE INTEREST
Assume you have put $5,000 in a savings account yielding 6% simple interest. After 3 years, how much money would you have?
Interest income each year of $5,000 * .06 = $300
After 3 years, $5,000 + $300 +300 + 300 = $5,900
COMPOUND
INTEREST
Assume you put the same $5,000 in a savings account yielding 6% interest compounded yearly. After 3 years, how much money would you have?
With compound interest, you earn interest on both the “principal” (the original investment) and any previous “interest payments”. Here’s how it works:
Year 1: Interest earned: $5,000 *.06 = $300 à now you have $5,300
Year 2: Interest earned: $5,300 *.06 = $318 à now you have $5,618
Year 3: Interest earned: $5,618 * .06= $337 à now you have $5,955
NOTICE: After 3 years, you have $5,955 with Compound Interest, and, only $5,900 with simple interest. The difference between the two increases dramatically as time goes on.
IV. ESTMATING COMPOUND INTEREST - The Rule
of 72
Money doubles on 72 – when the interest rate and time are multiplied together.
(n) x (R) = 72; money doubles
Example: How long will it take money to double if invested at 6%?
Answer: 12 years. (because 6*12=72).
This rule is for estimating; it is not exact.
V. DOING CALCULATIONS – EXCEL MAKES IT EASY
Of course, estimates aren’t always good enough. We need a way to calculate exact amounts given compound interest.
Fortunately, we don’t have to do the calculations by hand (as we did above). Excel has “functions” to calculate:
Present
Value – PV
Future
Value – FV
Annuity
amount – (amount of car note or house note) – AMT
Number
of Periods – NPER
Interest
Rate - RATE
Microsoft Excel – five different functions
1. enter interest rate as a decimal as decimal
2. notice if PV is positive, then FV is negative, and vice versa.
3. FUNCTIONS:
1ST FV =
= FV (rate, n, pmt, PV)
2ND PV
= = PV (i, n,
pmt, FV)
3RD i = = RATE (n, pmt, PV, FV)
4TH n = = NPER (rate, pmt, PV, FV)
5th PMT =PMT(rate,nper,pv,fv)
A.Practical Examples of Present Value,
Future Value, interest rate, and number of periods (annuity, a bit harder, is
covered below – this is the house note or car note).
In PV and FV problems, you will begin by
setting the PMT to zero. (There is no
monthly or yearly payment).
In all of these problems, you will know all BUT ONE variable, and, you will be solving for the unknown.
Examples:
If I deposit $5,000 into a savings account that earns 6% compounded annually, how much will I have in 3 years?
I need to have $100,000 in 14 years to send my child to college. I can earn 10% in the stock market. How much must I invest today (PV) so that I will have $100,000 in 14 years? (notice, this is ONE deposit made today, not periodic annual or monthly deposits).
I bought a house for $100,000. I sold it five years later for $200,000. What return did I earn on my investment? (Return is the interest rate)
I have $20,000. How long will it take me before I have $30,000 if I invest the money at 5% compounded annually?
B.
Annuity Calculations – will be equal monthly or yearly payments. Common examples are car notes or house notes,
or, periodic payments into a retirement account.
The
annuity is the PMT that we have been setting to zero.
In
an Annuity problem, the PMT is not zero.
We are either solving for the value of PMT.
Or we know the PMT, and we are solving for one of the other variables.
Note: The annuity payment is always considered to happen on the last day.
An “annuity due”, which you are not responsible for, happens on the first day.
So, the PMT is the 5th variable. The first four we learned were PV, FV, N, i.
IN EXCEL:
=PMT(rate,nper,pv,fv)
Using calculator:
Put in knowns to solve for unknowns.
Example: Joe wants to have $1,000 in 5 years. He can earn 6%.
He wants to make equal payments each year into a savings account.
How much must he put up at the end of each year?
N=5 (number of periods
I=.06 (or 6, if using the calculator) (interest rate)
PV=0 (he currently has no savings)
FV=$1,000 (he wants to have $1,000)
Solve for PMT.
Answer: -$177.40 (if he puts up $177.40 at the end of each year,
He’ll have his $1,000 at the end of five years.)
C.
Period is not a Year.
Up until now, all problems have assumed “compounded annually”.
What if we said, “compounded monthly”, such as a car note, house note.
Then, in 1 year, we have 12 periods.
In 2 years, we have 24 periods.
For a typical house note, 30 years equals 360 periods.
We must also adjust the interest rate.
The interest rate is usually stated in terms of a year.
But with monthly compounding, we must divide that interest rate by 12.
The best example is a house note.
Example: You borrow $120,000 to buy a house. You take out a
30 year mortgage, and the bank quotes that the interest rate of 6%.
Question: What is your monthly house note?
PV = $120,000
FV= $0 (you pay off the note over 30 years)
N=30*12=360 (you will make 360 monthly payments)
I=6%/12 or .5% per month
(in Excel, enter .005 0r to be safe enter .06/12 )
(in calculator, enter .5)
Solve for PMT: -$719.46 per month is your monthly house note.