Finance Savings; Amortization; Future Value

What to read: Read pages 780–792 and 809–820 in the For All Practical Purposes textbook about
introducing types of savings accounts and loans, amortization, and annuities. For alternative expla-
nations and examples, see pages 197–=204 and 204–221 in the Math in Society textbook.

Key terms:

• Savings/Payment formula
• Annuity
• Present value
• Present value formula
• Amortize/Amortization
• Amortization payment formula
• Conventional loan
• Equity
• Fixed-rate mortgage
• Adjustable rate mortgage (ARM)
• Inflation; annual rate of inflation
• Relative purchasing power
• Current/constant dollars
• Consumer Price Index (CPI)

Learning goals:

• Understand the difference between the formulas for savings for a lump sum deposit versus
making regular deposits.
• Know how to use the formulas for regular deposits or payments to calculate both the total
amount of savings or the loan and the amount needed for regular deposits or payments.
• Know the basics of mortgages and items of consideration.
• Understand relative purchasing power and the effect that inflation has on investments.

Page 1
Finance Savings; Amortization; Future Value

Motivation and goals

We want to first adjust the previous formulas for savings to account for regular deposits or pay-
ments rather than only one initial lump sum. The other basic idea of in this batch of notes is that the
value of money changes over time, so we need to be able to compare how much an amount is worth
now to how much it will be worth later. By discussing these ideas, you’ll be more knowledgeable
about what considerations to make regarding loans like mortgages and long-term savings plans
like for retirement.

Present Value and Savings

The interest formulas that we saw tell us how much money we’ll have in the future if we deposit
a specific amount P right now. However, we commonly want to know the other direction of this
question: in order to have a certain amount of money in the future, such as when we retire, how
much money do we need to deposit right now? If we deposit a lump sum right now and wait until
the specified time, then the previous formulas can immediately be solved to tell us what this lump
sum is.

Remark: As a reminder, while the discussion in these notes is primarily for investments and
depositing money, very similar formulas and examples can be used for borrowing money, some of
which we’ll see. For the previous formulas, we would simply make the rates negative because we
values should be decreasing.

Present Value
The present value of an amount to be paid or received at a specific time in the future is what the
future payment would be worth today, as determined from a given interest rate and compounding
period.

The present value P of an amount A to be paid t years in the future, earning a nominal annual rate
of interest r compounded n times per year, is

(
P=

!
Example 1: How much money would need to be deposited into /future
a high-yield savings account with
an APR of 4.25%, compounded daily, if the goal is to have $50, 000 after five years?
>
-
~ z -

~ = 4 25
. % =, 0425
n=
3652=50 000 += 5

unknown = P =
principal

↑= 40 ,
428

Page 2
 Finance Savings; Amortization; Future Value

While this formula is certainly useful, it doesn’t capture the full details of a typical investment
plan: the formula above is only used if we deposit a lump sum at the beginning and never make
any further deposits or withdrawals. This might be true for investments like buying bonds, but it is
rarely true for savings accounts, retirement investments, or paying off loans.
As such, we need to come up with a new formula that accurately describes the changes that come
with regular deposits or payments. The important detail is that the later deposits will earn less
compound interest than the earlier deposits. We skip the mathematical details and simply present
the formula to you.

Savings Formula for a Sinking Fund

A sinking fund is a savings plan to accumulate a fixed sum by a particular date, usually through
equal periodic deposits. The formula for such a fund is

where
A =

d[[
• A is the future amount to be accumulated
• d is the regular deposit at the end of each period
• r is the annual interest rate (APR)
• n is the number of compound periods per year
• t is the number of years
When the same amount of money is deposited regularly like this, the sinking fund is called an
annuity.

regular deposit is somings formula

Example 2: Suppose you deposit- $100 each month into an investment account that you project will
pay interest at an annual rate of 10%. What will be the value of your investment:
(a) after 10 years? theses !
pover

100 [0)
1)]
d= 100 -

~= 10 %=, 10
= 120 484, .
50
t= 10
n= 12 (b) after 30 years?

1007] Ro -1226
(c) after 50 years?
,
08

100(10)
-1 2 "1 ,
732 4390
,

Page 3
 * 12
(a) total amount deposited =
, 000
"
total interestened = 8
,
483 50 .

(b) total amount deposited I

$36 000 ,
$
total interesterned = 190
,
048 79 .

*
(1) total amount deposited = 60
, 000

total interesterned =
$1 672
, ,
439 .
08
Finance Savings; Amortization; Future Value

Remark: There are five variables in the above formula. In the way that we’ve written the formula,
if we know d, r, n, and t, then we can plug these in to find A. However, given any four of the five
variables, we can mathematically solve for the fifth variable, providing a slightly different formula
that expresses the same relationship between them. For instance, if you knew the amount A that you
wanted to end with, the regular deposit d, the interest rate r, and the compounding period n, then
you can solve for t to determine how long it will take to end with the amount A. We, however, won’t
worry about algebraically solving these equations.

Payment Formula

Solving for d in the savings formula above gives us the regular amount that we need to deposit
each compounding period to earn the amount A after t years:
value
d =
[] ~ present

12 - = 0 10
:
.

Example 3: How much would you need to invest each month at an annual rate of 10% to have a
-
million
~
dollars: a = 1, 000, 000
(a) after 10 years?

[inco -1]
+ = 10
*
d = 1, 000000 ~ 4, 081 34 .

(b) after 30 years?

[incased -]
t =30
"
d = 1, 000000 me 342 38 .
-

(c) after 50 years?

+ = 50
d = 1, 000000
[21 J -
,
=
*
52 72
.

The previous example really illustrates the importance of retirement planning at an early age.
A 401(k) is a typical safe and long-term retirement investment which allows employees to make
monthly contributions to a retirement account. The plan has the advantage that income tax on the
contributions is deferred until the employee withdraws the money during retirement. This means,
for example, that an employee making a $100 monthly contribution may see a reduction in take-home
pay of only $75 or less since taxes are not withheld on the contribution. Making even small contri-
butions starting in your 20s is significantly more important than making moderate contributions in
your 50s.

Page 4
Finance Savings; Amortization; Future Value

Amortization and Conventional Loans

What if we were paying off a loan? We certainly don’t want the total amount that we owe to
increase if we’re making payments! The idea for loans versus investments is that we typically make
the exponent and/or the rate a negative number. This is not a hard rule, however, so some care
should be taken because we’re not going through the full mathematical details.
When you take out a loan you might make the same payment each compounding period at the
same interest rate, however, the interest that you’re paying is only on the amount of the principal that
you have not paid off yet.

As the principal is reduced, there is less interest owed, so less payment goes to the interest and more toward
paying off the principal.

A conventional loan is a loan in which you make regular payments that pays off the current
interest plus part of the principal. Your payments are said to amortize (pay back) the loan.

Amortization Payment Formula

To pay off a conventional loan amount P at a nominal APR of r with n compounding periods per
year for t years, the regular amount that must be paid at the end of each compounding period is

d=
P[] -
=> nt

Example 4: Suppose that you want to buy a car, but don’t have enough money to buy it outright.
After an initial down payment, you finance (borrow) an additional $12, 000 at an interest rate of 4.9%.
-
payment? How much interest did you pay?
O
compounded monthly. If you want to pay off the car at the end of 4 years, what is your monthly
-
-

[- &
p = 12 000
,

4 9% = 0 049
d= 12
, 000
r = .
.

049 (12)(4)
(
- -

Th
n = 12

t= 4 2275 .
8)

d =?
*
total =
(275 81) (28)
.
= 13 238 88
,
.

08 238 88
interest 13 238 12 000 1
-

= .
,
: .

,
,

Page 5
 Finance Savings; Amortization; Future Value

Example 5: You’re wanting to buy your first home and have been looking into mortgages. The
median price of a home in Athens is currently around $340, 000. Unfortunately, mortgage rates are
rather high right now, being around 7.9% for a 30-year fixed-rate mortgage. You’ve been saving your
money and can afford a down payment of $50, 000 (including the closing costs). What will be the
monthly
-
payment of your mortgage? How much will a median home cost you over the term of the
mortgage?

p 340 50, 000

1290 000
, 000
= - =
,

r = 0 079

[
.

10)
t = 30

12
d = , 000
290
)
1=
-

d = T

2 $2 ,
107 Ty .

monthly payment
total amount =
(12) (30) (2 ,
107 .
73) ** 758 ,
784 84
.

total interest =$468 784 , .

84
Example 6: What if you had instead decided to buckle down on extravagances in the future and
decided to go with a 15-year fixed-rate mortgage? What would your monthly payment have been
-
and how much will a median home cost you over the term of this mortgage?

some but now + = 15

d = , 000
290
[# & *
(4)
-
12105)

↳ 2
,
754 68 .
E 650 more per month
A
total mount = (2)(15) (2754 68) .

2495, 841 57 .

*
total interest = 205, 841 . 5)
There are different types of mortgages, including fixed-rate mortgages, where the rate is exactly
the same the entire length of the mortgage, and adjustable-rate mortgages (ARMs), where the rates
can go up or down depending on interest rates in the economy. Usually, the rate can be raised or
lowered only every year or two, and then by a limited percentage. An ARM typically has a decently
lower interest rate than a fixed-rate mortgage though are only available for shorter time periods and
commonly has an initial fixed-rate period. For instance, a 10�6 ARM means a mortgage that has a
fixed rate of 10 years and then the rate is adjusted every 6 months for the duration of your loan. There
are many different types of mortgages and many different factors that goes into determining what
the best option is for the consumer.

Page 6
Finance Savings; Amortization; Future Value

The nice thing about a loan like a mortgage or for a car is that you can regard paying off the
principal as gaining a certain amount of ownership of the item. While you’re paying both part of the
principal and the accumulated interest with each payment, it’s really only the interest payment for
which you’re getting nothing in return. Equity is term for the amount of principal of a loan that has
been repaid. One says that you “building equity” in a house as you pay off the mortgage. Because
you are initially paying a significant amount of interest, over time more and more of your monthly
payment goes to the principal, meaning you are earning significantly more equity later in your loan
period.

Example 7: Suppose that your parents bought their home in 1980 with a 30-year fixed-rate mortgage
for $100, 000 at an 8% interest rate. After 21 years, how much equity did they have in the house,
meaning how much of the principal had been repaid?

When you take out a loan you are always asked about what equity you have. Even if you don’t
own the entirety of your home, you can put up the equity that you do have in your home as collateral.
One example of this is for a second mortgage or a reverse mortgage.

Page 7
Finance Savings; Amortization; Future Value

Future Value
In all of our discussion so far we’ve neglected one very important consideration when investing:
inflation. Inflation is a rise in prices from a set base year. The fact is that $1 today is not the same as
$1 next year because the purchasing power of money decreases over time. If the inflation rate is r,
then goods that cost $1 today will cost $(1 + r) next year, and this will grow exponentially. We then
need to account for the future value of money.

Example 8: Suppose that there is a constant 3% annual inflation rate for a four year period. If an item
costs $100 today, how much will it cost in four years?

Example 9: Any time that you buy an asset like a car, it will depreciate in value due to things like its
use and less consumer demand for it. Suppose that you bought a car for $24, 000 today (in current
dollars) and it depreciates at a rate of 15% per year. What will its value be in three years?

This is why when you’re considering jobs and salaries, it’s always important to consider cost of
living increases or opportunities for raises and bonuses. A starting salary of $100, 000 might sound
like a lot right now (and it is), but if you’re still earning this same salary in 20 years time, you will be
making significantly less money, in fact, if the inflation rate is constant at 3% for these 20 years, you
will only be earning roughly $55, 400 in the future!

Relative Purchasing Power

For an annual inflation rate of r, the relative purchasing power, meaning how much a current
amount of money will be able to purchase at some fixed point in the future, of an amount P after t
years will be
A=

Page 8
Finance Savings; Amortization; Future Value

In the discussion above we supposed that inflation stayed constant over a period of time. This is
not generally the case. However, based on regularity of changes, we can still determine the equiv-
alent today of a price in an earlier year or how much a dollar in that year would be worth today in
purchasing power. The Consumer Price Index (CPI) is the official measure of inflation. Each month,
the Bureau of Labor Statistics (BLS) determines the average cost of several goods, including food,
housing, transportation, clothing, etc, and it compares this cost to the same or comparable goods in
a base period. By doing so, we can compare year-to-year purchasing power proportionally:

cost in year A CPI for year A

= .
cost in year B CPI for year B

What this means is that a CPI of, say, 195.3 in 2005 tells us that inflation has increased 95.3% since
1982 when the base amount was set at 100 (actually it was averaged between 1982 and 1984).

Page 9
Finance Savings; Amortization; Future Value

Example 10: You’ve no doubt heard about how much cheaper housing was when your parents
bought their first home. In mid-1992, the median home price in the US was around $133, 000. Today,
the median home price in Athens is around $340, 000. This is over a $200, 000 difference! However,
this simple calculation neglects purchasing power due to inflation. Taking this into account and us-
ing the fact that the CPI in March 2024 was about 312, how much more expensive is buying a median
home today?

Page 10