Present Value Notes
Present Value is an important concept for everyone to understand. It relates to the
fact that money today is worth more than money tomorrow. If we have the money today,
we can spend it or lend it out at interest. If we do not have it today, we can do neither.
IN order to use money today, we must forego using it in the future. IN order to have
money in the future, we must forgo the use of this same money (less any accumulated
interest that this money can earn).
The key formula is
Future Value = Present Value * (1 + interest rate) * (1 + interest rate) * …
Where
Future Value is the value of the money in the future
Present Value is the value of this same money in the present
Interest rate is the rate of discount or interest related to this money
The number of terms of (1 + interest rate) depends on the number of
periods (usually years) that the present value and the future value are separated by.
Example:
What is the future value of $100 in two (2) years when the interest rate is 5%?
(This same problem could be stated what would $100 today be worth in two years when
the interest rate is 5% per year.)
FV = PV (1+int)((1+int) = 100*1.05*1.05 = $110.25
This future value consists of the original $100 which we get back (hence the 1 in 1+int)
and interest. The interest consists of $5 for the first year plus $5 in the second year on the
principle (original investment) plus $0.25 interest on the interest earned during the first
year.
Year
0
1
2
Value
$100
105
110.25
Interest on principle Interest on interest
$5
0
5
0.25
5
0.5125
etc.
Example 2:
Now what if you want to find out what the present value of $100 paid in the third year
would be today if the interest rate were 10$? This would be of value if you wanted to
know how much to save today to have $100 for a purchase in the third year. It would
also would be of value when determining what to pay someone who has lost the
opportunity to earn $100 three years hence in order to compensate them for the loss. This
is a very common question in law as it is how we compensate victims of accidents
because we are paying them today for the loss of earning power that would be used in the
future. It is also valuable in determining whether to invest in a project that has future
returns (such as going to college to increase your earning power).
FV = PV (1+int) (1+int) (1+int)
PV =
FV
_
(1+int) (1+int) (1+int)
PV =
100
(1.10) (1.10) (1.10)
_
=
100/1.331
=
$ 75.13
What determines the interest rate to be used? One possible way to determine it is to ask
what interest rate could be gotten on savings or “risk free” investments. Why “risk
Free?” This is because risk is greater for money to be received in the future than for
money in your hand. If there is no risk, the return necessary to convince someone to lend
money is lower. IBM gets a lower interest rate than you do because they are less likely
not to pay the money back. House mortgages have lower interest rates than do credit
card debt because the bank can take the house if you do not pay the mortgage
(“collateral” for the loan). If you do not pay your credit card debt, the bank must sue
you, which is harder to do and costs them money.
Lets look at some implications of this type of analysis.
1. Do the rich or the poor get lower interest rates on their borrowing? The rich get lower
interest rates because they are less likely to not pay the money back. Thus, the poor are
less likely to go to college. They have a greater need for money today to finance the
college tuition and they would have a higher interest rate. Thus the present value of their
college degree would be lower due to the higher opportunity cost of money.
2. Should you plan for your retirement today or just before your retirement? TODAY! If
you put $100 away when you are age 50 at 5% interest, this money will be worth
$207.89. If you put that same $100 away at 5% at age 20, it will be worth $898.50!
Waiting until age 25 to put it away decreases this total to $703.99! This is because the
last few years are the most valuable, with the interest rate being applied to all the
accumulated interest! If the interest rate were higher, these numbers would be much
higher due to the compounding effect.
Compounding is the effect of the interest rate being successively applied to the principle
AND the accumulated interest. In the 41st year of the example above the interest is
applied to the 100 initial investment AND the 603.99 accumulated interest. The longer
the investment is held, the more interest is being added to. In this example, the
accumulated interest is six times the initial investment. Interest on interest in the 41st
year is 6 times as great as the interest on the initial investment.
RULE of 72:
Compound interest is so powerful that the initial investment will double over time
(potentially again and again). The rule of 72 says that if you divide 72 by the interest
rate, this will tell you the number of years it will take for the money to double.
Example:
$100 at 6% will double in 12 years (72/6=12) (actually = $201.22)
$100 at 8% will double in 9 years (72/8 = 9)(actually = 199.90)
The stock market over a long period of time has averaged 10% rate of return. His means
that on average, if you leave your money in safe investments your money would double
each 7.2 years.
Sample problems:
1. What would $144 paid two years hence be worth today at an interest rate of 20%?
2. What would $1,000 be worth in three years at 10%?
Notice the wording. The first is a present value problem. The second is a future value
problem.
LEVERAGE:
Why borrow? Because the return one expects exceeds the cost of the funds you are
borrowing. If I borrow at 6% and earn 10%, I make 4% on this money (and it is someone
else’s money!).
Example:
Buy a house, borrow 90% of the purchase price of $100,000 (that is borrow $90,000 and
invest $10,000 of your own money) at 8%, sell the house one year later at a 10% profit
(for $110,000). What is the profit? What is the rate of profit on your investment
($10,000)?
Total Price
Borrowed
Interest
Investment
Net
Purchase
100,000
90,000
10,000
0
Sale
110,000
90,000
7,200
10,000
2,800
Rate of (net) return on the total investment 2,800/100,000 = 2.8%
This is calculated as 110,000 revenue, 97,200 payback to the lender, 10,000 return
of principle, leaving 2,800.
Rate of (net) return on your investment (of 10,000) 2,800/10,000 = 28%
The 7,000 is the return to the lender of interest on their money. The 90,000 is
return of their principle. The remaining 12,800 comes to you. 10,000 of this is return of
your principle. The remaining 2,800 is return (interest) on your investment. Since you
only put 10,000 of your own money up, all 2,800 applies only to 10,000 initial
investment leading to 28% return. This 2,800 consists of 2% on 90,000 (10%-8%) or
$1,800 plus 10% on 10,000 or $1,000 for a total gain of $2,800.
Change the above to a slight loss in the sale price of the house:
Total Price
Borrowed
Interest
Investment
Net
Purchase
100,000
90,000
10,000
0
Sale
99,000
90,000
7,200
10,000
- 8,200
Net return on total investment –8,200/100,000 = -8.2%
Net return on your money is -8,200/10,000 = -82%!
Leverage can work positively or negatively.
The -8,200 consists of -9% on 90,000 (-1%-8%) or 8,100 plus –1% on 10,000 or –100 for
a total loss of $8,200.