Different debt instruments have very different streams of payment with very different
timing. Thus we first need to understand how we can compare the value of one kind
of debt instrument with another before we see how interest rates are measured. To do
this, we make use of the concept of present value.
The concept of present value (or present discounted value) is based on the commonsense
notion that a dollar paid to you one year from now is less valuable to you than
a dollar paid to you today: This notion is true because you can deposit a dollar in asavings account that earns interest and have more than a dollar in one year.
Economists use a more formal definition, as explained in this section.
Let’s look at the simplest kind of debt instrument, which we will call a simple
loan. In this loan, the lender provides the borrower with an amount of funds (called
the principal) that must be repaid to the lender at the maturity date, along with an
additional payment for the interest. For example, if you made your friend, Jane, a simple
loan of $100 for one year, you would require her to repay the principal of $100
in one year’s time along with an additional payment for interest; say, $10. In the case
of a simple loan like this one, the interest payment divided by the amount of the loan
is a natural and sensible way to measure the interest rate. This measure of the socalled
simple interest rate, i, is:
If you make this $100 loan, at the end of the year you would have $110, which
can be rewritten as:
$100 (1 0.10) $110
If you then lent out the $110, at the end of the second year you would have:
$110 (1 0.10) $121
or, equivalently,
$100 (1 0.10) (1 0.10) $100 (1 0.10)2 $121
Continuing with the loan again, you would have at the end of the third year:
$121 (1 0.10) $100 (1 0.10)3 $133
Generalizing, we can see that at the end of n years, your $100 would turn into:
$100 (1 i )n
The amounts you would have at the end of each year by making the $100 loan today
can be seen in the following timeline:
This timeline immediately tells you that you are just as happy having $100 today
as having $110 a year from now (of course, as long as you are sure that Jane will pay
you back). Or that you are just as happy having $100 today as having $121 two years
from now, or $133 three years from now or $100 (1 0.10)n, n years from now.
The timeline tells us that we can also work backward from future amounts to the present:
for example, $133 $100 (1 0.10)3 three years from now is worth $100
today, so that:
The process of calculating today’s value of dollars received in the future, as we have
done above, is called discounting the future. We can generalize this process by writing
$100
$133
(1 0.10)3
$100 (1 0.10)n
Year
n
Today
0
$100 $110
Year
1
$121
Year
2
$133
Year
3
i
$10
$100
0.10 10%
62 PA RT I I Financial Markets
today’s (present) value of $100 as PV, the future value of $133 as FV, and replacing
0.10 (the 10% interest rate) by i. This leads to the following formula:
timing. Thus we first need to understand how we can compare the value of one kind
of debt instrument with another before we see how interest rates are measured. To do
this, we make use of the concept of present value.
The concept of present value (or present discounted value) is based on the commonsense
notion that a dollar paid to you one year from now is less valuable to you than
a dollar paid to you today: This notion is true because you can deposit a dollar in asavings account that earns interest and have more than a dollar in one year.
Economists use a more formal definition, as explained in this section.
Let’s look at the simplest kind of debt instrument, which we will call a simple
loan. In this loan, the lender provides the borrower with an amount of funds (called
the principal) that must be repaid to the lender at the maturity date, along with an
additional payment for the interest. For example, if you made your friend, Jane, a simple
loan of $100 for one year, you would require her to repay the principal of $100
in one year’s time along with an additional payment for interest; say, $10. In the case
of a simple loan like this one, the interest payment divided by the amount of the loan
is a natural and sensible way to measure the interest rate. This measure of the socalled
simple interest rate, i, is:
If you make this $100 loan, at the end of the year you would have $110, which
can be rewritten as:
$100 (1 0.10) $110
If you then lent out the $110, at the end of the second year you would have:
$110 (1 0.10) $121
or, equivalently,
$100 (1 0.10) (1 0.10) $100 (1 0.10)2 $121
Continuing with the loan again, you would have at the end of the third year:
$121 (1 0.10) $100 (1 0.10)3 $133
Generalizing, we can see that at the end of n years, your $100 would turn into:
$100 (1 i )n
The amounts you would have at the end of each year by making the $100 loan today
can be seen in the following timeline:
This timeline immediately tells you that you are just as happy having $100 today
as having $110 a year from now (of course, as long as you are sure that Jane will pay
you back). Or that you are just as happy having $100 today as having $121 two years
from now, or $133 three years from now or $100 (1 0.10)n, n years from now.
The timeline tells us that we can also work backward from future amounts to the present:
for example, $133 $100 (1 0.10)3 three years from now is worth $100
today, so that:
The process of calculating today’s value of dollars received in the future, as we have
done above, is called discounting the future. We can generalize this process by writing
$100
$133
(1 0.10)3
$100 (1 0.10)n
Year
n
Today
0
$100 $110
Year
1
$121
Year
2
$133
Year
3
i
$10
$100
0.10 10%
62 PA RT I I Financial Markets
today’s (present) value of $100 as PV, the future value of $133 as FV, and replacing
0.10 (the 10% interest rate) by i. This leads to the following formula:
No comments:
Post a Comment