Transcript Slide 1

2
The Mathematics
of Finance
Copyright © Cengage Learning. All rights reserved.
2.3
Annuities, Loans, and Bonds
Copyright © Cengage Learning. All rights reserved.
Annuities, Loans, and Bonds
A typical defined-contribution pension fund works as
follows: Every month while you work, you and your
employer deposit a certain amount of money in an account.
This money earns (compound) interest from the time it is
deposited. When you retire, the account continues to earn
interest, but you may then start withdrawing money at a rate
calculated to reduce the account to zero after some number
of years.
This account is an example of an annuity, an account
earning interest into which you make periodic deposits or
from which you make periodic withdrawals.
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Annuities, Loans, and Bonds
In common usage, the term “annuity” is used for an
account from which you make withdrawals.
There are various terms used for accounts into which you
make payments, based on their purpose. Examples include
savings account, pension fund, and sinking fund.
A sinking fund is generally used by businesses or
governments to accumulate money to pay off an
anticipated debt, but we’ll use the term to refer to any
account into which you make periodic payments.
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Sinking Funds
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Sinking Funds
Suppose you make a payment of $100 at the end of every
month into an account earning 3.6% interest per year,
compounded monthly. This means that your investment is
earning 3.6%/12 = 0.3% per month. We write
i = 0.036/12 = 0.003. What will be the value of the
investment at the end of 2 years (24 months)?
Think of the deposits separately. Each earns interest from
the time it is deposited, and the total accumulated after
2 years is the sum of these deposits and the interest they
earn.
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Sinking Funds
In other words, the accumulated value is the sum of the
future values of the deposits, taking into account how long
each deposit sits in the account.
Figure 1 shows a timeline with the deposits and the
contribution of each to the final value.
Figure 1
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Sinking Funds
For example, the very last deposit (at the end of month 24)
has no time to earn interest, so it contributes only $100.
The very first deposit, which earns interest for 23 months, by
the future value formula for compound interest contributes
$100(1 + 0.003)23 to the total.
Adding together all of the future values gives us the total
future value:
FV = 100 + 100(1 + 0.003) + 100(1 + 0.003)2 + ··· + 100(1 + 0.003)23
= 100[1 + (1 + 0.003) + (1 + 0.003)2 + ··· +(1 + 0.003)23]
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Sinking Funds
Fortunately, this sort of sum is well-known and there is a
convenient formula for its value:
In our case, with x = 1 + 0.003, this formula allows us to
calculate the future value:
It is now easy to generalize this calculation.
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Sinking Funds
Future Value of a Sinking Fund
A sinking fund is an account earning compound interest
into which you make periodic deposits. Suppose that the
account has an annual rate of r compounded m times per
year, so that i = r/m is the interest rate per compounding
period.
If you make a payment of PMT at the end of each period,
then the future value after t years, or n = mt periods, will be
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Sinking Funds
Quick Example
At the end of each month you deposit $50 into an account
earning 2% annual interest compounded monthly. To find
the future value after 5 years, we use i = 0.02/12 and
n = 12  5 = 60 compounding periods, so
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Example 1 – Retirement Account
Your retirement account has $5,000 in it and earns
5% interest per year compounded monthly. Every month
for the next 10 years you will deposit $100 into the account.
How much money will there be in the account at the end of
those 10 years?
Solution:
This is a sinking fund with PMT = $100, r = 0.05, m = 12, so
i = 0.05/12, and n = 12  10 = 120.
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Example 1 – Solution
cont’d
Ignoring for the moment the $5,000 already in the account,
your payments have the following future value:
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Example 1 – Solution
cont’d
What about the $5,000 that was already in the account?
That sits there and earns interest, so we need to find its
future value as well, using the compound interest formula:
FV = PV(1 + i)n
= 5,000(1 + 0.05/12)120
= $8,235.05.
Hence, the total amount in the account at the end of
10 years will be
$15,528.23 + 8,235.05 = $23,763.28.
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Sinking Funds
Payment Formula for a Sinking Fund
Suppose that an account has an annual rate of r
compounded m times per year, so that i = r/m is the interest
rate per compounding period.
If you want to accumulate a total of FV in the account after
t years, or n = mt periods, by making payments of PMT at
the end of each period, then each payment must be
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Annuities
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Annuities
Suppose we deposit an amount PV now in an account
earning 3.6% interest per year, compounded monthly.
Starting 1 month from now, the bank will send us monthly
payments of $100. What must PV be so that the account
will be drawn down to $0 in exactly 2 years?
As before, we write i = r/m = 0.036/12 = 0.003, and we
have PMT = 100. The first payment of $100 will be made
1 month from now, so its present value is
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Annuities
In other words, that much of the original PV goes toward
funding the first payment. The second payment, 2 months
from now, has a present value of
That much of the original PV funds the second payment.
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Annuities
This continues for 2 years, at which point we receive the
last payment, which has a present value of
and that exhausts the account.
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Annuities
Figure 2 shows a timeline with the payments and the
present value of each.
Figure 2
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Annuities
Because PV must be the sum of these present values, we
get
PV = 100(1 + 0.003)–1 + 100(1 + 0.003)–2 + · · · + 100(1 + 0.003)–24
= 100[(1 + 0.003)–1 + (1 + 0.003)–2 + · · · + (1 + 0.003)–24].
We can again find a simpler formula for this sum:
x –1 + x –2 + · · · + x –n =
(x n – 1 + x n – 2 + · · · + 1)
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Annuities
So, in our case,
or
If we deposit $2,312.29 initially and the bank sends us
$100 per month for 2 years, our account will be exhausted
at the end of that time.
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Annuities
Generalizing, we get the following formula:
Present Value of an Annuity
An annuity is an account earning compound interest from
which periodic withdrawals are made.
Suppose that the account has an annual rate of r
compounded m times per year, so that i = r/m is the interest
rate per compounding period.
Suppose also that the account starts with a balance of PV.
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Annuities
If you receive a payment of PMT at the end of each
compounding period, and the account is down to $0 after
t years, or n = mt periods, then
Quick Example
At the end of each month you want to withdraw $50 from
an account earning 2% annual interest compounded
monthly.
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Annuities
If you want the account to last for 5 years (60 compounding
periods), it must have the following amount to begin with:
Note
If you make your withdrawals at the end of each
compounding period, you have an ordinary annuity.
If, instead, you make withdrawals at the beginning of each
compounding period, you have an annuity due.
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Annuities
Because each payment occurs one period earlier, there is
one less period in which to earn interest, hence the present
value must be larger by a factor of (1 + i) to fund each
payment.
So, the present value formula for an annuity due is
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Annuities
Payment Formula for an Ordinary Annuity
Suppose that an account has an annual rate of r
compounded m times per year, so that i = r/m is the interest
rate per compounding period.
Suppose also that the account starts with a balance of PV.
If you want to receive a payment of PMT at the end of each
compounding period, and the account is down to $0 after
t years, or n = mt periods, then
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Installment Loans
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Installment Loans
In a typical installment loan, such as a car loan or a home
mortgage, we borrow an amount of money and then pay it
back with interest by making fixed payments (usually every
month) over some number of years.
From the point of view of the lender, this is an annuity.
Thus, loan calculations are identical to annuity calculations.
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Example 6 – Home Mortgages
Marc and Mira are buying a house, and have taken out a
30-year, $90,000 mortgage at 8% interest per year. What
will their monthly payments be?
Solution:
From the bank’s point of view, a mortgage is an annuity. In
this case, the present value is PV = $90,000, r = 0.08,
m = 12, and n = 12  30 = 360.
To find the payments, we use the payment formula:
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Example 6 – Solution
cont’d
The word “mortgage” comes from the French for “dead
pledge.” The process of paying off a loan is called
amortizing the loan, meaning to kill the debt owed.
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Bonds
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Bonds
Suppose that a corporation offers a 10-year bond paying
6.5% with payments every 6 months.
If we pay $10,000 for bonds with a maturity value of
$10,000, we will receive 6.5/2 = 3.25% of $10,000, or $325,
every 6 months for 10 years, at the end of which time the
corporation will give us the original $10,000 back.
But bonds are rarely sold at their maturity value. Rather,
they are auctioned off and sold at a price the bond market
determines they are worth.
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Bonds
For example, suppose that bond traders are looking for an
investment that has a rate of return or yield of 7% rather
than the stated 6.5% (sometimes called the coupon
interest rate to distinguish it from the rate of return).
How much would they be willing to pay for the bonds above
with a maturity value of $10,000? Think of the bonds as an
investment that will pay the owner $325 every 6 months for
10 years, and will pay an additional $10,000 on maturity at
the end of the 10 years..
We can treat the $325 payments as if they come from an
annuity and determine how much an investor would pay for
such an annuity if it earned 7% compounded semiannually.
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Bonds
Separately, we determine the present value of an
investment worth $10,000 ten years from now, if it earned
7% compounded semiannually.
For the first calculation, we use the annuity present value
formula, with i = 0.07/2 and n = 2  10 = 20.
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Bonds
For the second calculation, we use the present value
formula for compound interest:
PV = 10,000(1 + 0.07/2)–20
= $5,025.66.
Thus, an investor looking for a 7% return will be willing to
pay $4,619.03 for the semiannual payments of $325 and
$5,025.66 for the $10,000 payment at the end of 10 years,
for a total of $4,619.03 + 5,025.66 = $9,644.69 for the
$10,000 bond.
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Example 8 – Bonds
Suppose that bond traders are looking for only a 6% yield
on their investment. How much would they pay per $10,000
for the 10-year bonds above, which have a coupon interest
rate of 6.5% and pay interest every six months?
Solution:
We redo the calculation with r = 0.06. For the annuity
calculation we now get
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Example 8 – Solution
cont’d
For the compound interest calculation we get
PV = 10,000(1 + 0.06/2)–20
= $5,536.76.
Thus, traders would be willing to pay a total of
$4,835.18 + $5,536.76 = $10,371.94 for bonds with a
maturity value of $10,000.
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