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2 The Mathematicsof Finance

Copyright © Cengage Learning. All rights reserved.

2.3 Annuities, Loans, and Bonds

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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 writei = 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

Ignoring for the moment the $5,000 already in the account,

your payments have the following future value:

cont’d

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Example 1 – Solution

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.

cont’d

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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

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.

cont’d

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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

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.

cont’d