TVM Concepts Explained

.
Time Value of Money
. QTM1300: Quantitative Methods for Business

Dr. Ji Li

Babson College

November - December, Fall 2010

. . . . . .

Simple Interest and Compound Interest
Sinking Funds, Annuities, and Bonds Simple Interest
More on Finance Compound Interest
Notations and Formulas

. Time Value of Money
1. Simple Interest and Compound Interest
Simple Interest
Compound Interest
2. Sinking Funds, Annuities, and Bonds
Sinking Funds
Annuities
Amortization Schedule
More Examples
3. More on Finance
Sinking Funds and Annuities: New Formulas
Perpetuities
Net Present Value
4. Notations and Formulas
Notations
Sinking Funds, Annuities, and Perpetuities
Net Present Value
. . . . . .

Dr. Ji Li Time Value of Money


. Simple Interest
An investment of PV dollars growing with simple interest rate of
r after t years is worth FV dollars:

FV = PV (1 + r t).

. . . . . .



. Simple Interest

FV = PV (1 + r t).

.
Example
.
The Megabucks Corporation is issuing 10-year bonds paying
an annual rate of 6.5%. If you buy $10,000 worth of bonds, how
much interest will you earn every six months? How much
interest will you earn over the life of the bonds?
.

. . . . . .



. Simple Interest

FV = PV (1 + r t).

.
Example
.
A stock fund costs $900 in July 2001 and sells for $892 in July
2002. What is the annual percentage loss of this stock?
.

. . . . . .



. Simple Interest

FV = PV (1 + r t).

.
Example
.
You hear the following on your local radio station’s business news:
The economy last year grew by 1%. This was the second year
in a row in which the economy showed a 1% growth.

Because the rate of growth was the same two years in a row, this represents
a
. simple interest growth, right?
. . . . . .



. Example
Suppose that $20, 000.00 is invested in a bank account. Assume that there is
no other deposits or withdrawals. How much is in the account after 10 years
if
(a) the bank pays 6% simple interest rate once a year?
(b) the bank pays 6% interest rate compounded annually?
(c) the bank pays 6% interest rate compounded monthly?

. . . . . .



. Example
if

.
Answer to (a)
.
The account earns
20, 000 × 0.06 = $1, 200
interest every year. In 10 years, the account becomes

. FV = PV (1 + rt) = 20, 000(1 + 0.06 × 10) = $32, 000.

. . . . . .



. Example
if

.
Answer to (b)
.
In 10 years, the account becomes

. FV = 20, 000 (1 + 0.06)10 = $35, 816.95.

. . . . . .



. Example
if

.
Answer to (c)
.
In 10 years, the account becomes
( )12×10
0.06
FV = 20, 000 1 + = $36, 387.93.
. 12

. . . . . .



. Compound Interest
An investment of PV dollars earning interest at an annual rate
of r compounded (reinvested) m times per year for a period of t
years is worth FV dollars:
( r )mt
FV = PV 1+ .
m

. . . . . .



. Compound Interest
( r )mt
FV = PV 1+ .
m

.
Example
.
Determine the amount of money you must invest at 5% per
year, compounded monthly, so that you will be a millionaire in
30 years.
.

. . . . . .



. Compound Interest
( r )mt
FV = PV 1+ .
m

.
Effective Rate (APY)
.
You deposit $100,000 in an account earning interest of 5%
compounded annually. What is the APY (Annual Percentage
Yield) of your account?
.

. . . . . .



. Compound Interest
( r )mt
FV = PV 1+ .
m

.
Constant Dollars
.
You deposit $100,000 in an account earning interest of 5%
compounded annually. What is the APY (Annual Percentage
Yield) of your account?

Suppose also that inﬂation is running 3% when you make the
deposit. How much money will you have two years from now?
. . . . . . .



. Compound Interest
( r )mt
FV = PV 1+ .
m

.
Bonds Part I
.
How much do you have to pay for a 20-year zero coupon bond
with maturity value of $100,000 and a yield of 5.65% annually?
.

. . . . . .



. Compound Interest
( r )mt
FV = PV 1+ .
m

.
Bonds Part II
.
How much do you have to pay for a 20-year zero coupon bond with maturity
value of $100,000 and a yield of 5.65% annually?
Once purchased, bonds can be sold in the secondary market. How much
money would you have received if you sold your bond 5 years before maturity
to
. an investor looking for a return of 5% annually?

. . . . . .



. Compound Interest
( r )mt
FV = PV 1+ .
m

.
Bonds Part III
.
How much do you have to pay for a 20-year zero coupon bond with maturity
value of $100,000 and a yield of 5.65% annually?
Once purchased, bonds can be sold in the secondary market. How much
money would you have received if you sold your bond 5 years before maturity
to an investor looking for a return of 5% annually?
What is your annual yield on your 15-year investment?
.
. . . . . .


Simple Interest and Compound Interest Sinking Funds
Sinking Funds, Annuities, and Bonds Annuities
More on Finance Amortization Schedule
Notations and Formulas More Examples

Simple Interest
Compound Interest
Sinking Funds
Annuities
More Examples
3. More on Finance
Perpetuities
Net Present Value
Notations
Net Present Value
. . . . . .



. Example

Suppose you make a deposit of $1000 at the end of every
month into an account earning 5% interest per year,
compounded monthly. What will be the value of the investment
at the end of 30 years?

. . . . . .



. Sinking Funds
A sinking fund is worth FV dollars if you make a payment of PMT at
the end of each compounding period into an account earning interest
at an annual rate of r compounded (reinvested) m times per year for
t years:

(1 + r /m)mt − 1
FV = PMT .
r /m

. . . . . .



. Sinking Funds
t years:

(1 + r /m)mt − 1
FV = PMT .
r /m

.
Example
.
At the end of each month you deposit $100 into an account earning
3% annual rate compounded monthly. How much is the account
worth in the end of one year?
.

. . . . . .



. Sinking Funds
t years:

(1 + r /m)mt − 1
FV = PMT .
r /m

.
Retirement Account
.
Your retirement account has $10,000 in it and ears 5% interest per
year compounded monthly. Every month for the next 20 years you will
deposit $500 into the account. How much money will be there at the
end of those 20 years?
.

. . . . . .



. Sinking Funds
t years:

(1 + r /m)mt − 1
FV = PMT .
r /m

.
Example
.
If $2,000 is deposited in an account at the end of each year for the
next 12 years, how much will be in the account at the time of the ﬁnal
deposit if interest is 5% compounded annually?
.

. . . . . .



. Example

Suppose you deposit an amount PV now in an account earning
5% interest per year, compounded monthly. Starting 1 month
from now, the bank will send you monthly payments of $5,000.
What must PV be so that the account will be drawn down to $0
in exactly 10 years?

. . . . . .



. Annuities
An annuity is an account earning compound interest from which
periodic withdrawals are made. If the starting balance is PV dollars,
you receive a payment of PMT at the end of each compounding
period, and the account is down to $0 after for t years, then

1 − (1 + r /m)−mt
PV = PMT .
r /m

. . . . . .



. Annuities

1 − (1 + r /m)−mt
PV = PMT .
r /m

.
Example
.
At the end of each month you want to withdraw $100 from an account
earning 3% annual rate compounded monthly. How much is it worth
right now if you want the account to last for 5 years?
.

. . . . . .



. Annuities

1 − (1 + r /m)−mt
PV = PMT .
r /m

.
Car Loan Part I
.
Mira bought a car worth $30,000 with an initial payment of $6,000.
How much does she have to pay in the end of each month for the
5-year car loan with an interest rate of 4% compounded monthly?
.

. . . . . .



. Annuities

1 − (1 + r /m)−mt
PV = PMT .
r /m

.
Car Loan Part II
.
Mira bought a car worth $30,000 with an initial payment of $6,000 on
a 5-year car loan with an interest rate of 4% compounded monthly.
After making monthly payments over 3 years, she decided to end the
loan earlier. How much does she have to pay in her last payment?
.

. . . . . .



. Amortization Schedule

Mira bought a car worth $30,000 with an initial payment of
$6,000 on a 5-year car loan with an interest rate of 4%
compounded monthly.
How much interest does she have to pay in the end of the
ﬁrst month?
How much outstanding principal is left after Mira makes the
ﬁrst payment?
How much interest does she have to pay in the end of the
second month?
How much interest does Mira have to pay in total?

. . . . . .



. Example

Find the monthly payment on the mortgage if you are buying a
$300,000 apartment with a down payment of $60,000 for 30
years at 9% interest rate compounded monthly.

Find the total amount you will pay in interest.

Produce an amortization schedule for the ﬁrst 12 payments.

. . . . . .



. Bonds

Suppose that a corporation offers a 20-year bond paying coupon interest
rate 4.5% with semiannual coupons. If someone pays $5,000 for bonds with
a maturity value of $5,000, he will receive a coupon every 6 months for 20
years for the interest. At the end of the 20 years, he will get the $5,000 back.
How much is each coupon worth?
Think of the bonds as an investment that will pay the owner a certain
amount every 6 months for 20 years, at the end of which it will pay
$5,000. How much will a bond trader be willing to pay for the bond if
he’s looking for a yield (also called “rate of return”) of 7%?
Another trader is looking for 6% yield on her investment. How much will
she pay for the same bond?

. . . . . .



. Multi-Step Example
You wish to provide yourself with an income of $5,000 every 6
months, starting 15 and a half years from now and ending 35
years from now.

You deposit $25,000 in the account now, and a guaranteed
inheritance of $10,000 which you will receive 10 years from
now.

You know that these sums will not provide the income you want,
so you plan to make periodic deposits to the account at the end
of every 6 months for 15 years to make up the difference.

How much should the periodic deposits be if all interest is
computed at 6% compounded semiannually?
. . . . . .



. The Effect of Inflation

(Turner Example 4, Finance.xlsx) An entrepreneur borrows $10,000 under
the following terms:
Loan interest rate: 12%
Term: 6 years
Payment schedule: Monthly
Determine the cost of the loan in today’s dollars if inflation average 5% over
the term of the loan.

There are two steps involved to solve this problem:
Step 1: Find the monthly payment PMT by ignoring the inflation rate.
Step 2: Find the present value using the PMT found in an annuity
situation.

. . . . . .


Sinking Funds, Annuities, and Bonds
Perpetuities
More on Finance
Net Present Value

Simple Interest
Compound Interest
Sinking Funds
Annuities
More Examples
3. More on Finance
Perpetuities
Net Present Value
Notations
Net Present Value
. . . . . .


Perpetuities
More on Finance
Net Present Value

. Example on Annuities

Sally has just received a loan to finance the purchase of a
pretty blue convertible. The amount of the loan is $25,000.

Sally is required to transfer to the lending institution a fixed
amount at the end of each month starting at the end of the first
month after she receives her loan.

The interest rate on the loan is 9% compounded daily, and the
term of the loan is three years.

What will be Sally’s monthly payment?

. . . . . .


Perpetuities
More on Finance
Net Present Value

. Sinking Funds and Annuities — New Formulas
Suppose that the number of compounding periods per year, m, is different
from the number of payments per year, ppy . Then the interest rate per
payment period becomes
( )m/ppy
r
j = 1+ −1
m

and the sinking fund and annuity formulas become
( )
(1 + j)ppy·t − 1
Sinking Fund FV = PMT ,
j
( )
1 − (1 + j)−ppy ·t
Annuity PV = PMT ,
j

In calculator, input the following:
N = ppy · t
I%=j%
. . . . . .


Perpetuities
More on Finance
Net Present Value

. Examples on New Formulas

( )
(1 + j)ppy·t − 1
j
( )
1 − (1 + j)−ppy ·t
Annuity PV = PMT ,
j

. . . . . .


Perpetuities
More on Finance
Net Present Value


( )
(1 + j)ppy·t − 1
j
( )
1 − (1 + j)−ppy ·t
Annuity PV = PMT ,
j

.
Example
.
Mira bought a car worth $30,000 with an initial payment of
$6,000. How much does she have to pay in the end of each
month for the 5-year car loan with an interest rate of 4%
compounded daily?
.
. . . . . .


Perpetuities
More on Finance
Net Present Value


( )
(1 + j)ppy·t − 1
j
( )
1 − (1 + j)−ppy ·t
Annuity PV = PMT ,
j

.
Example
.
Find the monthly payment on the mortgage if you are buying a
$300,000 apartment with a down payment of $60,000 for 30
years at 9% interest rate compounded daily.
.

. . . . . .


Perpetuities
More on Finance
Net Present Value

. Perpetuities: Example 1

(Turner Example 5)
Determine the initial deposit that must be placed in an account that bears 5%
interest compounded monthly in order to withdraw $1,000 every month for
ever.

Now in the annuity formula

1 − (1 + r /m)−mt
PV = PMT ,
r /m
by setting t −→ ∞, the perpetuity formula follows
1
PV = PMT .
r /m

. . . . . .


Perpetuities
More on Finance
Net Present Value


Determine the initial deposit that must be placed in an account that bears
5.49% interest compounded monthly in order to withdraw $10,000 every 6
months forever, starting a month from now.
( )m/ppy ( )
r 1 − (1 + j)−ppy·t
j= 1+ −1 PV = PMT
m j

Setting t −→ ∞, the perpetuity formula is
1 PMT
PV = PMT = ( )m/ppy
j r
1+ m −1

. . . . . .


Perpetuities
More on Finance
Net Present Value


1 PMT
PV = PMT =( )m/ppy
j
r
1+ m −1

.
Example
.
Determine the initial deposit that must be placed in an account
that bears 2.75% interest compounded daily in order to
withdraw $8,000 quarterly forever, starting 3 months from now.
.

. . . . . .


Perpetuities
More on Finance
Net Present Value

. Issuing a Bond

(Turner Example 6, Finance.xlsx) A company ﬂoats a $10,000,000 bond
issue with a 20 year term. The interest rate on the bond is 3%. How much is
each bond interest payment (to the bond holders) made semiannually?

The company has set up a sinking fund for the accumulation and dispersion
of all funds related to the bond issue and wants to make equal quarterly
payments to the fund. Note that the fund will be used both to pay the bond
interest due each six months and the bond face value of $10,000,000 twenty
years hence. Determine the minimum quarterly payment to the fund that
would meet the needs of the company if the interest on the fund is 8%
compounded quarterly.

. . . . . .


Perpetuities
More on Finance
Net Present Value

. Net Present Value
Assume r is the APR and m is the number of compounding periods per year.
The net present value NPV of a sequence of equally-spaced end-of-period
payments (negative) and income (positive) with same compounding and
payment/income periods, with an initial cash ﬂow v0 and the payment/income
sequence v1 , v2 , . . . , vn is
v1 v2 vn
NPV = v0 + + + ··· + .
1 + r /m (1 + r /m)2 (1 + r /m)n

. . . . . .


Perpetuities
More on Finance
Net Present Value

. Net Present Value
v1 v2 vn
NPV = v0 + + + ··· + .
1 + r /m (1 + r /m)2 (1 + r /m)n

.
Example 1
.
Suppose you are considering an investment in which you pay $10,000 one
year from today and receive an annual income of $3,000, $4,200, and $6,800
at the end of the three years that follow, respectively. Assuming an annual
interest rate of 10%, what is the net present value of this investment?
.

. . . . . .


Perpetuities
More on Finance
Net Present Value

. Net Present Value
v1 v2 vn
NPV = v0 + + + ··· + .
1 + r /m (1 + r /m)2 (1 + r /m)n

.
Example 1: Continue
.
Suppose you are considering an investment in which you pay $10,000 one
year from today and receive an annual income of $3,000, $4,200, and $6,800
at the end of the three years that follow, respectively.
Determine the Internal Rate of Return, or the interest rate per period which
would provide a net present value of $0, after all four years.
.

. . . . . .


Perpetuities
More on Finance
Net Present Value

. NPV: Example 2

You run a takeout pizza business and believe you could improve your return
by buying a van to add delivery service.

You can buy a van today for $15,000 and belive you will use it for ﬁve years
and then sell it for $5,000. After expenses, you estimate that your business
will make $4,000 annually by adding the delivery service, starting a year from
now.

You could make 7.5% by investing in a U.S. Treasure bill over ﬁve years, but
you decide that the added risks of the pizza delivery business mean you
should earn at least twice that rate. So you set the annual interest rate at
15%.

Should you buy the van?

. . . . . .


Simple Interest and Compound Interest Notations
Sinking Funds, Annuities, and Bonds Simple Interest and Compound Interest
More on Finance Sinking Funds, Annuities, and Perpetuities
Notations and Formulas Net Present Value

Simple Interest
Compound Interest
Sinking Funds
Annuities
More Examples
3. More on Finance
Perpetuities
Net Present Value
Notations
Net Present Value
. . . . . .



. Notations
PV — Present Value
FV— Future Value
r — Nominal Rate (also called APR)
t — Number of years
m — Number of compounding periods per year
n = mt — Total number of compounding periods
iper = r /m — Interest rate per compounding period
PMT — The amount of payment
ppy — Number of payments per year in an ordinary annuity
( )m/ppy
r
j = 1+ − 1 — Interest rate per payment period for an annuity
m
NPV — Net Present Value
. . . . . .



. Simple Interest


FV = PV (1 + r t).

. . . . . .



. Compound Interest

( r )mt
FV = PV 1+ .
m

. . . . . .



. Sinking Funds

A sinking fund is worth FV dollars if you make a payment of
PMT at the end of each compounding period into an account
earning interest at an annual rate of r compounded
(reinvested) m times per year for t years:

(1 + r /m)mt − 1
FV = PMT .
r /m

. . . . . .



. Annuities

An annuity is an account earning compound interest from
which periodic withdrawals are made. If the starting balance is
PV dollars, you receive a payment of PMT at the end of each
compounding period, and the account is down to $0 after for t
years, then

1 − (1 + r /m)−mt
PV = PMT .
r /m

. . . . . .



. New Formulars for Sinking Funds and Annuities
Suppose that the number of compounding periods per year, m,
is different from the number of payments per year, ppy . Then
the interest rate per payment period becomes
( )m/ppy
r
j= 1+ −1
m

and the sinking fund and annuity formulas become
( )
(1 + j)ppy·t − 1
j
( )
1 − (1 + j)−ppy ·t
Annuity PV = PMT ,
j
. . . . . .



. Perpetuities

A perpetuity is basically an annuity that lasts for ever.

1
PV = PMT .
r /m

If the compounding periods is not the same as payment
periods, then we use the following formula

1 PMT
PV = PMT =( )m/ppy
j
r
1+ m −1

. . . . . .



. Net Present Value

Assume r is the APR and m is the number of compounding
periods per year. The net present value NPV of a sequence of
equally-spaced end-of-period payments (negative) and income
(positive) with same compounding and payment/income
periods, with an initial cash ﬂow v0 and the payment/income

v1 v2 vn
NPV = v0 + + + ··· + .
1 + r /m (1 + r /m)2 (1 + r /m)n

. . . . . .


TVM Concepts Explained

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