LECTURE 2:
TIME VALUE OF MONEY
MSc. Ngo Nguyen Bao Tran
Interest: Money paid (earned)
for the use of money.
In this lecture, we will study how to deal with uncertainty (risk).
Focusing on:
•
the time value of money
•
the ways in which the rate of interest can be used to adjust the
value of cash flows to a single point in time.
Learning
Outcomes
T H E I N T E R E S T R AT E
SIMPLE INTEREST
COMPOUND INTEREST
• Single Amounts
• Annuities
• Mixed Flows
COMPOUNDING MORE THAN ONCE A YEAR
• Semiannual and Other Compounding Periods
• Continuous Compounding
• Effective Annual Interest Rate
The Interest Rate
1. Interest (I)
Money paid (earned) for the use of money.
There are 02 types of interest:
•
Simple interest
•
Compound interest
2. Principal (P)
The amount of money borrowed or invested
The Interest Rate
3. Rate of interest (i)
the percentage on the principal that the borrower pays the lender per
time period as compensation for forgoing other investment or
consumption opportunities.
4. Future Value (FV)
The value at some future time of a present amount of money
5. Present Value (PV)
The current value of a future amount of money
Simple Interest
Simple Interest: Interest paid (earned) on only the original amount, or
principal, borrowed (lent).
SI = P0 * (i ) * (n)
where:
SI = simple interest in dollars
P0 = principal, or original amount borrowed (lent) at time period 0
i = interest rate per time period
n = number of time periods
Simple Interest
Simple Interest Exercises:
1. What is the simple interest on $100 at 10 percent per annum for
six months?
Simple Interest
Simple Interest Exercises:
2. If Isaiah Williams bought a house and borrowed $30,000 at a 10
percent annual simple interest rate, what would be his first
month’s interest payment?
Simple Interest
Simple Interest Exercises:
3. Mary Schiller receives $30 every three months from a bank account that
pays a 6% annual simple interest rate. How much is invested in the
account?
Simple Interest
Simple Interest Exercises:
4. Raymond Gomez borrows $1,000 for 9 months at a simple rate of 8
percent per annum. How much will he have to repay at the end of the 9month period?
Simple Interest
Simple Interest Exercises:
5. Marie Como agrees to invest $1,000 in a venture that promises to pay 10
percent simple interest each year for two years. How much money will she
have at the end of the second year?
Compound Interest
Compound Interest: is interest that is paid not only on the principal but
also on any interest earned but not withdrawn during earlier periods.
FVn = P0 * (1+i )n
where:
FV = Future Value
P0 = principal, or original amount borrowed (lent) at time period 0
i = interest rate per time period
n = number of time periods
Compound Interest
Compound Interest (cont):
FVn = P0 * (1+i )n
or: FVn = P0(FVIFi,n)
where:
FVIFi,n= the future value interest factor at i% for n periods, equal (1 + i)n
Compound Interest
Compound Interest Exercises:
1. Jay Ritter has put $500 in a saving account at the BIDV Bank. The
account earns 7 percent, compounded annually. How much will Mr. Ritter
have at the end of three years?
Compound Interest
Compound Interest Exercises:
2. In 2015 John Jacob Astor bought approximately an acre of land on the
east side of Manhattan Island for $58,000. How much would his
descendants have in 2021, if instead of buying the land, Astor had invested
the $58,000 at 7% compound annual interest?
Compound Interest
Compound Interest Exercises:
2. Fernando Zapetero, who recently won $10,000 in the lottery, wants to
buy a car in five years; therefore, he deposited the money in his account in
ABC Bank. Fernando estimates that the car will cost $16,105 at the time.
What compound interest rate must he earn to be able to afford the car?
Compound Interest
Compound Interest Exercises:
3. If you invest $1,000 today, you will receive $3,000 in exactly 8 years.
What is the compound interest (or discount) rate implicit in this situation
Compound Interest
Compound Interest Exercises:
4. How long would it take for an investment of $9,000 to grow to $15,000 if
we invested it at a compound annual interest rate of 8%?
Compound Interest
Annuities: is the payment or receipt of equal cash flows per period
for a specified amount of time
Ordinary annuity: is one in which the payments or receipts occur at
the end of each period
Annuity due: is one in which payments or receipts occur at the
beginning of each period
Compound Interest
Annuities (cont)
Compound Interest
Ordinary Annuities Exercises:
1. Ms. Jefferson receives a 3-year ordinary annuity of $1,000 per
year and deposits the money in a savings account at the end of
each year. The account earns interest at a rate of 6 percent
compounded annually. How much will her account be worth at
the end of the 3-year period?
Compound Interest
Ordinal Annuities Formula
FVAi,n = PMT x
where:
𝟏+𝒊 𝒏 −𝟏
𝒊
FVAi,n = Future Value of the Annuity
PMT = equal payment
i = interest rate per time period
n = number of time periods
Compound Interest
Ordinary Annuities Formula
PVAi,n = PMT x
where:
𝟏
𝟏−
𝟏+𝒊 𝒏
𝒊
PVAi,n = Present Value of the Annuity
PMT = equal payment
i = interest rate per time period
n = number of time periods
Compound Interest
Ordinary Annuities Exercises:
2. Suppose you have a house for rent which gives you an income
of $1500 at the end of each year in 5 years. The account earns
interest at a rate of 8% compounded annually. How much will
your total income from your rental house be worth at the end of
the 5-year period?
Compound Interest
Ordinary Annuities Exercises:
3. You have the savings account with the withdrawals cash flows
of $1,000 at the end of each year in 4 years, earning 8% compound
annual interest. How much money would you have to place in the
account right now (time period 0) such that you would end up
with a zero balance after the last $1,000 withdrawal?
Compound Interest
Ordinary Annuities Exercises:
4. Suppose that you need to have at least $10,500 at the end of 8
years in order to send your parents on a luxury cruise. To
accumulate this sum, you have decided to deposit $1,000 at the
end of each of the next 8 years in a bank savings account. If the
bank compounds interest annually, what minimum compound
annual interest rate must the bank offer for your savings plan to
work?
Compound Interest
Ordinary Annuities - Perpetuity: An ordinary annuity whose payments
or receipts continue forever.
PVA∞ = PMT /i
where:
PVAN∞ = Present Value of the Annuity
PMT= equal payment
i = interest rate per time period
Compound Interest
Annuities Due Formula
FVADi,n = PMT x
𝟏+𝒊 𝒏 −𝟏
x (1+i)
𝒊
= FVAi,n x (1+i)
where:
FVADi,n = Future Value of the Annuity Due
PMT = equal payment
i = interest rate per time period
n = number of time periods
Compound Interest
Annuities Due Formula
PVADi,n = PMT x
𝟏
𝟏−
𝟏+𝒊 𝒏
𝒊
+PMT
= PVAi,n + PMT
where:
PVADi,n = Present Value of the Annuity Due
PMT = equal payment
i = interest rate per time period
n = number of time periods
Compound Interest
Annuities Due Exercises:
1. Find the present value of an ordinary $1,000 annuity received at
the end of each year for five years discounted at a 6 percent rate,
Compound Interest
Annuities Due Exercises:
2. Ms. Jefferson deposits $1,000 in a savings account at the
beginning of each year for the next three years and the account
earns 6 percent interest, compounded annually, how much will be
in the account at the end of three years?
Compound Interest
Annuities Due Exercises:
3. Calculation the future value of a $1,000 annuity due for four
years at 8 percent (FVAD3). Notice that the cash flows for the
annuity due are perceived to occur at the beginning of periods 2,
3, and 4.
Compound Interest
Annuities Exercises:
Solving for FVA, PVA, PV of Annuity and Perpetuity:
• What is future value and present value of 3-Year Ordinary
Annuity of $5000 at 10%?
• What is future value and present value of 3-Year Annuity Due of
$5000 at 10%?
• What is present value of perpetuity of $5000 at 10%?
Compound Interest
Mix Flow Formula
Compound Interest
Mix Flow Exercises:
1. What is the present value of $5,000 to be received annually at
the end of years 1 and 2, followed by $6,000 annually at the end of
years 3 and 4, and concluding with a final payment of $1,000 at the
end of year 5, all discounted at 5 percent?
Compound Interest
Mix Flow Exercises:
2. What is the present value of $100,000 to be received annually at
the end of years 1 and 5, followed by $15,000 annually at the end
of years 2, the payment of $50,000 at the end of year 3, the
payment of $50,000 at the end of year 4, all discounted at 10
percent?