Time Value of Money
 The sample problems are not in order
of topic or difficulty.
Part 1: Example 1 - PV
 If you invest $15,000 for ten years, you receive $30,000.
What is your annual return?
Example 2 - The Multi-Period Case
 Assume that the average college tuition
costs $20,000 dollars per annum (paid at
the end of the year). For a freshman just
starting college, what is the present value
of the cost of a four year degree when the
interest rate is 10%?
 If the tuition is paid at the beginning
of the year, what is the PV?
Ex. 3 - Computing Payments with
 Suppose you want to buy a new computer system and the
store is willing to sell it to allow you to make monthly
payments. The entire computer system costs $3500. The loan
period is for 2 years and the interest rate is 16.9% (APR) with
monthly compounding. What is your monthly payment? An
annuity problem...
 Monthly rate =
 Number of months =
 3500 =
Ex. 4 - Future Values with Monthly Compounding
 Suppose you deposit $50 a month
into an account that has an APR of
9%, based on monthly compounding.
How much will you have in the
account in 35 years?
Monthly rate =
Number of months =
FV =
Ex. 5 - Present Value with Daily
 You need $15,000 in 3 years for a new car.
If you can deposit money into an account
that pays an effective annual rate of 5.63%
based on daily compounding, how much
would you need to deposit?
 Daily rate =
 Number of days =
 FV =
Example 6 - Pure Discount Loans
 A T-bill promises to repay
$10,000 in 6
months. If the bill sells for $9,600 in the
market, what is the annual rate of return
(simple and compound)?
 HPR =
 Simple annual return=
 Compound annual return=
Example 7 - Investment
 Your broker calls you and tells you that he
has this great investment opportunity. If you
invest $100 today, you will receive two cash
flows: $40 next year and $75 in two years. If
you require a 15% return on investments of
this risk, should you take the investment?
Ex. 8 - Saving For Retirement
 You are offered the opportunity to put some
money away for retirement now. You will
receive ten annual payments of $25,000 each
beginning in 30 years. How much would you
be willing to invest today if your money can
earn an interest rate of 12% per annum over
the years.
Ex. 8 - Saving For Retirement Timeline
0 1 2
? 0 0 …
25K 25K 25K
Notice that the year 0 cash flow = CF0 = ?
The cash flows years 1 – 29 are 0
The cash flows years 30 – 39 are 25,000
25K 25K
Example 9 - Buying a House
 Fixed rate mortgage loans are basically annuities.
They promise a specified stream of cash payments
to a lender.
 Suppose, you want to take out a 30-year mortgage
loan for 500,000$ at an interest rate of 7.5% per
 A) What would be your monthly payment?
 B) How much would you owe to the bank after third
monthly payment? Show the amortization schedule
for the first three months.
 C) How much would you owe to the bank after the
60th payment?
Definitions – EAR – Birleşik Faiz
 Effective Annual Rate (EAR)
This is the amount of interest you
would earn in one year assuming
that you rollover the loan and
reinvest all interest payments as
often as is allowed by the terms of
the loan, that is, the loan is
compounded as often as possible
during the year.
Definitions – APR – Basit Faiz
 Annual Percentage Rate (APR)
 This is the amount of interest you
would earn in one year assuming
that you rollover the loan but do
not reinvest any interest payment
paid during the year, that is, the
loan is not compounded.
Ex. 10 - Computing EARs
Suppose you can earn 1% per month on $1 invested today.
 What is the APR?
 How much are you effectively earning?
Suppose you put it in another account, you earn 3% per
 What is the APR?
 How much are you effectively earning?
Which account would you prefer? Monthly or quarterly
Ex. 11 – EAR’s
 You are looking at two savings accounts. One
pays 5.25%, with daily compounding. The other
pays 5.3% with semiannual compounding. Which
account should you use?
 First account:
 EAR =
 Second account:
 EAR = ...................... =
 Which account?
Ex. 11 – EAR’s
 Let’s verify the choice.
Suppose you invest $100
in each account. How much will you have in each
account in one year?
 First Account:
 Second Account:
Example 11/A – EAR’s
 Your money is promised a return of 100% over 15
years. Is this a lot or a little?
 How does this compare to 1% over 3 months?
Example 12 - APR
 Suppose you want to earn an effective rate of
12% and you are looking at an account that
compounds on a monthly basis. What APR
must the account pay?
Example 14 - APR
 If a 3 month bond has a 8% APR,
how much interest will I earn over the
life of the bond?
Example 15 - EAR
 If a 3 month bond has a 8% EAR, how much
interest will I earn over the life of the bond?
 Since the EAR quote does include
interest on interest and since a 3 month
bond can be reinvested 4 times during the
Why is this rate lower?
Ex. 16 - Finding the Number of Payments
 Suppose you borrow $2000 at 5% and you
are going to make annual payments of
$734.42. How long before you pay off the
Ex. 17 - Amortized Loan with Fixed Principal2-22
 Consider a $50,000, 10 year loan at
8% interest. The loan agreement
requires the firm to pay $5,000 in
principal each year plus interest for
that year.
 Construct the amortization schedule
on an excel file.
Ex. 18 - Amortized Loan with Fixed Payment
 Each payment covers the interest expense plus
reduces principal
 Consider a 4 year loan with annual payments. The
interest rate is 8% and the principal amount is
 What is the annual payment?
 ..................................................
 PMT =
 Construct
the amortization table
Example 19 - Perpetuity
 You made your fortune in the dot-com
boom (and got out in time!). As part of
your legacy, you want to endow an annual
MBA graduation party at your alma matter.
You want it to be a memorable, so you
budget $30,000 per year for the party. If
the university earns 8% per year on its
investments, and if the first party is in one
year’s time, how much will you need to
donate to endow the party?
Example 19 - Cont’d
 Suppose instead the first party was
scheduled to be held 2 years from today.
How would this change the amount of the
donation required?
Non-standard TVM Problems
 Sometimes the problem we face is not
stated as a typical time value of money
problem and so does not exactly fit
any formula.
 In these cases, we often have to “back
into” a solution.
Example 24– Deceptive Ads (1)
Motown Autos (MA) Advertisement:
American Classic Cars! Finance Special!
Sprite Conversion! Now Only $15,000!
Just $1,000 Down, 0% interest, and 3years to pay with easy monthly payments!
What is the true interest rate in this
deal? You can use the additional info
Classic Car News has an almost identical
car advertised for $9,000, but it needs
$3,000 of work to match the condition of
the car offered by MA.
Example 25
 You have $30,000 in student loans that call for
monthly payments over 10 years.
 $15,000 is financed at seven percent APR
 $8,000 is financed at eight percent APR and
 $7,000 at 15 percent APR
 What is the interest rate on your portfolio of debt?
Hint: don’t even think about doing this:
= 15,000 × 7% + 8,000 × 8% + 7,000 × 15%
Example 27
 You are considering the purchase of a prepaid
tuition plan for your 8-year old daughter. She will
start college in exactly 10 years, with the first
tuition payment of $12,500 due at the start of the
year. Sophomore year tuition will be $15,000;
junior year tuition $18,000, and senior year tuition
$22,000. How much money will you have to pay
today to fully fund her tuition expenses? The
discount rate is 14%
Example 28
You are thinking of buying a new car. You
bought your current car exactly 3 years ago
for $25,000 and financed it at 7% APR for 60
months. You need to estimate how much you
owe on the loan to make sure that you can
pay it off when you sell the old car.
Example 29
You have just landed a job and are going to start
saving for a down-payment on a house. You
want to save 20 percent of the purchase price
and then borrow the rest from a bank.
You have an investment that pays 10 percent APR.
Houses that you like and can afford currently
cost $100,000. Real estate has been
appreciating in price at 5 percent per year and
you expect this trend to continue.
How much should you save every month in order to
have a down payment saved five years from
Part 2: Some Problems for practice:
 Q1)
 On January 1 you deposit $100 in an account
that pays a nominal interest rate of
11.33463%, with daily compounding (365
 How much will you have on October 1, or
after 9 months (273 days)? (Days given.)
Q2) What’s the value at the end of Year 3 of
the following CF stream if the quoted
interest rate is 10%, compounded
Q3) You are offered a note which pays
$1,000 in 15 months (or 456 days) for
You have $850 in a bank which pays a
6.76649% nominal rate, with 365 daily
compounding, which is a daily rate of
0.018538% and an EAR of 7.0%.
You plan to leave the money in the bank
if you don’t buy the note.
The note is riskless. Should you buy it?
Q4) Future Values
 Suppose you just started a new job at a
current annual salary of $25,000. Average
expected inflation rate is 4% for the next 40
years. If you receive annual cost-of-living
raises tied to the inflation rate, what would
be your ending salary?
Also today’s $20,000 car will cost $96,020
under the same assumptions.
Q5) Future Values
 Suppose you had a relative deposit $10 at
5.5% interest 200 years ago. How much
would the investment be worth today?
 What is the effect of compounding?
Q6) Present Values
 You want to begin saving for your
daughter’s college education and you
estimate that she will need $150,000 in
17 years. If you feel confident that you
can earn 8% per year, how much do
you need to invest today?
Q7) Discount Rate
 Suppose you are offered an investment
that will allow you to double your
money in 6 years. You have $10,000 to
invest. What is the implied rate of
Q8) Number of Periods
 Suppose you want to buy a new house.
You currently have $15000 and you figure
you need to have a 10% down payment plus
an additional 5% of the loan amount for
closing costs.
 Assume the type of house you want will
cost about $150,000 and you can earn 7.5%
per year, how long will it be before you have
enough money for the down payment and
closing costs?