


Lesson 4  Time Value of Money

This lecture teaches financial analysts the time value of money. The lecture starts with the present value of a single future payment and evolves into the present value of multiple future withdrawals and payments. Finally, an amortization table is constructed for fixed payment mortgage.

Single Investment

1.
The Present Value Formula places a value of future cash flows in terms of money
today.�
 Emphasizes the present
 For example, I deposit $100 into a bank at 5%
interest rate.
 After one year, I earn 0.05($100) = $5 in
interest.� My balance
is $105.00
 After two years, I earn 0.05($105.00) = $5.25.� My balance is
$110.25
If let the money earn interest after n years, then I can build the sequence
In onehundred years, $100 grows into $13,150.13 at 5% interest.
 Notation
 FV is Future Value
 PV is Present Value
 n or t refer to the time
 i is the discount rate or interest rate
 subscripts refer to time
2. Present value  rearrange equation and solve for PV_{0}
 One hundred years is very far away.�
 I rather have the money today.�
 The present value of $13,150.13 in one hundred
years is worth $100 to today.�
 I can take that $100
today, invest it in a savings account with 5% interest, and let it grow to
$13,150.13
If I receive a payment in the future, then the
present value is:
3. Solving for the interest rate
 Example
 You have $10,000 to invest
 You want to earn $15,000
 You want your money in 4 years
 What is the minimum interest rate you need to earn?
 You need to earn at least 10.66% interest to meet your goals
4. The Rule of 72  an easy way to determine how long it takes something to double in size
The Equation
 Examples
 Bank Account
 If your bank deposit is earning 4% per year, divide 72 by 4, and your bank account will double in 18 years
 If your bank deposit is earning 7% per year, divide 72 by 7, and your bank account will double in 10.3 years
 If you plan to put money into a savings account for 5 years, divide 72 by 5, and your interest needs to be 14.4% to double
 Economic Growth
 China's economy is growing 10% per year; divide 72 by 10, which means China's economy will double every 7.2 years
 U.S.A. is growing 1% per year, divide 70 by 1 and the U.S. economy will double in 70 years

Multiple Investments

1. Let’s change the analysis, so we receive
multiple future payments.
 Every year, I invest $500 into the bank account at
6% interest.
 After the first year, I earn $500(0.06) = �$30�
My balance is $500 + $30 + $500 = $1,030
 After the second year, I earn $1,030( 0.06 ) = $61.80.� My balance is $1,030 + $61.80 + $500 = $1,591.80
 After the third year, I
earn $1,591.80(0.06) = $95.508.� My
balance is $1,591.80 + $95.508 + $500 = �$2,187.308
 If I wrote an equation
 How much is it worth to me today, if I receive $500 today, $500 in one year, $500 in
two years, and $500 in three years?
 If I received $1,836.51 today, I can invest in a
savings account and earn $2,187.31 in three years. (Rounding error)
2. Uneven withdrawals and investments
 This formula is flexible.�
 I can withdraw or invest any amounts.
 If the interest rate is 14% and investment time is four years
 For example
Year 
Activity 
Amount 
Interest + Balance 
0 
Deposit 
$100 
$148.15

1

Deposit 
$300

$389.88

2

Withdrew 
$50

$57

2

Deposit 
$100

$114.00

3

Withdrew 
$75

$75



Total 
$520.03 
 How much are these cash flows worth to me today if
interest rate is 14%?
 If I invest $351.01 today at 14% interest, then in
3 years, I will have $520.04.

Compounding Frequency

1. Interest rates are defined as Annual Percentage
Rate (APR)
 For example, is 1% a good interest rate for a borrower?
 There is no time units.�
 If 1% is annual,
then it is a good rate.�
 If it is daily,
then the rate is terrible.�
 The borrower
borrowed money from a loan shark.
 All interest rates are defined in annual terms,
unless otherwise stated.
 Usually loan payments and bank interests are calculated
monthly.�
 Loans could be
semiannually (two payments per year)
 Loans coudl be quarterly (four payments per year).
Adjust the equation as follows
 m is compounding
 Example
 You put $10 in your bank account for 20 years
 Earns 8% interest (APR)
 Compounded monthly
 Your savings grow into:
Note: if this was compounded annually, then it would be $46.61
2. Effective Annual Rate (EFF)  the equivalent interest rate if the compounding were once a year.
 Example: Convert the interest rate 8% APR compounded monthly into an interest that is compounded annnually
 Example: If you put $10 in your bank account for 20 years that earn 8.3% APR compounding yearly, then your savings grow into:
3. Present and future value problems with multiple cash flows can be compounded monthly, semiannually, or quarterly.
 Example
 What is the present value if I receive $50 within a month, $100 within six
months, $75 in exactly one year and one month, and the interest rate is 10%.
 The smallest time unit is the month, so we need to adjust the time units for
interest percent and time.�
 Time is monthly
 Interest rate is 0.1 divided by 12
4. Continuous compounding
Special case
If m approaches infinity, then the compouding equation turns into
 The number e is a constant and equals approximately 2.1828.
 The number e is similar to pi and has not pattern in the number.
 Example
 You deposit $50 into your bank and forget about it for 70 years.
 The bank used continuous compounding, then your savings grow into:
 If you used monthly compounding, then your savings would be $9,373.90

Annuities and Mortgages

1. Annuity  an investment for people planning for retiring
 Annuities are two types
 Ordinary Annuity  Payment is due at the end of period
 Annuity Due  Payment is due at the beginning of the period
 We will stick to ordinary annuities
 Example  you are investing into an annuity
 Interest rate is 9% APR that is compounded annually
 You are investing in $20,000 per year for five years
 What is the Future Value of your annuity?
 Did you notice the exponents?
 A formula exist that calculates the future value of annuities
 Example
 You plan to retire and you want $40,000 ordinary annuity
 You will pay 4 annual payments at the end of the period
 The interest rate is 3%
 What are your annual payments?
All future payments are the same, so
2. Calculate a home mortgage. Start with the formula.
 All FV_{t} are future mortgage payments
 r is interest rate (loan rate) and fixed throughout life of the loan.
 PV_{0} is the bank loan, when you bought the house
All loan payments are the same, so FV = FV_{1} = FV_{2} = FV_{3} = ... = FV_{t}
Incorporate into the equation
 Example
 Mortgage: $60,000
 Interest rate: 12%
 Sixyear loan
 Paid yearly
 Solving for FV, your payment yearly payment is $14,594.
 Build an Amortization Table. This table show the breakdown of interest and principal paid for each payment.
 Example
 At the end of Year 1, you have $60,000
outstanding.
 Your interest is 12% multiplied by $60,000, which is $7,200.
 Your payment
is
$14,594, so interest is $7,200, the remainder reduces the loan
balance.
 Year 2, and beyond, repeat the sequence.

Payment 
Interest 
Principal Paid 
Loan Balance 
Year 0 
 
 
 
$60,000 
Year 1 
$14,594 
$7,200 
$7,394 
$52,606 
Year 2

$14,594 
$6,313

$8,281 
$44,325 
Year 3 
$14,594 
$5,319 
$9,275 
$35,050 
Year 4 
$14,594 
$4,206 
$10,388 
$24,662 
Year 5 
$14,594 
$2,959 
$11,635 
$13,027 
Year 6 
$14,594 
$1,563 
$13,027 
$0 
If you pay the mortgage monthly, divide interest rate by 12 and multiply the number of years by 12.
A 20 year mortgage will have 240 payments. I have a program that calculates the amortization table for long time periods.
The amortization table can also handle balloon payments and variable interest rate mortgages. 
Comparing Different Investments

1. Net Present Value (NPV)  Calculate the net present value
 Change the present value equation into the form
 You paid out PV_{0} for the investment
 Any FV's will be negative if it is a payout
 The return to the project is r
 Invest in the project with the highest NPV
 NPV has to equal or greater than zero
 Example
 Your brother wants you to invest $10,000 into his business
 He will promise you $12,000 in two years
 The projected rate of return is 10%
 You buy a Tbill for $9,000
 A year later, the U.S. gov. will pay back $10,000
 The projected rate of return is 4%
 Calculate the NPV's for both situations
Brother's business
Tbill investment
 Conclusion  invests in the Tbill
2. Yield to Maturity  set the Net Present Value to zero and solve for the return
 Also called Internal Rate of Return (IRR)
 Use numerical techniques
 I have a program that uses two techniques
 Grid Search  try various r values until NPV equals zero
 Find the Root  an algorithm that finds roots to the equation
 Example:
 Your brother wants you to invest $10,000 into his business
 He will promise you $12,000 in two years
 You can invest in a CD that pays 3% APR
 If you trust your brother, then you earn a higher rate of return.
3. Foreign Currencies  more difficult to deal with
 Exchange rates are continuously changing
 We assume we know the exchange rate at every point in time
 Convert exchange rate to home currency
 Exchange rates are E_{0}, E_{1}, E_{2}, and E_{n}
 Example
 You invest 10,000 euros into Greece
 You expect to earn 12,000 euros in two years
 Projected rate of return is 5%
 Exchange rate at time 0 is $1.5 per 1 euro
 Projected exchange rate in two years is $1.6 per 1 euro
 The NPV is
 Instead we had the Greek Financial Crisis
 In Year 2, the exchange rate decreases to $1 per 1 euro
 The NPV is:
 You were harmed by the depreciating Euro
4. Inflation  we can compute a real net present value and a nominal present value.
 Both results in the same number, because the inflation term falls out of the equation.
 Start with real FV and the real interest rate
 Convert the cash flow into nominal by substituting the Fisher Equation into the equation:
 Note  FV's are still in real, but are converted into nominal by multiplying it by expected inflation

Compounding Different Rates of Return

1. What do you do, if over the life of a project, you have different rates of return?
 Example 1
 Year 1, r_{1} = 10%
 Year 2, r_{2} = 5%
 Year 3, r_{3}=20%
 Calculating the rate of return is using the geometric average
 r bar is the average rate of return in annualized return
 Example 2
 Year 1, r_{1} = 50%
 Year 2, r_{2} = 75%
 Calculating the rate of return is using the geometric average
 r bar is the average rate of return in annualized return
2. What is the present value of the following cash flows from a project with different returns for each year?
 You invest $2,000 today
 Year 1, you receive $1,500 with a 10% interest rate
 Year 2, you receive $1,600 with a 12% interest rate
 Year 3, you receive $500 with a 5% interest rate
 The net present value is:

You should only invest in a project, if the net present value is greater than zero. 


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