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 one-hundred 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 PV0
- 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
semi-annually (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, semi-annually, 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 FVt are future mortgage payments
- r is interest rate (loan rate) and fixed throughout life of the loan.
- PV0 is the bank loan, when you bought the house
All loan payments are the same, so FV = FV1 = FV2 = FV3 = ... = FVt
Incorporate into the equation
- Example
- Mortgage: $60,000
- Interest rate: 12%
- Six-year 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 PV0 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 T-bill 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
T-bill investment
- Conclusion - invests in the T-bill
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 E0, E1, E2, and En
- 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, r1 = 10%
- Year 2, r2 = 5%
- Year 3, r3=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, r1 = 50%
- Year 2, r2 = 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. |
|