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# 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
1. 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
2. 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).

• m is compounding
• Example
• You put \$10 in your bank account for 20 years
• Earns 8% interest (APR)
• Compounded monthly

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

• 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
• Invest in the project with the highest NPV
• NPV has to equal or greater than zero
• Example
• 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

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:
• 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

• 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.

• 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.

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