Discounted Cash Flow and Time Value of Money

Understanding DCF is crucial in finance for decisions related to:
- Personal finance
- Financial markets
- Real estate
- Various business disciplines
Focus: Making decisions based on recurring payment streams.
Core Elements of Time Value of Money with Annuities:
- N (Number of periods)
- I (Interest rate)
- Present Value
- Payment
- Future Value

Tools for retirement savings.
Calculating Effective Annual Rate (EAR).
Understanding perpetuities (cash flow streams that go on forever).
Required Tools:
- Spreadsheet software like Microsoft Excel.
- Calculator (if required by instructor).

Scenario: Winning a $400,000,000$ lottery.
Choice: Receiving the total amount over 20 years at $20,000,000$ per year or taking a lump sum today.
Using time value of money to decide which option is more beneficial.
Inputs for Present Value Calculation:
- N = 20 years
- I = 5% (discount rate)
- Payment = $20,000,000$ per year
- Future Value = 0
Formula: Present Value = Payment / (1 + Discount Rate)^Number of Periods
- Using a 5% discount rate, the present value of the $20,000,000$ payment stream over 20 years is about $249,000,000$ .
The actual lump sum value is approximately $232,000,000$ .
Investment Analogy: The lottery is like any other investment, requiring participation to win.
"You miss 100% of the shots you don't take."

Inputs:
- N = 20 (years)
- I = 5 (interest rate)
- PMT = $20,000,000$ (payment)
- FV = 0 (future value)
Compute Present Value (PV) = - $249,000,000$ .

Average Student Loan:
- Amount = $30,000$
- Interest Rate = 6.5%
- Repayment Period = 10 years
Calculating Monthly Payment:
- N = 10 years * 12 months/year = 120 months
- I = 6.5% / 12 (monthly interest rate)
Excel Function: PMT(rate, nper, pv, [fv], [type])
- Rate = 6.5% / 12
- Nper = 10 * 12
- PV = - $30,000$
- FV = 0
Monthly payment = 340.64

Scenario: Saving for a pony that costs 8,000.
Saving 300 $per month with a 4% interest rate.</li>\n<li>Inputs:<ul>\n<li>I = 4% (interest rate)</li>\n<li>PV = 0 (present value)</li>\n<li>PMT =$ 300 (payment)
FV = -8,000 (future value)

Excel Function: NPER(rate, pmt, pv, [fv], [type])

N = 25.6 months

Excel Formula: RATE(nper, pmt, pv, [fv], [type], [guess])
- nper = 40
- pmt = 28000
- pv = -120000
- guess = 0.25 (25%)
Return on investment in education: 23.3%
Calculator:
- N = 40
- I = ?
- PV = -120000
- PMT = 28000
- FV = 0
Compute I = 23.3%

Scenario 1: Saving from age 45 to 65 (20 years).
- Saving 2,500 $per year with a 7% return.</li>\n<li>N = 20 years</li>\n<li>I = 7%</li>\n<li>PV = 0</li>\n<li>PMT = 2500</li></ul></li>\n<li>Excel Function: FV(rate, nper, pmt, pv, [type])<ul>\n<li>FV =$ 102,000
Scenario 2: Saving from age 25 to 65 (40 years).
- N = 40 years
- I = 7%
- PV = 0
- PMT = 2500
- FV = 500,000
Conclusion: Starting to save earlier significantly increases the retirement nest egg.

Definition: An investment that pays a constant cash flow forever, such as stocks or real estate.
Formula for Present Value of a Perpetuity: $PV = Payment / r$
Example: Receiving $100$ per year at an 8% yield.
- PV = 100 / 0.08 = 1,250

APR: The nominal annual interest rate without considering compounding.
EAR: The actual compounded annual interest rate.
The formula to solve for EAR is: EAR = (1 + \frac{APR}{m})^m - 1
Example: Earning 1% a month.
- APR = 1% * 12 = 12%
- EAR = (1 + 0.12/12)^12 - 1 = 12.68%
Credit Card Example: 18% APR compounded monthly.
- EAR = (1 + 0.18/12)^12 - 1 = 19.56%

Payday Loan: 20 $fee on a$ 200 $loan for two weeks (10% every two weeks).</li>\n<li>APR = 10% * 26 (bi-weekly periods) = 260%</li>\n<li>EAR = (1 + 2.60/26)^26 - 1 = 10.92 (1092%)</li>\n</ul>\n<h3 id="homeownershipmortgageratesandaffordability">Homeownership: Mortgage Rates and Affordability</h3>\n<ul>\n<li>Scenario: Buying a home with a$ 1,500 monthly mortgage payment.
Original Mortgage Rate: 4%
Standard mortgage term: 30 years
Calculation: Present Value (loan amount) at 4%
- N = 30 * 12 = 360 months
- I = 4% / 12
- PMT = 1500
- FV = 0
Present Value = $315,191.86

Scenario: Selling the home after 5 years when mortgage rates have risen to 8.5%.
New buyer's affordability with 1,500 monthly payment:
- N = 30 * 12
- I = 8.5% / 12
- PMT = 1500
- FV = 0
New Present Value = 195,080.47
Scenario: What happens if rates rise to 18%?
- New buyer's affordability with $1,500$ monthly payment:
- N = 30 * 12
- I = 18% / 12
- PMT = 1500
- FV = 0
- New Present Value = $$99,529.86
Consequence: Higher interest rates significantly reduce affordability and the value of a home.