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Lecture 7: Time Value of Money

Introduction to Time Value of Money​

Core Principle: Money today is worth more than the same amount of money in the future.

Why This Matters: In finance, we need to compare cash flows that happen at different times. We cannot simply add $100 today + $100 next year = $200. We must account for the time value of money.

Simple Example:

  • Would you rather have $100 today or $100 in one year?
  • Most people choose $100 today because they can invest it and have more than $100 in one year.

Why Money Has Time Value​

1. Opportunity Cost​

Definition: The cost of giving up the next best alternative.

Example: If you have $100 today, you can:

  • Buy something now
  • Invest it and earn interest
  • Save it for future use

Simple Analogy: Like choosing between eating a cookie now or saving it for later. The cookie now has more value because you can enjoy it immediately.

2. Consumption Preference​

Definition: People prefer to consume goods and services now rather than later.

Example: A child asking for gum:

  • "Give me one gum now, and I'll give you two gums tomorrow"
  • This shows natural preference for current consumption

Real Rate of Return: The compensation for delaying consumption, even without inflation or risk.

3. Inflation Risk​

Definition: The risk that prices will increase over time, reducing purchasing power.

Example:

  • Today: $100 can buy 100 apples at $1 each
  • Next year: $100 can only buy 50 apples at $2 each
  • Inflation rate: 100%

Simple Analogy: Like a shrinking shopping cart - the same amount of money buys fewer items over time.

Components of Required Return​

Required Return=Real Rate+Expected Inflation+Risk Premium\text{Required Return} = \text{Real Rate} + \text{Expected Inflation} + \text{Risk Premium}

1. Real Rate of Return​

Definition: Compensation for delaying consumption, without inflation or risk.

Characteristics:

  • Usually stable over time
  • Varies by country and time period
  • Determined by people's risk preferences

Example: 2-3% real rate is common in developed countries.

2. Expected Inflation​

Definition: The expected increase in prices over the investment period.

Types of Inflation:

  • Supply-side inflation: When production decreases (e.g., bad weather reduces apple harvest)
  • Demand-side inflation: When money supply increases (e.g., government prints more money)

Example: If expected inflation is 3%, you need 3% higher return to maintain purchasing power.

3. Risk Premium​

Definition: Additional return required for taking on risk.

Types of Risk:

  • Credit risk: Risk of default (corporate bonds vs government bonds)
  • Inflation risk premium: Uncertainty about future inflation
  • Market risk: General market fluctuations

Example: Corporate bonds pay higher interest than government bonds because of credit risk.

Simple vs Compound Interest​

Simple Interest​

Definition: Interest calculated only on the principal amount.

Interest=Principal×Rate×Time\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}

Example: $100 at 10% simple interest for 2 years

  • Year 1: 100×10%=10100 \times 10\% = 10 interest
  • Year 2: 100×10%=10100 \times 10\% = 10 interest
  • Total: 100+20=120100 + 20 = 120

Compound Interest​

Definition: Interest calculated on principal plus previously earned interest.

Future Value=Principal×(1+Rate)Time\text{Future Value} = \text{Principal} \times (1 + \text{Rate})^{\text{Time}}

Example: $100 at 10% compound interest for 2 years

  • Year 1: 100×1.10=110100 \times 1.10 = 110
  • Year 2: 110×1.10=121110 \times 1.10 = 121
  • Total: 121121

Key Difference: Compound interest earns "interest on interest"

Present Value and Future Value​

Future Value (FV)​

Definition: The value of money at a future date.

FV=PV×(1+r)t\text{FV} = \text{PV} \times (1 + r)^t

Where:

  • PV = Present Value (money today)
  • r = Interest rate (required return)
  • t = Time period

Example: $100 today at 10% for 2 years

  • FV=100×(1.10)2=100×1.21=121\text{FV} = 100 \times (1.10)^2 = 100 \times 1.21 = 121

Present Value (PV)​

Definition: The value today of money to be received in the future.

PV=FV(1+r)t\text{PV} = \frac{\text{FV}}{(1 + r)^t}

Example: $121 in 2 years at 10% discount rate

  • PV=121(1.10)2=1211.21=100\text{PV} = \frac{121}{(1.10)^2} = \frac{121}{1.21} = 100

Key Insight: Discounting is the reverse of compounding.

Practical Example: Office Lease Decision​

Scenario: Infosoft company needs office space

  • Option 1: Pay $500,000 in 10 years
  • Option 2: Pay some amount today
  • Required return: 10%

Question: What amount today makes you indifferent between the two options?

Solution:

  • PV=500,000(1.10)10\text{PV} = \frac{500,000}{(1.10)^{10}}
  • PV=500,0002.594\text{PV} = \frac{500,000}{2.594}
  • PV=192,772\text{PV} = 192,772

Decision Rule:

  • If asked to pay less than $192,772 today → Choose today's payment
  • If asked to pay more than $192,772 today → Choose future payment

Annuities​

Definition​

Annuity: A series of equal payments made at regular intervals.

Examples:

  • Monthly rent payments
  • Annual insurance premiums
  • Retirement account withdrawals
  • Loan payments

Types of Annuities​

1. Ordinary Annuity: Payments made at the end of each period 2. Annuity Due: Payments made at the beginning of each period

Present Value of Annuity​

PV of Annuity=Payment×1−(1+r)−nr\text{PV of Annuity} = \text{Payment} \times \frac{1 - (1 + r)^{-n}}{r}

Where:

  • Payment = Regular payment amount
  • r = Interest rate per period
  • n = Number of periods

Practical Example: Copier Purchase Decision​

Scenario: Infosoft needs a copier

  • Option 1: Pay $10,000 cash today
  • Option 2: Pay $3,000 per year for 5 years
  • Required return: 12%

Question: Which option is cheaper?

Solution - Option 2 (Annuity):

  • Payment = $3,000
  • r=12%=0.12r = 12\% = 0.12
  • n=5n = 5 years

Step 1: Calculate (1+r)−n(1 + r)^{-n}

  • (1.12)−5=0.567(1.12)^{-5} = 0.567

Step 2: Calculate 1−(1+r)−n1 - (1 + r)^{-n}

  • 1−0.567=0.4331 - 0.567 = 0.433

Step 3: Divide by rr

  • 0.4330.12=3.605\frac{0.433}{0.12} = 3.605

Step 4: Multiply by payment

  • 3,000×3.605=10,8153,000 \times 3.605 = 10,815

Decision: Choose Option 1 ($10,000) because it's cheaper than Option 2 ($10,815).

Practical Examples​

Example 1: Retirement Planning​

Scenario: 25-year-old planning for retirement

  • Current investment: $100
  • Time horizon: 40 years
  • Investment options:
    • Stocks: 12.4% average return
    • Government bonds: 5.3% average return
    • Cash/T-bills: 3.8% average return

Future Value Calculations:

  • Stocks: 100×(1.124)40=10,000+100 \times (1.124)^{40} = 10{,}000+
  • Bonds: 100×(1.053)40=750100 \times (1.053)^{40} = 750
  • Cash: 100×(1.038)40=400100 \times (1.038)^{40} = 400

Key Insight: The power of compound interest over long periods is enormous.

Example 2: Loan vs Lease Decision​

Scenario: Business equipment financing

  • Equipment cost: $50,000
  • Option 1: Bank loan at 8% for 5 years
  • Option 2: Lease at $12,000 per year for 5 years

Analysis:

  • Loan: Calculate monthly payments using annuity formula
  • Lease: Calculate present value of lease payments
  • Decision: Choose option with lower present value

Example 3: Investment Comparison​

Scenario: Comparing two investment opportunities

  • Investment A: $1,000 today, $1,500 in 3 years
  • Investment B: $1,000 today, $200 per year for 8 years
  • Required return: 10%

Analysis:

  • Investment A: PV=1,500(1.10)3=1,127\text{PV} = \frac{1,500}{(1.10)^3} = 1,127
  • Investment B: PV=200×annuity factor=1,067\text{PV} = 200 \times \text{annuity factor} = 1,067
  • Decision: Choose Investment A (higher present value)

Key Takeaways​

  1. Time Value of Money: Money today is worth more than the same amount in the future.

  2. Three Components of Required Return:

    • Real rate (compensation for waiting)
    • Expected inflation (purchasing power protection)
    • Risk premium (compensation for uncertainty)

  3. Compound Interest: Interest on interest creates exponential growth over time.

  4. Present Value: The value today of future cash flows, calculated by discounting.

  5. Future Value: The value in the future of money invested today, calculated by compounding.

  6. Annuities: Regular payments that can be valued using present value formulas.

  7. Decision Making: Always compare alternatives using present value analysis.

  8. Practical Applications: Time value of money is used in retirement planning, loan decisions, and investment analysis.

  9. Key Formulas:

    • Future Value: FV=PV×(1+r)t\text{FV} = \text{PV} \times (1 + r)^t
    • Present Value: PV=FV(1+r)t\text{PV} = \frac{\text{FV}}{(1 + r)^t}
    • Annuity PV: PV=Payment×1−(1+r)−nr\text{PV} = \text{Payment} \times \frac{1 - (1 + r)^{-n}}{r}

  10. Real-World Impact: Understanding time value of money helps make better financial decisions in both personal and business contexts.