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Time Value of Money Models

Time Value of Money Models & Formulas

Complete mathematical framework for time value of money calculations with detailed formulas, derivations, and practical applications

Mathematical ModelsFormula DerivationsPractical Examples

1. Simple Interest Model

Interest calculated only on the principal amount

Model Conditions & Assumptions

  • Interest calculated on original principal only
  • No compounding of interest
  • Constant interest rate throughout the period
  • Common for short-term loans and basic calculations

Variables:

  • • PV: Present Value (Principal amount)
  • • i: Interest rate per period (decimal)
  • • n: Number of periods
  • • FV: Future Value
  • • I: Simple interest amount

Formulas:

I = PV × i × n

FV = PV + I = PV × (1 + i × n)

Example & Solution

Question: You invest $10,000 at 5% simple interest for 3 years. What is the future value?

Given Conditions:

  • • Principal (PV): $10,000
  • • Interest rate (i): 5% or 0.05 per year
  • • Time period (n): 3 years
  • • Simple interest (no compounding)

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Calculate interest amount**: I = PV × i × n = $10,000 × 0.05 × 3 = $1,500 2. **Calculate future value**: FV = PV + I = $10,000 + $1,500 = $11,500 3. **Alternative formula**: FV = PV × (1 + i × n) = $10,000 × (1 + 0.05 × 3) = $10,000 × 1.15 = $11,500 4. **Verify calculation**: - Total interest: $1,500 - Principal returned: $10,000 - Total received: $11,500

Answer: $11,500

Explanation: The investment grows by $1,500 in interest over 3 years, resulting in a total future value of $11,500. Simple interest does not compound, so the growth is linear.

2. Compound Interest Model

Interest calculated on both principal and accumulated interest

Model Conditions & Assumptions

  • Interest compounds at regular intervals
  • Interest earned becomes part of the principal
  • Compounding can be annual, semi-annual, quarterly, etc.
  • Most realistic model for long-term investments

Variables:

  • • PV: Present Value (Principal amount)
  • • i: Interest rate per compounding period (decimal)
  • • n: Number of compounding periods
  • • FV: Future Value
  • • m: Compounding frequency per year

Formulas:

FV = PV × (1 + i)ⁿ

i = r / m (where r is annual rate)

n = t × m (where t is years)

Example & Solution

Question: You invest $5,000 at 8% annual interest compounded annually for 5 years. What is the future value?

Given Conditions:

  • • Principal (PV): $5,000
  • • Annual interest rate (r): 8% or 0.08
  • • Compounding frequency: Annual (m = 1)
  • • Time period (t): 5 years
  • • Total compounding periods (n): 5

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Identify variables**: - PV = $5,000 - r = 0.08 (annual rate) - m = 1 (annual compounding) - t = 5 years - n = t × m = 5 periods - i = r/m = 0.08 (per period rate) 2. **Apply compound interest formula**: FV = PV × (1 + i)ⁿ = $5,000 × (1 + 0.08)⁵ 3. **Calculate (1 + i)ⁿ**: (1.08)⁵ = 1.4693 4. **Calculate future value**: FV = $5,000 × 1.4693 = $7,346.50 5. **Alternative year-by-year calculation**: - Year 1: $5,000 × 1.08 = $5,400.00 - Year 2: $5,400 × 1.08 = $5,832.00 - Year 3: $5,832 × 1.08 = $6,298.56 - Year 4: $6,298.56 × 1.08 = $6,802.44 - Year 5: $6,802.44 × 1.08 = $7,346.64 6. **Total interest earned**: $7,346.64 - $5,000 = $2,346.64

Answer: $7,346.64

Explanation: The investment grows to $7,346.64 after 5 years with compound interest. The total interest of $2,346.64 is significantly higher than the $2,000 that would be earned with simple interest over the same period.

3. Present Value Model

Discounting future cash flows to determine current value

Model Conditions & Assumptions

  • Future cash flow is known with certainty
  • Discount rate reflects risk and time preference
  • Single future payment (not annuity)
  • Time value of money principle applies

Variables:

  • • FV: Future Value (amount to be received)
  • • r: Discount rate per period (decimal)
  • • n: Number of periods until payment
  • • PV: Present Value (current worth)

Formulas:

PV = FV / (1 + r)ⁿ

Example & Solution

Question: You will receive $15,000 in 4 years. If the discount rate is 6%, what is the present value?

Given Conditions:

  • • Future value (FV): $15,000
  • • Discount rate (r): 6% or 0.06 per year
  • • Time period (n): 4 years
  • • Single future payment

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Identify variables**: - FV = $15,000 - r = 0.06 - n = 4 2. **Apply present value formula**: PV = FV / (1 + r)ⁿ 3. **Calculate (1 + r)ⁿ**: (1.06)⁴ = 1.2625 4. **Calculate present value**: PV = $15,000 / 1.2625 = $11,883.02 5. **Alternative calculation**: - Year 1 discount factor: 1 / 1.06 = 0.9434 - Year 2 discount factor: 0.9434 / 1.06 = 0.8899 - Year 3 discount factor: 0.8899 / 1.06 = 0.8396 - Year 4 discount factor: 0.8396 / 1.06 = 0.7921 - PV = $15,000 × 0.7921 = $11,881.50 (slight rounding difference)

Answer: $11,883.02

Explanation: The present value of $15,000 to be received in 4 years at a 6% discount rate is $11,883.02. This means $11,883.02 invested today at 6% would grow to $15,000 in 4 years.

4. Ordinary Annuity Present Value

Present value of a series of equal payments received at the end of each period

Model Conditions & Assumptions

  • Equal payments made at the end of each period
  • Payments continue for a fixed number of periods
  • Constant interest rate throughout
  • Ordinary annuity (payments at end of period)

Variables:

  • • PMT: Payment amount per period
  • • r: Interest rate per period (decimal)
  • • n: Number of periods
  • • PV: Present Value of annuity

Formulas:

PV = PMT × [1 - 1/(1+r)ⁿ] / r

Example & Solution

Question: You will receive $1,000 at the end of each year for 5 years. If the discount rate is 7%, what is the present value?

Given Conditions:

  • • Payment (PMT): $1,000 per year
  • • Discount rate (r): 7% or 0.07 per year
  • • Number of periods (n): 5 years
  • • Ordinary annuity (payments at end of year)

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Identify variables**: - PMT = $1,000 - r = 0.07 - n = 5 2. **Apply annuity PV formula**: PV = PMT × [1 - 1/(1+r)ⁿ] / r 3. **Calculate 1/(1+r)ⁿ**: 1/(1.07)⁵ = 1/1.4026 = 0.7130 4. **Calculate [1 - 1/(1+r)ⁿ]**: 1 - 0.7130 = 0.2870 5. **Calculate PV**: PV = $1,000 × 0.2870 / 0.07 = $1,000 × 4.100 = $4,100 6. **Verify with individual calculations**: - Year 1: $1,000 / 1.07 = $934.58 - Year 2: $1,000 / 1.07² = $873.44 - Year 3: $1,000 / 1.07³ = $816.30 - Year 4: $1,000 / 1.07⁴ = $762.90 - Year 5: $1,000 / 1.07⁵ = $712.99 - Total: $4,100.21 (matches)

Answer: $4,100

Explanation: The present value of $1,000 annual payments for 5 years at 7% discount rate is $4,100. This represents the lump sum needed today to generate the same cash flows.

Time Value of Money Framework

Core Concepts

Time Preference

Money today is worth more than money tomorrow

Opportunity Cost

Money could be invested elsewhere during the waiting period

Risk Consideration

Future cash flows are uncertain and must be discounted

Applications

Investment Analysis

Compare projects with different time horizons

Loan Pricing

Determine fair interest rates for loans

Retirement Planning

Calculate required savings for future needs

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