2.5 Time Value of Money

The time value of money (TVM) is one of the most fundamental concepts in finance: a dollar today is worth more than a dollar in the future because of its earning potential. Investment advisers use TVM concepts daily when analyzing investments and planning for client goals.

Why a Dollar Today Is Worth More

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Core Concepts

Present Value (PV)

Present value is the current worth of a future sum of money, discounted at a given rate.

Question it answers: "What is $10,000 received in 5 years worth today?"

Future Value (FV)

Future value is the value of a current sum of money at a future date, compounded at a given rate.

Question it answers: "What will $10,000 invested today be worth in 5 years?"

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Key Formulas

Future Value Formula

$$FV = PV \times (1 + r)^n$$

Example: $10,000 invested at 8% for 5 years:

• FV = $10,000 × (1.08)^5 = $10,000 × 1.469 = $14,693

Present Value Formula

$$PV = \frac{FV}{(1 + r)^n}$$

Example: What is $14,693 received in 5 years worth today at 8%?

• PV = $14,693 / (1.08)^5 = $14,693 / 1.469 = $10,000

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The Rule of 72

A quick way to estimate how long it takes to double your money:

$$\text{Years to Double} \approx \frac{72}{\text{Interest Rate}}$$

The Rule of 72 works in reverse too: At what rate do you need to invest to double money in 10 years? 72 / 10 = 7.2%

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Compounding Frequency

More frequent compounding results in higher effective returns:

Key Point: For the same stated (nominal) rate, more frequent compounding = higher effective annual rate (EAR).

Effective Annual Rate (EAR)

The effective annual rate converts any compounding frequency to an annual equivalent:

$$EAR = \left(1 + \frac{r}{m}\right)^m - 1$$

Where m = number of compounding periods per year

Example: 8% compounded monthly:

• EAR = (1 + 0.08/12)^12 - 1 = 8.30%

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Annuities

An annuity is a series of equal payments at regular intervals. Investment advisers encounter annuities in retirement planning, loan analysis, and income products.

Types of Annuities

Key Difference: Annuity due payments occur one period earlier, so an annuity due is worth more than an otherwise identical ordinary annuity.

Annuity Applications

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Discount Rate Selection

The discount rate used in TVM calculations significantly affects results:

Important: Higher discount rates result in lower present values. A risky investment's cash flows should be discounted at a higher rate than a safe investment's cash flows.

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In Practice: How Investment Advisers Apply This

Client planning applications:

• Calculate how much to save monthly for retirement
• Determine if a lump sum or annuity is more valuable
• Compare investment alternatives with different timing
• Evaluate whether early payment of debt makes sense

Investment analysis:

• Calculate present value of expected cash flows
• Compare bonds with different maturities
• Evaluate NPV of investment opportunities

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On the Exam

The Series 65 exam tests your ability to:

1. Understand that a dollar today is worth more than a dollar tomorrow
2. Apply the Rule of 72 to estimate doubling time
3. Recognize that more frequent compounding increases effective returns
4. Distinguish between ordinary annuity and annuity due
5. Understand how discount rate affects present value

Common formats include Rule of 72 calculations and understanding compounding frequency effects.

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Key Takeaways

• Time value of money: A dollar today is worth more than a dollar in the future
• Future value: $FV = PV \times (1 + r)^n$ — grows current money at a given rate
• Present value: $PV = \frac{FV}{(1 + r)^n}$ — discounts future money back to today
• Rule of 72: Years to double = 72 / interest rate
• More frequent compounding = higher effective annual rate
• Ordinary annuity: Payments at end of period; Annuity due: Payments at beginning
• Higher discount rates result in lower present values