🩻 Hardcore Finance: Lesson 2 – How to Value the Future in Today's Terms [FREE]

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Welcome back to Hardcore Finance, where we delve deep into the foundational principles of finance.

In today's lesson, we're tackling a concept that sits at the heart of all financial decision-making: how to value future cash flows in today's terms.

This practical skill will empower you to make smarter investment choices, evaluate business opportunities, and understand the financial world more clearly.

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You can find the free introductory lesson here:

🩻 Hardcore Finance: Lesson 1 – An Introduction

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The Parable of the Two Envelopes: A Lesson in Value and Information

Imagine you're presented with two sealed envelopes. You're told that one contains a check for a certain amount, and the other holds a promissory note for a different amount, payable in one year. You have to bid on these envelopes without knowing their exact contents.

Envelope A sold for $75 at an auction, while Envelope B went for $120. Later, it was revealed that Envelope A contained a check for $100, and Envelope B held a promissory note promising $150 in one year.

Why did Envelope B fetch a higher price despite the uncertainty of future payment? The bidders assigned value based on the face value and considering the time delay and potential risks. This scenario illustrates the power of information (or lack thereof) in determining value and highlights the need to evaluate future cash flows in present terms.

Redefining Assets: Beyond Physical Possessions

In finance, an asset isn't just something you can hold in your hand. It's any resource that is expected to provide future economic benefits. Let's consider some unconventional assets:

• A Viral Social Media Account: Generates advertising revenue and brand partnerships.

• An Exclusive License Agreement: Grants rights to sell a popular product in a specific region.

• An Online Course Platform: Offers educational content that subscribers pay for over time.

• Reputation and Brand Loyalty: Encourages repeat business and customer retention.

What unifies these assets? They all promise a sequence of future cash flows.

The Fundamental Definition of an Asset

An asset, at any given point in time, is defined by its expected future cash flows:

Asset at time:

\(t = \{ \text{CF}_{t}, \text{CF}_{t+1}, \text{CF}_{t+2}, \dots \}\)

Where:

• CFt represents the cash flow at time t.

This definition shifts our focus from an asset's physical characteristics to its economic essence—the cash flows it generates over time.

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The Time Line: Visualizing Cash Flows Over Time

To accurately value an asset, we must understand when its cash flows occur. This is where the timeline becomes an essential tool.

Constructing a Time Line

Suppose you're evaluating an investment with the following cash flows:

• Initial Investment (Time 0): −$5,000

• Year 1 Cash Flow: $2,000

• Year 2 Cash Flow: $2,500

• Year 3 Cash Flow: $3,000

The timeline helps you:

• Identify Cash Flow Timing: Know exactly when money moves in or out.

• Avoid Confusion: Prevent mixing up cash flows from different periods.

• Facilitate Calculations: Serve as a reference for discounting cash flows.

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The Incompatibility of Cash Flows at Different Times

Attempting to add cash flows from different periods without adjustment is like adding kilograms to kilometers—it doesn't make sense.

The Currency-Time Analogy

Consider this: You have $100 and €100. Can you say you have $200? No, because dollars and euros are different currencies. Similarly, cash flows at different times are denominated in different "time currencies.”

To make future cash flows comparable, we must convert them into today's dollars. This process is known as discounting.

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Discounting: Converting Future Cash Flows to Present Value

Discounting adjusts future cash flows to reflect their value today, accounting for the time value of money.

The Time Value of Money (TVM)

The TVM principle states that a dollar today is worth more than a dollar tomorrow because:

• Opportunity Cost: Money today can be invested to earn returns.

• Inflation: Erodes purchasing power over time.

• Risk: Future payments carry uncertainty.

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Calculating Present Value Using Discount Rates

The discount rate is our "exchange rate" between future and present dollars.

Determining the Discount Rate

The discount rate (r) reflects:

• Risk-Free Rate: Typically based on government bond yields.

• Risk Premium: Additional return required for taking on risk.

• Inflation Expectations: The anticipated rise in prices over time.

• Opportunity Cost of Capital: The return foregone by investing in this asset instead of the next best alternative.

Present Value Formula

To calculate the present value (PV) of a future cash flow (CF) occurring in n periods:

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

Where:

• r is the discount rate (expressed as a decimal).

• n is the number of periods until the cash flow occurs.

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Applying Present Value to Our Investment Example

Let's return to our investment with cash flows over three years and assume a discount rate of 6%.

Calculating Present Values

Initial Investment (Time 0):

\(PV_0 = -\$5,000 \quad \text{(No discounting needed)}\)

Year 1 Cash Flow:

\(PV_1 = \frac{\$2,000}{(1 + 0.06)^1} = \$1,886.79\)

Year 2 Cash Flow:

\(PV_2 = \frac{\$2,500}{(1 + 0.06)^2} = \$2,228.20\)

Year 3 Cash Flow:

\(PV_3 = \frac{\$3,000}{(1 + 0.06)^3} = \$2,518.98\)

Calculating Net Present Value (NPV)

\(NPV = PV_0 + PV_1 + PV_2 + PV_3\)

\(NPV=−$5,000+$1,886.79+$2,228.20+$2,518.98=$1,633.97\)

Since the NPV is positive, the investment is expected to generate value above the required return of 6%.

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The Concept of Exchange Rates Between Time Periods

As currency exchange rates convert one currency to another, discount factors convert future cash flows to present value.

Discount Factors as Exchange Rates

The discount factor for a period n is:

\(\text{Discount Factor}_n = \frac{1}{(1 + r)^n}\)

This factor tells us the present value of $1 received in n periods.

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Moving Money Through Time: Compounding and Discounting

Understanding how money grows or shrinks over time is essential.

Future Value (FV) with Compounding

To find out how much an investment today will be worth in the future:

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

Example: Investing $1,000 at 5% interest for 3 years.

\(FV = \$1,000 \times (1 + 0.05)^3 = \$1,157.63\)

Present Value (PV) with Discounting

To find out how much a future amount is worth today:

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

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The Power of Discounting: An Everyday Example

Suppose you're offered $10,000 to be received in 5 years. The bank offers a 4% annual interest rate. What's the present value of that $10,000?

Calculating Present Value

\(PV = \frac{\$10,000}{(1 + 0.04)^5} = \frac{\$10,000}{1.21665} = \$8,208.50\)

This means that, given a 4% discount rate, receiving $10,000 in 5 years is equivalent to having $8,208.50 today.

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Annuities and Perpetuities: Special Cases of Cash Flows

Annuity: Equal Cash Flows for a Fixed Period

An annuity is a series of equal payments made regularly for specific periods.

Present Value of an Annuity Formula:

\(PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)\)

Where:

• PMT = Payment per period

• r = Discount rate

• n = Number of periods

Example: Calculating the PV of receiving $500 annually for 10 years at a 5% discount rate.

\(PV = \$500 \times \left( \frac{1 - (1 + 0.05)^{-10}}{0.05} \right) = \$3,852.38\)

Perpetuity: Equal Cash Flows Forever

A perpetuity is an infinite series of equal payments.

Present Value of a Perpetuity Formula:

\(PV = \frac{PMT}{r}\)

Example: Calculating the PV of receiving $1,000 annually forever at a 5% discount rate.

\(PV = \frac{\$1,000}{0.05} = \$20,000\)

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Real-World Application: Mortgage Payments

Understanding annuities is crucial for calculating loan payments.

Calculating Monthly Mortgage Payments

Suppose you're taking out a $300,000 mortgage at a 4% annual interest rate, to be repaid over 30 years with monthly payments.

First, convert the annual rate to a monthly rate:

\(r_{monthly} = \frac{0.04}{12} = 0.003333...\)

Total number of payments:

\(n = 30 \times 12 = 360\)

Monthly payment (PMT) calculation:

\(PMT = PV \times \left( \frac{r_{monthly}}{1 - (1 + r_{monthly})^{-n}} \right)\)

\(PMT = \$300,000 \times \left( \frac{0.003333...}{1 - (1 + 0.003333...)^{-360}} \right) \approx \$1,432.25\)

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The Role of Financial Markets in Valuation

Financial markets provide the exchange rates (discount rates) needed to value cash flows.

Sourcing Discount Rates from the Market

• Government Bonds: Provide risk-free rates for various time horizons.

• Corporate Bonds: Reflect default risk premiums.

• Stock Market Returns: Indicate expected returns for equity investments.

These market rates help us align our valuations with prevailing economic conditions and investor expectations.

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Inflation and Real vs. Nominal Rates

Inflation affects the purchasing power of future cash flows.

Nominal vs. Real Interest Rates

• Nominal Rate (r nominal): The stated interest rate is not adjusted for inflation.

• Real Rate (r real): Adjusted for inflation, reflecting true purchasing power.

Fisher Equation:

\(1 + r_{nominal} = (1 + r_{real})(1 + i)\)

Where:

• i = Inflation rate

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Understanding the Opportunity Cost of Capital

The opportunity cost of capital is the return foregone by investing in a project instead of the next best alternative.

Incorporating Opportunity Cost

When evaluating an investment:

• Compare Expected Returns: Ensure the investment offers returns at least equal to alternatives.

• Adjust for Risk: Higher-risk projects should offer higher expected returns.

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Simplifying Assumptions and Their Implications

Our models often rely on simplifying assumptions:

1. No Uncertainty: Cash flows are known with certainty.

2. Constant Discount Rate: The rate doesn't change over time.

3. No Transaction Costs: Ignoring fees, taxes, or other frictions.

4. Immediate and Costless Currency Conversion: No delays or costs in moving money through time.

While these assumptions help us build foundational understanding, real-world scenarios require adjustments to account for the following:

• Risk and Uncertainty: Incorporate probability and risk premiums.

• Changing Rates: Adjust for fluctuating interest rates and economic conditions.

• Transaction Costs: Factor in fees, taxes, and other expenses.

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The Psychological Aspect: Time Preference

Human behavior often shows a preference for immediate gratification.

Impatience and Discounting

• Positive Time Preference: Valuing present consumption over future consumption.

• Implications for Discount Rates: Higher impatience leads to higher discount rates.

Understanding this helps explain why people might accept less money today rather than waiting for more in the future.

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A Deeper Dive into Discount Rates

Risk-Free Rate

• Based on government securities with virtually no default risk.

• Serves as the foundation for building discount rates.

Risk Premium

• Additional return is required to take on extra risk.

• Varies by asset class and individual investment.

Market Rate of Return

• Reflects the average return expected by Investors.

• Influenced by economic conditions, Investor sentiment, and market volatility.

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The Mathematics of Continuous Compounding

In some cases, interest compounds continuously.

Continuous Compounding Formula

• Future Value:

\(FV = PV \times e^{(r \times t)}\)

• Present Value:

\(PV = FV \times e^{-(r \times t)}\)

Where:

• e is the base of the natural logarithm (≈2.71828)

• r is the annual interest rate

• t is the time in years

Example: What is the future value of $1,000 after 2 years at a 5% continuous interest rate?

\(FV = $1,000 \times e^{(0.05 \times 2)} = $1,000 \times e^{0.1} \approx $1,105.17\)

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Practical Considerations: Taxes and Regulations

Taxes

• Impact on Cash Flows: Taxes reduce net cash inflows.

• After-Tax Discount Rate: Adjust the discount rate to reflect tax implications.

Regulations

• Compliance Costs: Expenses associated with meeting legal requirements.

• Impact on Risk: Regulatory changes can affect the risk profile of an investment.

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Case Study: Valuing a Tech Startup

Suppose you're evaluating an investment in a tech startup expected to generate the following cash flows (in millions):

• Year 0: −$2 (initial investment)

• Year 1: $0.5

• Year 2: $1

• Year 3: $2

• Year 4: $3

• Year 5: $5

Given the high risk, you decide on a discount rate of 15%.

Calculating NPV

Year 1:

\(PV_1 = \frac{$0.5}{(1 + 0.15)^1} = $0.4348\)

Year 2:

\(PV_2 = \frac{$1}{(1 + 0.15)^2} = $0.7561\)

Year 3:

\(PV_3 = \frac{$2}{(1 + 0.15)^3} = $1.3168\)

Year 4:

\(PV_4 = \frac{$3}{(1 + 0.15)^4} = $1.7130\)

Year 5:

\(PV_5 = \frac{$5}{(1 + 0.15)^5} = $2.4859\)

Total PV of Cash Inflows:

\(\text{Total PV} = $0.4348 + $0.7561 + $1.3168 + $1.7130 + $2.4859 = $6.7066\)

NPV:

\(NPV = -$2 + $6.7066 = $4.7066\)

Despite the high discount rate, the NPV is significantly positive, indicating a potentially lucrative investment.

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Sensitivity Analysis: Varying Discount Rates

Understanding how changes in the discount rate affect NPV is crucial.

Example with Different Rates

• At 10% Discount Rate:

  NPV increases reflect lower required returns due to perceived lower risk.

• At 20% Discount Rate:

  NPV decreases, signaling higher required returns for increased risk.

Implications

• Investor's Risk Appetite: Higher discount rates imply higher risk aversion.

• Project Viability: Projects may be acceptable at one discount rate but not another.

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Conclusion: The Art and Science of Valuing Future Cash Flows

Valuing future cash flows in today's terms is both an art and a science:

• Science: Mathematical models and formulas provide the tools for calculation.

• Art: Assumptions about discount rates, risk, and future cash flows require judgment and experience.

By mastering these concepts, you're better equipped to:

• Evaluate Investments: Make informed decisions based on rigorous analysis.

• Understand Market Dynamics: Grasp how interest rates and risk affect asset values.

• Plan Financially: Align personal or corporate financial goals with realistic expectations.

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Next Steps: Embracing Uncertainty and Risk Management

In our next lesson, we'll:

• Introduce Uncertainty: Incorporate risk into cash flow projections.

• Explore Risk Management: Learn techniques to mitigate financial risks.

• Delve into Market Behavior: Understand how market fluctuations impact valuations.

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Thank you for joining this comprehensive exploration of valuing future cash flows. Continue your journey with Hardcore Finance to deepen your financial acumen and apply these principles to real-world challenges.

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all the best,

Jack Roshi, PhD