Evaluating Choices: Time, Risk, and Value - Notes

Introduction

• Financial decisions are future-oriented and involve speculation due to uncertainty.

• Evaluation involves understanding risks and opportunities presented by time.

4.1 The Time Value of Money

Learning Objectives

1. Explain the value of liquidity.

2. Demonstrate how time creates distance, risk, and opportunity cost.

3. Demonstrate how time affects liquidity.

4. Analyze how time affects value.

Time and Value

• Evaluating future results is part of financial planning.

• Time affects the probability and value of future benefits and costs.

• Time affects value because it affects liquidity.

• Liquidity is valuable; more liquid assets are generally better.

Liquidity

• The time value of money is how time affects the liquidity of money and thus its value.

• Example: Exchanging dollars for pesos in Mexico involves transaction costs, opportunity costs, and risk to obtain liquidity.

• Transforming not-so-liquid wealth into liquid wealth incurs transaction costs, opportunity costs, and risk, which detract from wealth's value.

• Liquidity has value because it can be used without additional costs.

• Past cash flows are sunk, present cash flows are liquid, and future cash flows are not yet liquid.

• Choices can only be made with liquid wealth.

Time and Opportunity Cost

• Separation from liquidity creates opportunity costs: having liquidity now allows for consumption or investment.

• Risk arises from uncertainty about future cash flows and their worth.

• The further in the future cash flows are, the more opportunity cost and risk exist, reducing the present value (PV) of wealth.

• Time creates distance from liquidity, which incurs costs that diminish value.

• Financial decisions rely on values at a distance in time.

• Comparing cash flows requires understanding the relationships between nominal or face values in the future and their equivalent present values.

• Present values are less than future values due to the cost of time.

Key Takeaways

• Liquidity enables choice and thus has value.

• Time creates distance or delay from liquidity.

• Distance or delay creates risk and opportunity costs.

• Time affects value by creating distance, risk, and opportunity costs.

• Time discounts value.

4.2 Calculating the Relationship of Time and Value

Learning Objectives

1. Identify the factors needed to relate a present value to a future value.

2. Write the algebraic expression for the relationship between present and future value.

3. Discuss the use of the algebraic expression in evaluating the relationship between present and future values.

4. Explain the importance of understanding the relationships among the factors that affect future value.

Financial Calculation

• Understanding financial calculations is crucial for grasping the relationships between time, risk, opportunity cost, and value.

• Required information:

  • Future cash flows (CF)

  • Timing of future cash flows

  • Rate at which time affects value (discount rate)

• Forecasting timing and amounts of future cash flows can be uncertain.

Discount Rate

• The discount rate represents the opportunity cost of not having liquidity; it is the rate at which time discounts value.

• Opportunity cost comes from forgone choices.

• Choosing the discount rate is an important judgment call as it directly affects valuation.

Calculating Future Value

• If you had $1,000 today, you could deposit it in an interest-bearing account. At a 4 percent annual interest rate:

  • After one year: 1,000 + (1,000 × 0.04) = 1,000 × (1 + 0.04) = 1,040

  • After two years: 1,040 + (1,040 × 0.04) = 1,040 × (1 + 0.04) = [1,000 × (1 + 0.04)] × (1 + 0.04) = 1,000 × (1 + 0.04)^2 = 1,081.60

General Formula

• If PV is the present value, r is the discount rate, and t is the number of time periods, then the future value (FV) is:

  • PV × (1 + r)^t = FV

Present Value

• Assuming negligible risk, the cost of not having liquidity is the opportunity cost of delayed consumption or not earning interest.

• The cost of delayed consumption is subjective and depends on the utility or satisfaction derived from consumption.

• If you would save the money, the cost of waiting to receive the $1,000 is the $40 in interest you miss out on.

• To calculate present value (PV) given future value (FV):

  • Since PV × (1 + r)^t = FV, then PV = FV / [(1 + r)^t]

• The gift is worth $961.5385 today (its present value).

• Receiving $961.5385 today is equivalent to receiving $1,000 one year from now.

Relationships

• Understanding the relationships between time, risk, opportunity cost, and value is essential.

• PV × (1 + r)^t = FV

• 'r' is the discount rate, including opportunity costs and risk.

• 't' is the time until liquidity.

Insights from the Equation

• The more time (t) separating you from liquidity, the more time affects value.

• The less time separating you from liquidity, the less time affects value.

• As t increases, the PV of your FV liquidity decreases.

• As t decreases, the PV of your FV liquidity increase

• The greater the rate at which time affects value (r), or the greater the opportunity cost and risk, the more time affects value.

• The less your opportunity cost or risk, the less your value is affected.

• As r increases, the PV of your FV liquidity decreases.

• As r decreases, the PV of your FV liquidity increases.

Strategy Implications

• It is more valuable to have liquidity sooner rather than later (positive cash flow) and to pay out later rather than sooner (negative cash flow).

• Accelerate incoming cash flows and decelerate outgoing cash flows.

Key Takeaways

• To relate a present (liquid) value to a future value, you need to know:

  • What the present value is or the future value will be.

  • When the future value will be.

  • The rate at which time affects value: the costs per time period or the magnitude of the effect of time on value.

• The relationship of present value (PV), future value (FV), risk and opportunity cost (the discount rate, r), and time (t) may be expressed as:

  • PV × (1 + r)^t = FV

• The above equation yields valuable insights into these relationships:

  • The more time (t) creates distance from liquidity, the more time affects value.

  • The greater the rate at which time affects value (r), or the greater the opportunity cost and risk, the more time affects value.

  • The closer the liquidity, the less time affects value.

  • The less the opportunity cost or risk, the less value is affected.

• To maximize value, get paid sooner and pay later.

4.3 Valuing a Series of Cash Flows

Learning Objectives

1. Discuss the importance of the idea of the time value of money in financial decisions.

2. Define the present value of a series of cash flows.

3. Define an annuity.

4. Identify the factors you need to know to calculate the value of an annuity.

5. Discuss the relationships of those factors to the annuity’s value.

6. Define a perpetuity.

Series of Cash Flows

• It is common in finance to value a series of future cash flows (CF).

• The present value (PV) of the series of cash flows is equal to the sum of the present value of each cash flow.

• When there are regular payments at regular intervals and each payment is the same amount, that series of cash flows is an annuity.

• Examples of annuities: loan repayments, installment purchases, mortgages, retirement investments, savings plans, and retirement plan payouts.

Annuity Calculation

• To calculate the present value of an annuity, you need to know:

  • The amount of the future cash flows (the same for each).

  • The frequency of the cash flows.

  • The number of cash flows (t).

  • The rate at which time affects value (r).

• Calculators and software applications can do the math, but understanding the relationships between time, risk, opportunity cost, and value is important.

Lottery Example

• Winning the lottery often involves choosing between a lump sum or annual payments over twenty years.

• Lottery agencies prefer annual payments because they retain liquidity longer.

• The lump-sum option is discounted to reflect the present value of the annuity.

• The discount rate is chosen at the discretion of the lottery agency.

• Example: Winning $10 million with a choice of $500,000 per year for 20 years or a lump sum of $6,700,000.

• The value of the annuity isn't $10 million because time affects liquidity and value.

• Your discount rate or opportunity cost determines the annuity’s value to you.

Discount Rates

• The present value of the annuity is less if your discount rate is higher.

• The annuity would be worth the same as the lump-sum payout if your discount rate were 4.16 percent.

• If your discount rate is about 4 percent or less, the annuity is worth more to you.

• If your discount rate is higher than 4 percent, the lump sum is worth more to you.

Key Points

• The greater the rate at which time affects value, the greater the effect on the present value.

• When opportunity cost or risk is low, waiting for liquidity doesn’t matter as much as when opportunity costs or risks are higher.

• Liquidity is valuable because it allows you to make choices.

• The higher the rate at which time affects value, the more it costs to wait for liquidity, and the more choices pass you by.

• When risk is low, getting liquidity sooner is not as important as when risk is high.

Annuity Value

• As r increases, the PV of the annuity decreases.

• As r decreases, the PV of the annuity increases.

• Amount of each payment or cash flow affects the value of the annuity.

• As CF increases, the PV of the annuity increases.

• As CF decreases, the PV of the annuity decreases.

• The more periods in the annuity, the more cash flows and the more liquidity there are, thus increasing the value of the annuity.

• As t increases, the PV of the annuity increases.

• As t decreases, the PV of the annuity decreases.

Calculating Future Value

• It is common in financial planning to calculate the FV of a series of cash flows when saving for a goal.

• Going forward, the rate at which time affects value (r) is the rate at which value grows, or the rate at which your value compounds.

• As r increases, the FV of the annuity increases.

• As r decreases, the FV of the annuity decreases.

• Amount of each payment or cash flow affects the value of the annuity.

• As CF increases, the FV of the annuity increases.

• As CF decreases, the FV of the annuity decreases.

• The more periods in the annuity, the more cash flows, and the greater the effect of time, thus increasing the future value of the annuity.

• As t increases, the FV of the annuity increases.

• As t decreases, the FV of the annuity decreases.

Perpetuities

• A perpetuity is an annuity that goes on forever.

• Dividends from a share of corporate stock are a perpetuity.

• The perpetuity represents the maximum value of the annuity.

Life as Cash flow

• These are the fundamental relationships that structure so many financial decisions, most of which involve a series of cash inflows or outflows. Understanding these relationships can be a tool to help you answer some of the most common financial questions about buying and selling liquidity.

Mortgage Example

• Loans are usually designed as annuities, with regular periodic payments that include interest expense and principal repayment. Using these relationships, you can see the effect of a different amount borrowed (PVannuity), interest rate (r), or term of the loan (t) on the periodic payment (CF).

Saving Example

• Saving to reach a goal is often accomplished by a plan of regular deposits to an account for that purpose. The savings plan is an annuity, so these relationships can be used to calculate how much would have to be saved each period to reach the goal (CF), or given how much can be saved each period, how long it will take to reach the goal (t), or how a better investment return (r) would affect the periodic savings, or the time needed (t), or the goal (FV).

Financial Calculations

• Modern tools such as calculators, spreadsheets, and software make financial calculations easier.

• Calculations are discussed so that you can understand them, and most importantly, so that you can understand the relationships that they describe.

Key Takeaways

• The idea of the time value of money is fundamental to financial decisions.

• The present value of the series of cash flows is equal to the sum of the present value of each cash flow.

• A series of cash flows is an annuity when there are regular payments at regular intervals and each payment is the same amount.

• To calculate the present value of an annuity, you need to know:

  • The amount of the identical cash flows (CF).

  • The frequency of the cash flows.

  • The number of cash flows (t).

  • The discount rate (r) or the rate at which time affects value.

• The calculation for the present value of an annuity yields valuable insights:

  • The more time (t), the more periods and the more periodic payments, that is, the more cash flows, and so the more liquidity and the more value.

  • The greater the cash flows, the more liquidity and the more value.

  • The greater the rate at which time affects value (r) or the greater the opportunity cost and risk or the greater the rate of discounting, the more time affects value.

• The calculation for the future value of an annuity yields valuable insights:

  • The more time (t), the more periods and the more periodic payments, that is, the more cash flows, and so the more liquidity and the more value.

  • The greater the cash flows, the more liquidity and the more value.

  • The greater the rate at which time affects value (r) or the greater the rate of compounding, the more time affects value.

• A perpetuity is an infinite annuity.

4.4 Using Financial Statements to Evaluate Financial Choices

Learning Objectives

1. Define pro forma financial statements.

2. Explain how pro forma financial statements can be used to project future scenarios for the planning process.

Financial Statements

• Projected or pro forma financial statements can show the consequences of choices.

• To project future financial statements, you need to be able to envision the expected results of all the items on them.

• Predictions always contain uncertainty, so projections are always, at best, educated guesses.

• Alice can actually project how her financial statements will look after each choice is followed.

Importance in Decisions

• When making financial decisions, it is helpful to be able to think in terms of their consequences on the financial statements, which provide an order to our summary of financial results.

• Choices were to continue to pay it down gradually as she does now; to get a second job to pay it off faster; or to go to Vegas, hit it big (or lose big), and eliminate her debt altogether (or wind up with even more).

Vegas Example

• While Vegas yields the largest increase in net income or personal profit if she wins, it creates the largest decrease if she loses; it is clearly the riskiest option. The pro forma cash flow statements reinforce this observation.

• If Alice has a second job, she will use the extra cash flow, after taxes, to pay down her student loan, leaving her with a bit more free cash flow than she would have had without the second job. If she wins in Vegas, she can pay off both her car loan and her student loan and still have an increased free cash flow. However, if she loses in Vegas, she will have to secure more debt to cover her losses.

• Assuming she borrows as much as she loses, she will have a small negative net cash flow and no free cash flow, and her other assets will have to make up for this loss of cash value.

• To reduce the risk from the strategy it is useful to research options such as tax implications, which will improve the quality of future decisions.

Summary of Outcomes

• If Alice has a second job, her net worth increases but is still negative, as she has paid down more of her student loan than she otherwise would have, but it is still larger than her asset value. If she wins in Vegas, her net worth can be positive; with her loan paid off entirely, her asset value will equal her net worth. However, if she loses in Vegas, she will have to borrow more, her new debt quadrupling her liabilities and decreasing her net worth by that much more.

• A summary of the critical “bottom lines” from each pro forma statement (Figure 4.18 "Alice’s Pro Forma Bottom Lines") most clearly shows Alice’s complete picture for each alternative.

Critical Choices

• Going to Vegas creates the best and the worst scenarios for Alice, depending on whether she wins or loses. While the outcomes for continuing or getting a second job are fairly certain, the outcome in Vegas is not; there are two possible outcomes in Vegas. The Vegas choice has the most risk or the least certainty.

• The Vegas alternative also has strategic costs: if she loses, her increased debt and its obligations—more interest and principal payments on more debt—will further delay her goal of building an asset base from which to generate new sources of income. In the near future, or until her new debt is repaid, she will have even fewer financial choices.

• The strategic benefit of the Vegas alternative is that if she wins, she can eliminate debt, begin to build her asset base, and have even more choices (by eliminating debt and freeing cash flow).

Probabilities

• The next step for Alice would be to try to assess the probabilities of winning or of losing in Vegas.

• Once she has determined the risk involved—given the consequences now illuminated on the pro forma financial statements—she would have to decide if she can tolerate that risk, or if she should reject that alternative because of its risk.

Key Takeaway

• Pro forma financial statements show the consequences of financial choices in the context of the financial statements.

4.5 Evaluating Risk

Learning Objectives

1. Explain the basic dynamics of probabilities.

2. Discuss how probabilities can be used to measure expected value.

3. Describe how probabilities can be used in financial projections.

4. Analyze expected outcomes of financial choices.

The idea of independence

• Behavioral finance is significantly affected by risk in financial decision making.

• The study of risk and the interpretation of probabilities are complex.

• An independent event is one that happens by chance. It cannot be willed or decided upon.

• The probability or likelihood of an independent event can be measured, based on its frequency in the past, and that probability can be used to predict whether it will recur.

Alice and Vegas

• Alice does research and determines she cannot choose to win; there is always uncertainty. Alice can choose whether or not to go to Vegas.

• The probability of any one outcome for an event is always stated as a percentage of the total outcomes possible.

• An independent or risky event has at least two possible outcomes: it happens or it does not happen. There may be more outcomes possible, but there are at least two; if there were only one outcome possible, there would be no uncertainty or risk about the outcome.

Coin Flip

• Example: you have a “50-50 chance” of “heads” when you flip a coin, or a 50 percent probability. On average “heads” comes up half the time.

• There are only two possible outcomes when you flip a coin, and there is a 50 percent chance of each.

• The probabilities of each possible outcome add up to 100 percent, because there is 100 percent probability that something will happen.

• Probabilities can be used in financial decisions to measure the expected result of an independent event.

Probabilities and Results

• You have a 50 percent chance of $1.00 and a 50 percent chance of −$1.00. Half the time you can expect to gain a dollar, and half the time you can expect to lose a dollar.

• Example: The expected result for each outcome is its probability or likelihood multiplied by its result. The expected result or expected value for the action, for flipping a coin, is its weighted average outcome, with the “weights” being the probabilities of each of its outcomes.

Expected Return

• There is zero-sum return on the activity since the returns can be modeled as (0.50×1.00)+(0.50×−1.00)=0.50+−0.50=0

• The expected value (E(V)) of an event is the sum of each possible outcome’s probability multiplied by its result, or E(V)=Σ(pn×rn), where Σ means summation, p is the probability of an outcome, r is its result, and n is the number of outcomes possible.

Vegas continued

• When faced with the uncertainty of an alternative that involves an independent event, it is often quite helpful to be able to at least calculate its expected value. Then, when making a decision, that expectation can be weighed against or compared to those of other choices.

• Alice can calculate the expected result of going to Vegas if she knows the probabilities of its two outcomes, winning and losing. Alice does a bit of research and has a friend show her a few tricks and decides that for her the probability of winning is 30 percent, which makes the probability of losing 70 percent. (As there are only two possible outcomes in this case, their probabilities must add to 100 percent.)

Calculating Vegas

• Her expected result in Vegas, then, is (0.30×100,000)+(0.70×−100,000)=30,000+−70,000=−40,000.

• Alice can also calculate what the probability of winning would have to be to make it a worthwhile choice at all, that is, to give her at least as good a result as either of her other choices

Results for Alice

• If she only has a 30 percent chance of winning in Vegas, then going there at all is the worst choice for her in terms of her net income and net worth.

• Her net cash flow (CF) actually seems best with the Vegas option, but that assumes she can borrow to pay her gambling losses, so her losses don’t create net negative cash flow.

• To be the best choice in terms of all three bottom lines, Alice would have to have a 78 percent chance of winning at Vegas.

• Alice would have to determine if she can tolerate the risk that she might not and see if Vegas would be a viable choice for her.

Summary

• Using probabilities to derive the expected value of a choice provides a way to evaluate an alternative with uncertainty.

• It requires projecting the probabilities and results of each possible outcome or independent event.

• It cannot remove the uncertainty or the risk that independence presents, but it can at least provide a way to measure and then compare with other measurable, certain or uncertain, choices.

Key Takeaways

• Probabilities can be used in financial decisions to measure the expected result of an independent