Time Value of Money: Present Value of Lump Sums

Present Value Calculation: Lump Sum Cases (Chapter 4)

Introduction to Present Value

• Chapter Focus: Chapter 4 primarily focuses on lump sum cases, which involve a single cash flow. This serves as a foundation before moving to multiple cash flow scenarios in Chapter 5.

• Definition: Present Value ($PV$) is defined as the current value of an amount to be received in the future.

• Practical Application: This concept is highly effective for both daily personal financial planning and corporate strategic planning, enabling individuals or companies to determine how much to invest or set aside today to achieve a future financial goal.

Why Future Value is Generally Less Valuable Than Present Value

• Opportunity Cost: When you lend money today or forgo its immediate use, you lose the opportunity to utilize that money for other purposes. This missed opportunity is a cost.

  • Analogy: Being in this class means giving up an opportunity to be in another class or do something else.

• Risk and Uncertainty: The further into the future an amount is to be received, the greater the uncertainty associated with it. Factors like inflation, market volatility, and unforeseen events make future money less certain and, consequently, less valuable today.

• Proverbial Wisdom: The concept is encapsulated by the saying, "A bird in the hand is much better than two birds in the bush." This means that a guaranteed, immediate benefit is often preferred over a potentially larger, uncertain future benefit.

• Real-World Application (Dividends):

  • Mature Corporations: Companies with stable future cash flows (e.g., utility companies) often pay high dividends. This appeals to investors, such as retirees, who prioritize fixed income and immediate cash flow over future growth.

  • Tech Companies: Many growth-oriented tech companies reinvest their earnings back into the business for faster expansion rather than paying large dividends. They expect this reinvestment to lead to significant future growth and higher stock values, appealing to investors focused on capital appreciation.

The Time Value of Money (TVM) Timeline

• Purpose: Drawing a timeline is a crucial approach for visualizing and summarizing given information in time value of money problems, making it easier to identify missing variables.

• Structure:

  • Today: Represented by Time $0$.

  • End of Period: Tick marks on the timeline typically represent the end of a period (e.g., Time $1$ is the end of Period $1$). This is the default assumption for when cash flows occur in Chapter 4.

• Sign Convention:

  • Cash Inflow: Indicated by a plus ($+$) sign in front of the cash flow amount.

  • Cash Outflow: Indicated by a negative ($-$) sign in front of the cash flow amount.

• PMT (Constant Cash Flow): This refers to a series of equal, periodic cash flows (annuity payments). While important for TVM, it is not the focus of Chapter 4, which deals exclusively with lump sums. PMT will be discussed in Chapter 5.

Present Value Formula and Calculation (Discounting)

• Future Value Formula (Review): The fundamental relationship between present and future value is expressed as: $FV = PV * (1 + r)^t$

  • Where:

    • $FV$ = Future Value

    • $PV$ = Present Value

    • $r$ = Interest rate (expressed as a decimal)

    • $t$ = Number of periods

• Present Value Formula (Rearranged): To calculate present value, the future value formula is rearranged:

  • $PV = FV / (1 + r)^t$

  • Alternatively: $PV = FV * (1 + r)^{-t}$ (This form highlights that one plus r is raised to a negative t power).

• Discounting: The process of calculating the present value of one or more future amounts is called discounting. It is the reverse operation of compounding (which is future value calculation).

Methods for Present Value Calculation

There are four primary methods for calculating present value:

1. General Formula Approach

• Concept: Directly applying the rearranged present value formula.

• Example 1: Present Value of $100 in 3 Years at$10%$

  • Given: $FV = \$100$, $t = 3$ years, $r = 10\%$ ($0.10$)

  • Calculation: $PV = \$100 / (1 + 0.10)^3 = \$100 / (1.10)^3 = \$100 / 1.331 = \$75.13$

  • Interpretation: An investment of $75.13 today at$10\%$interest for$3$years will grow to$100.

  • Significance: This method is effective for setting financial goals, determining how much to set aside today to reach a specific future amount.

2. Financial Calculator (TVM Keys)

• Concept: Utilizing dedicated time value of money functions on a financial calculator.

• Settings: Ensure the calculator is set to 'end of period' (default) and 'annual compounding'.

• Inputs for Example 1: (Assuming PMT is $0$ for lump sums)

  • N = $3$ (Number of periods)

  • I/Y = $10$ (Annual interest rate, usually entered as a whole number)

  • FV = $100$ (Future Value)

  • CPT PV (Compute Present Value)

• Output for Example 1: The calculator will display $-\$75.13$. The negative sign indicates a cash outflow (investment) from your perspective.

• Sign Convention Importance: The sign of PV and FV matters, especially when solving for r (interest rate) or N (number of periods), where one must be positive and the other negative to indicate cash inflow and outflow, respectively.

3. Excel Spreadsheet Function

• Concept: Using the built-in PV function in Excel or similar spreadsheet software.

• Formula: =PV(rate, nper, pmt, [fv], [type])

  • rate: Interest rate per period (e.g., $0.10$)

  • nper: Total number of payment periods (e.g., $3$)

  • pmt: Payment made each period (set to $0$ for lump sums in Chapter 4)

  • [fv]: Future value (e.g., $100$)

  • [type]: When payments are due (set to $0$ or omitted for end of period).

• Inputs for Example 1: =PV(0.10, 3, 0, 100, 0)

• Output for Example 1: $-\$75.13$. Again, the negative sign signifies an initial cash outflow.

4. Present Value Tables

• Concept: Using pre-calculated tables that provide Present Value Interest Factors (PVIFs) for lump sums, often labeled as "Present Value of $1". (Separate tables exist for annuities and annuities due, covered in Chapter 5).</p></li><li><p><strong>Process</strong>: Locate the factor corresponding to the given interest rate and number of periods.</p></li><li><p><strong>Example 1</strong>: For$10\%$interest and$3$years, the factor is approximately$0.75131$.</p></li><li><p><strong>Calculation</strong>:$PV = FV * PVIF = \$100 * 0.75131 = \$75.131$</p></li><li><p><strong>Note on Rounding</strong>: Calculations using tables may show slight differences due to rounding of the factors (often rounded to five decimal places), leading to minor discrepancies (e.g., a few cents) compared to formula or calculator methods.</p></li></ul><h4 id="504e7bd5-8d20-4d96-8723-63e498545cec" data-toc-id="504e7bd5-8d20-4d96-8723-63e498545cec" collapsed="false" seolevelmigrated="true">Further Examples and Applications</h4><h5 id="a9dace5f-fe03-4885-9dff-1763273042a3" data-toc-id="a9dace5f-fe03-4885-9dff-1763273042a3" collapsed="false" seolevelmigrated="true">Example 2: Down Payment for a Car</h5><ul><li><p><strong>Scenario</strong>: Need$10,000 in $1$ year for a car down payment; can earn $7\%$ annually. How much to invest today?

• Given: $FV = \$10,000$, $t = 1$ year, $r = 7\%$ ($0.07$)

• Formula: $PV = \$10,000 / (1 + 0.07)^1 = \$10,000 / 1.07 = \$9,345.79$

• TVM Calculator: $N=1$, $I/Y=7$, $FV=10000$, CPT PV $<br>ightarrow -\$9,345.79$

• PV Table: Factor for $7\%$ at $1$ year is $0.93458$. $PV = \$10,000 * 0.93458 = \$9,345.80$

• Excel: =PV(0.07, 1, 0, 10000, 0) $<br>ightarrow -\$9,345.79$

Example 3: Daughter's College Education Fund

• Scenario: Daughter needs $150,000 in$17$years; confident of earning$8\%$per year. How much to invest today?</p><ul><li><p><strong>Note</strong>: Earning a consistent$8\%$annual return over$17$years is ambitious and challenging.</p></li></ul></li><li><p><strong>Given</strong>:$FV = \$150,000$,$t = 17$years,$r = 8\%$($0.08$)</p></li><li><p><strong>Formula</strong>:$PV = \$150,000 / (1 + 0.08)^{17} = \$150,000 / 3.70001 = \$40,540.34$</p></li><li><p><strong>TVM Calculator</strong>:$N=17$,$I/Y=8$,$FV=150000$, <code>CPT PV</code>$
  ightarrow -\$40,540.34$</p></li><li><p><strong>PV Table</strong>: Factor for$8\%$at$17$years is$0.27027$.$PV = \$150,000 * 0.27027 = \$40,540.50$</p><ul><li><p><strong>Note</strong>: The$0.16$difference compared to other methods is due to the rounding of the factor in the table.</p></li></ul></li><li><p><strong>Practical Implication</strong>: This example highlights the significant impact of starting investments early. A longer investment horizon ($t$) means a smaller initial investment ($PV$) is required to reach the same future goal ($FV$). Investment experts emphasize starting as early as possible because "time matters."</p></li></ul><h5 id="97ae5068-c354-47b1-9274-794e1bdd8e54" data-toc-id="97ae5068-c354-47b1-9274-794e1bdd8e54" collapsed="false" seolevelmigrated="true">Example 4: Parents' Trust Fund</h5><ul><li><p><strong>Scenario</strong>: A trust fund was set up$10$years ago, now worth$19,671.51. It earned $7\%$ per year. How much did parents originally invest?

• Given: $FV = \$19,671.51$, $t = 10$ years, $r = 7\%$ ($0.07$)

• Formula: $PV = \$19,671.51 / (1 + 0.07)^{10} = \$19,671.51 / 1.96715 = \$10,000.00$

• Connection to Rule of 72: The Rule of 72 states that dividing $72$ by the annual interest rate approximates the number of years it takes for an investment to double. Here, $72 / 7\% \approx 10.28$ years. The fund nearly doubled in $10$ years (from $10,000 to$19,671.51), which aligns with the rule.

• TVM Calculator: $N=10$, $I/Y=7$, $FV=19671.51$, CPT PV $<br>ightarrow -\$10,000.00$

• PV Table: Factor for $7\%$ at $10$ years is $0.50835$. $PV = \$19,671.51 * 0.50835 = \$9,999.98$

• Excel: =PV(0.07, 10, 0, 19671.51, 0) $<br>ightarrow -\$10,000.00$

Key Relationships in Present Value Calculation (Review)

These relationships are fundamental and can be derived directly from the present value formula. They are the inverse of the relationships observed in future value calculations.

1. Relationship between Time Period ($t$) and Present Value ($PV$) (for a given interest rate)

• Principle: For a given interest rate ($r$), the longer the time period ($t$), the lower the present value ($PV$).

  • As $t$ increases, $PV$ decreases.

• Explanation: If you have more time for an investment to grow, you need to start with a smaller initial amount to reach the same future value.

• Example: $FV = \$500$, $r = 10\%$:

  • $t = 5$ years: $PV = \$500 / (1.10)^5 = \$310.46$

  • $t = 10$ years: $PV = \$500 / (1.10)^{10} = \$192.77$

• Practical Implication: This re-emphasizes the importance of starting investments early to leverage the power of compounding over time.

2. Relationship between Interest Rate ($r$) and Present Value ($PV$) (for a given time period)

• Principle: For a given time period ($t$), the higher the interest rate ($r$), the lower the present value ($PV$).

  • As $r$ increases, $PV$ decreases.

• Explanation: If your investment can earn a higher rate of return, you need to start with a smaller initial amount to reach the same future value.

• Example: $FV = \$500$, $t = 5$ years:

  • $r = 10\%$: $PV = \$500 / (1.10)^5 = \$310.46$

  • $r = 15\%$: $PV = \$500 / (1.15)^5 = \$248.59$

• Conclusion: These relationships are intuitive once the underlying time value of money concept is understood and can be easily visualized graphically. Memorization is not necessary if the formula and its implications are grasped.

Chapter 4 Summary and Future Topics

• Four Parts of the TVM Equation: Every time value of money problem involves four key variables:

  • Present Value ($PV$)

  • Future Value ($FV$)

  • Interest Rate ($r$)

  • Number of Periods ($t$)

• Solving for the Unknown: If any three of these variables are known, you can always solve for the fourth.

• Sign Convention (Crucial): Always remember the sign convention for cash flows when using financial calculators or spreadsheets:

  • Cash Inflow: Positive ($+$) sign.

  • Cash Outflow: Negative ($-$) sign.

  • This is especially critical when solving for r or N to avoid errors.

• Investment vs. Borrowing Decisions: Most TVM questions relate to investment decisions, but some may involve borrowing decisions. It's important to properly interpret the question context (e.g., if you're taking out a loan, that's an inflow to you today but requires future outflows).

• Upcoming Topics: Chapter 4 will conclude by discussing how to calculate the interest rate ($r$) and the number of periods ($t$) when they are the unknown variables. Chapter 5 will then introduce multiple cash flow scenarios.