Chapter 4.7® - Present Value & Discounting – We Know the Future Value, how do we find the Present Value? Single Period Case & PV Discounting for Multiple Periods

• Part 4.1 - Time Value of Money, Future Values of Compounding Interest, Investing for more than 1 Period & Examination of Original Investment & Growth of Investment
• Part 4.2 - Compounding Interest Homework Problem & Time Value of Money Continued - Future Value Formula, Growth of $100 & Future Value Comparisons
• Part 4.3 - How to Use a Financial Calculator BAII Plus to Perform Time Value of Money & Present / Future Value Calculations
• Part 4.4 - Changing Advanced Function Keys (BGN, C/Y, P/Y), Converting from Nominal Interest to Effective Interest Rates using BAII Financial Calculator
• Part 4.5 - Examples of Interest Rate Calculations & Practice Questions #1 - #7
• Part 4.6 - Nominal to Effective Interest Rate Calculations & Practice Questions #8 - #16
• Part 4.7 - Present Value & Discounting – We Know the Future Value, how do we find the Present Value? Single Period Case & PV Discounting for Multiple Periods
• Part 4.8 - An Example of Present Value Discounting & Saving Up Money & Deriving the Basic Present Value Equation
• Part 4.9 - Determining the Discount Rate using Basic Present Value equation & Finding the Number of Accounting Periods
• Part 4.10 - Summary of Time Value Formulas & Calculations
• Part 4.11 - Discounted Cash Flow Valuations - Future Value of Multiple Cash Flows & Designing the Cash Flows Timeline
• Part 4.12 - Compound the Accumulated Balance Forward One Year at a Time - Discounted Cash Flow Valuation - Determining Present Value of Multiple Future Cash Flows & Designing a Financial Timeline
• Part 4.13 - Determining Present Value of Multiple Future Cash Flows – Homework Example
• Part 4.14 - Calculating Present Value with Multiple Future Cash Flows – Example #2
• Part 4.15 - Valuing Annuity Level Cash Flows – Present & Future Value of Annuities
• Part 4.16 - Calculating Annuity Payments using Annuity Present Value Factor – Example
• Part 4.17 - Accounting for Perpetuities & Using the Annuity Present Value Factor Formula
• Part 4.18 - Accounting for Growing Perpetuities & Simplified Formula for Present Value of Growing Perpetuity - Example of Growing Perpetuity on Rental Cash Flows

When we discuss future value, we think of questions like “What will my $5,000 grow to if it earns 7% interest every year for the next 10 years?” The answer to this question is to find the future value of this $5,000 in 10 years, earning 7% return every year. The answer to this question is:

A similar type of question we could ask is, suppose we need to have $20,000 in 10 years, earning only 6.5% interest pear year, how much do we need to invest today to reach our goal? This is called present value discounting and we elaborate more on this topic below..

Single Period Case

From the calculations we did in the Future Value & compounding interest section, we know that the future value of $1 invested for one year at 10% is $1.10. We now modify this question slightly and ask the following: How much do we have to invest today at 10% to get $1 in one year? In other words, our desired future value is $1, what is the present value (PV)? The answer to this question is quite simple actually. Whatever we invest today will be 1.1 times bigger at the end of the year (due to the 10% interest). Since we need $1 at the end of the year:

The definition of this answer above is that, what amount invested today will grow to $1 in one year if the interest rate is 10%? Present value is thus the reverse of future value and instead of compounding the money in to the future; we discount it backwards to the present!

Real Life example of Single Period Present Value Discounting

Suppose you need $600 to buy textbooks for next academic year. You can earn 5% interest on your money, how much do you need to have today in order to save up that $600 for next year?

In other words, we need to know the PV of $600 in one year at 5% interest. Here are the calculations:

Notice in our examples above, we assume we need to grow our money 1 year in to the future. What if we need to grow our money 2 or 3 or more years in to the future? This is called present value discounting for multiple periods, and we discuss this topic below.

Present Value Discounting for Multiple Periods

Suppose you need to have $10,000 in two years. If you can earn 6% interest, how much do you have to invest today to make sure you have $10,000 in two years, when you will need it? In other words, what is the present value of $10,000 in two years if the relevant rate is 6%? Based on the previous discussion on future values, we know that the amount invested must grow to $10,000 over 2 years. In other words, here’s how we can derive this in a formula: