Chapter 4.11® - Discounted Cash Flow Valuations - Future Value of Multiple Cash Flows & Designing the Cash Flows Timeline

• 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

So far in the time value of money chapter, we have focused solely on the future value of a lump sum payment or present value of a single cash flow expected to be generated in the future. However more often than not, these valuations include multiple cash flow payments in the future. How do we account for those? For instance, if McDonalds is planning to open a new store, there will probably be a large cash outflow (investment) now and lots of cash inflows for many years in the future. In this chapter, we begin to explore investments of this nature.

Future Value of Multiple Cash Flows

Imagine you deposit $100 today in an interest bearing account paying 7%. In one year, you will contribute an additional $100. How much will you have in two years? This problem is easy to solve; at the end of the first year, you will have $107 plus the second $100 you deposit for a total of $207. Then, you will leave this $207 on deposit at 7% for the 2nd year. At the end of the 2nd year, this money will be worth:

2nd Year Worth = $207 x 1.07 = $221.49

1) Cash Flows Timeline

2) Calculating the Future Value

The table above is a financial time line that literally draws the process of calculating future value of these two $100 deposits in a format that is understandable by anyone. Drawing diagrams like this for solving complicated cash flows & future cash flow questions can be very helpful.

Here is how the set of calculations look like:

Here is how this data looks in graphical format:

This graph above shows the growth of $100 in year 1 to $221.49 in 2 years after making another $100 deposit in year 2, and earning an annual rate of return of 7% compounded annually.

From the diagrams, the first cash flow occurs today, which we have labelled as Time 0. This is why we have put $100 as the first contribution under Column 0 because that’s when it was deposited. To make things easy for you, always start financial time lines with a 0 and NOT 1. The second cash flow occurs in one year from today, so we wrote this down at the point labelled Time 1. At the end of Year 2, we receive interest on our compounded or accumulated balance of $207 and the calculation is:

We could have made these calculations a little easier because when we calculated the future value of the two $100 deposits, we simply calculated the balance as of the beginning of each year, then rolled that amount forward to the next year. The quicker way to do this is, the first $100 was on deposit for 2 years, at a deposit rate of 7%, thus its future value is calculated as (using the future value discount factor):

The second $100 is on deposit for 1 year at 7%, thus its future value is:

The total future value is equal to the sum of these two future value calculations: $114.49 + $107 = $221.49

Thus, from this example, there are two ways to calculate future values for multiple cash flows: