Question 1
Proficient-level: “Describe each component of a cash flow timeline with an initial cash inflow and a future cash outflow. Indicate which cash flows should be represented as negative or positive.”
Distinguished-level: Explain the purpose of including positive and negative amounts on the cash flow timeline.
The cash flow timeline serves as a visual representation of cash inflows and outflows over a specific period. It provides a structured depiction of the timing and magnitude of cash flows, with the timeline displayed horizontally. Above the timeline, the periods corresponding to the cash flows are indicated, which could be quarterly, semi-annually, or annually, depending on the context. The interest rate for each period may also be displayed to provide further context and analysis.
The cash flows themselves are represented horizontally below the timeline. Positive numbers represent cash inflows, indicating the receipt of money, while negative numbers describe cash outflows, showing the expenditure or contribution of funds. By utilizing this visual representation, the cash flow timeline allows for a clear understanding of the timing and direction of cash flows.
BUS 3062 Unit 2 Assignment 1 Time Value of Money Single Cash Flows
Including negative and positive amounts on the timeline is crucial as it identifies where money has been contributed or lost (negative) and where money has been earned (positive). This practice aligns with the fundamental accounting principles of debits and credits, where positive amounts signify credits or additions to an account, and negative amounts represent debits or subtractions from an account.
By utilizing a cash flow timeline, individuals and businesses can gain insights into their financial activities over time. It provides a comprehensive overview of cash flows, allowing for the identification of periods of high or low cash inflows or outflows. This information is valuable for financial planning, budgeting, and decision-making processes, as it helps assess an individual’s or organization’s overall financial health and performance.
In summary, the cash flow timeline is a powerful tool that visually organizes and presents cash inflows and outflows over a specific period. By representing cash flows horizontally below the timetable and incorporating positive and negative numbers, it facilitates a clear understanding of when and where money has been contributed or earned. This enables individuals and businesses to analyze their financial activities, make informed decisions, and effectively manage their cash flow.
Question 2
Proficient-level: “What is the relationship between present and future value?”
Distinguished-level: Explain why a dollar received today is worth more than one received a year from now.
The measure that connects present values to future values is the interest rate, denoted as “i.” A current value can be moved forward by applying interest to determine the future value. The formula for future importance in “N” years is FV = PV × (1 + i)^N. Similarly, a future value can be discounted back to the present by dividing the future value by the interest factor.
The future value represents the value of an investment after a certain period, while the present value indicates the worth of the future cash flow in today’s terms. The future value calculation considers the current value and adds the interest rate for the given period. The equation expresses this relationship: FV = PV × (1 + i). FV denotes the future value, and the subscript “1” represents the period. PV represents the present value of the account or investment.
A dollar received today holds greater value than one received a year from now due to factors such as interest rates and inflation. This assumption is based on the concept of the time value of money, which acknowledges that the value of money changes over time.
Question 3
Proficient-level: “How do changes in interest rates affect present values?”
Distinguished-level: Explain how changes in interest rates impact future values.
Higher interest rates lead to larger future values, while lower interest rates result in larger present values. Interest rates and present deals have an inverse relationship. When expected interest rates increase, present values decrease because future values are discounted at a higher rate, resulting in smaller current values. Conversely, when expected interest rates fall, present values increase because future values are ignored at a lower rate.
Question 4
Proficient-level: “If $150 is deposited today, how much will be in the savings account after 11 years with a 7 percent annual interest rate?” Recalculate the savings account balance using a 6 percent interest rate and an 8 percent interest rate.
Distinguished-level: Describe the relationship between changes in interest rates and subsequent changes in future values.
Based on the given information, when an initial deposit (PV) of $150 is made with an interest rate (I) of 7% and a period (N) of 11 years, the future value (FV) of the investment would be $315.73. This means that after 11 years, the account balance would grow to $315.73, taking into account the initial deposit and the accumulated interest.
BUS 3062 Unit 2 Assignment 1 Time Value of Money Single Cash Flows
Let’s delve into recalculating the future value using a 6% interest rate. Using the same formula, FV = PV × (1 + i)^N, where PV represents the initial deposit ($150), I represents the interest rate as a decimal (0.06 for 6%), and N represents the period (11 years), we can calculate the future value.
By substituting the values into the formula, we find:
FV = $150 × (1 + 0.06)^11
FV = $150 × (1.06)^11
FV ≈ $247.84
Thus, if the interest rate is reduced to 6%, the savings account balance after 11 years would be approximately $247.84. This shows that a lower interest rate leads to a smaller growth in the account balance over time.
Moving on to the recalculation with an 8% interest rate, we use the same formula with the updated values. In this case, PV remains $150, i is now 0.08 (8% as a decimal), and N remains 11:
FV = $150 × (1 + 0.08)^11
FV = $150 × (1.08)^11
FV ≈ $361.38
With an 8% interest rate, the savings account balance after 11 years would be approximately $361.38. This demonstrates that a higher interest rate leads to greater growth in the account balance over time.
In summary, the relationship between changes in interest rates and future values is significant. When the interest rate increases, the future value of an investment also increases. A higher interest rate allows for more substantial growth through compounding, resulting in a larger future value.
On the other hand, when the interest rate decreases, the future value also decreases. Lower interest rates lead to slower growth and less compounding, resulting in a smaller future value. Therefore, changes in interest rates directly impact the development and accumulation of future discounts in an investment or savings account.
Question 5
Calculating the third-year future value
We can calculate the future value using the given interest rates of 8% in the first year, 6% in the second year, and 5.5% in the third year.
FV = PV × (1 + i) × (1 + j) × (1 + k)
FV = $350 × (1 + 0.08) × (1 + 0.06) × (1 + 0.055)
FV = $350 × 1.08 × 1.06 × 1.055
FV ≈ $424.68
Therefore, the third-year future value would be approximately $424.68.
It is important to note that the future value is not calculated as the average of the annual interest rates because the compounding effect needs to be considered. Each year, the interest is applied to both the original amount and the accumulated interest from previous years. As a result, the compounding effect influences the future value, which leads to a higher value than simply taking the average of the interest rates. BUS 3062 Unit 2 Assignment 1 Time Value of Money Single Cash Flows
Question 6
Calculating the present value
Given a $850 payment to be made in 10 years and a discount rate of 12%, we can calculate the present value.
PV = FV / (1 + i)^N
PV = $850 / (1 + 0.12)^10
PV = $850 / 3.10585
PV ≈ $273.68
Recalculating the present value using an 11% discount rate:
PV = $850 / (1 + 0.11)^10
PV = $850 / 3.14554
PV ≈ $299.36
Recalculating the present value using a 13% discount rate:
PV = $850 / (1 + 0.13)^10
PV = $850 / 2.853116
PV ≈ $297.32
The relationship between changes in interest rates and present values is inverse. As the discount rate increases, the current value decreases. Conversely, as the discount rate decreases, the present value increases. This relationship is because higher discount rates discount future cash flows more heavily, leading to a lower current value. In comparison, lower discount rates result in less discounting and a higher present value.
Question 7
To calculate the annual rate of return, we are given a $5,000 investment that grows to $9,500 in five years. Let’s determine the annual rate of return in each scenario.
First, using the formula:
Future Value (FV) = Present Value (PV) × (1 + i)^N
We can substitute the given values into the formula:
$9,500 = $5,000 × (1 + i)^5
To isolate (1 + i)^5, divide both sides by $5,000:
(1 + i)^5 = $9,500 / $5,000
(1 + i)^5 = 1.9
Taking the fifth root of both sides, we get:
1 + i = 1.9^(1/5)
1 + i = 1.137
Subtracting one from both sides, we find:
i = 1.137 – 1
I ≈ 0.137 or 13.7%
Therefore, the annual rate of return for five years is approximately 13.7%.
Now, let’s recalculate the rate of return, assuming the growth occurred in four years:
$9,500 = $5,000 × (1 + i)^4
Dividing both sides by $5,000:
(1 + i)^4 = $9,500 / $5,000
(1 + i)^4 = 1.9
Taking the fourth root of both sides, we obtain:
1 + i = 1.9^(1/4)
i = 1.148 – 1
I ≈ 0.148 or 14.8%
Therefore, the annual rate of return for four years is approximately 14.8%.
Lastly, let’s calculate the rate of return assuming the growth occurred in six years:
$9,500 = $5,000 × (1 + i)^6
Dividing both sides by $5,000:
(1 + i)^6 = $9,500 / $5,000
(1 + i)^6 = 1.9
Taking the sixth root of both sides:
1 + i = 1.9^(1/6)
i = 1.122 – 1
I ≈ 0.122 or 12.2%
Hence, the annual rate of return for six years is approximately 12.2%.
The relationship between the amount of time and the annual rate of return can be observed in these calculations. As the period increases, the annual rate of return decreases. In the given examples, the rate of return is 13.7% for five years, 14.8% for four years, and 12.2% for six years. This relationship occurs due to compounding. A more extended period allows for more compounding, resulting in a higher future value and a lower annual rate of return. BUS 3062 Unit 2 Assignment 1 Time Value of Money Single Cash Flows
References
Cornett, M. M., Adair, T. A., & Nofsinger J. (2016). M: Finance (3rd ed.). New York, NY: McGraw-Hill.
