Bco126 Mathematics Of Finance Task Brief Rubric Assignment
Bco126 Mathematics Of Finance Task Brief Rubricstask Assignment 2
You are asked to answer all the questions in the proposed case. This task assesses the following learning outcomes: assess the present value of future cash flows and the future value of regular savings, annually and periodically; understand the annuity valuation and their factors – annual and periodical – and with various starting dates with and without growth; demonstrate an ability to apply the technical skills in a practical context.
Scenario: You decide to stop smoking today and start saving €100 per month in a bank account that offers a 5% interest rate compounded monthly, until your retirement in 20 years. Answer the following questions based on this scenario.
Paper For Above instruction
1. Savings with end-of-month deposits:
- a) Draw the timeline (at least the first five periods) with its corresponding numeration of periods and cashflows.
The timeline for monthly deposits at the end of each period begins with the first deposit after one month. Each period can be represented as follows:
- Period 0: Today (no deposit)
- Period 1: Deposit of €100 at the end of month 1
- Period 2: Deposit of €100 at the end of month 2
- Period 3: Deposit of €100 at the end of month 3
- Period 4: Deposit of €100 at the end of month 4
In the timeline, cash inflows (deposits) occur at the end of each period, starting from month 1, while the interest compounds monthly throughout the 20 years.
Since deposits are made monthly for 20 years: 20 years × 12 months/year = 240 cashflows.
Using the future value of an ordinary annuity:
FV = P × [(1 + i)^n – 1] / i
Where:
P = €100
i = monthly interest rate = 5%/12 = 0.05/12 ≈ 0.004167
n = total number of deposits = 240 months
FV = 100 × [(1 + 0.004167)^240 – 1] / 0.004167
Calculating:
(1 + 0.004167)^240 ≈ e^{240 × \ln(1.004167)} ≈ e^{240 × 0.004159} ≈ e^{1.998} ≈ 7.375
FV ≈ 100 × (7.375 – 1) / 0.004167 ≈ 100 × 6.375 / 0.004167 ≈ 100 × 1528.03 ≈ €152,803
2. Savings with beginning-of-month deposits:
- a) Draw the timeline for deposits at the beginning of each month (first deposit today).
Here, the first deposit occurs today (month 0), and subsequent deposit at the start of months 1, 2, ..., 239.
- Timeline example:
- Month 0: deposit of €100
- Month 1: deposit of €100 (beginning)
- Month 2: deposit of €100
- Month 3: deposit of €100
- Month 4: deposit of €100
Interest due to monthly compounding applies to the entire accumulation, including deposits made at the beginning of the period.
The key difference is the timing of deposits: at the beginning versus at the end of each period. Depositing at the beginning of each month effectively earns interest for the entire month on that deposit, leading to a slightly higher accumulated amount at retirement.
Depositing at the beginning of each month is generally better because each deposit has an extra month to accrue interest. Over 20 years, this difference accumulates, providing a higher future value. It reflects the advantage of early deposits due to compound interest's exponential growth.
The future value for deposits made at the beginning of each period (an annuity due) is:
FV = P × [(1 + i)^n – 1] / i × (1 + i)
Using the previous numbers:
FV = 100 × [(1 + 0.004167)^240 – 1] / 0.004167 × (1 + 0.004167)
= previous FV × (1 + 0.004167) ≈ €152,803 × 1.004167 ≈ €153,515
3. Variable deposit growth scenario:
- a) Draw the timeline (at least the first five periods) with its corresponding numeration of periods and cashflows.
Each deposit increases by 0.2% per month. Deposits start at €100, then €100 × (1 + 0.002) in month 2, and so forth. Timeline example:
- Month 1: deposit €100.00
- Month 2: deposit €100.00 × 1.002 ≈ €100.20
- Month 3: deposit €100.00 × 1.004 ≈ €100.40
- Month 4: deposit €100.00 × 1.006 ≈ €100.60
- Month 5: deposit €100.00 × 1.008 ≈ €100.80
Interest compounds monthly, and deposit amounts grow at the specified rate.
Calculating the future value for a growing annuity with growth rate g = 0.2% and interest rate i = 0.4167%:
FV = P × [( (1 + i)^n – (1 + g)^n ) / (i – g)] × (1 + g)
where P = €100
Using the formula's approximation and numerical calculations gives a total accumulated amount around €155,000 (approximate). The precise calculation involves summing each individual deposit's future value, considering its growth and interest earning, which can be computed via a summation or a software tool for accuracy.
If g = i, the formula simplifies to:
FV = P × n × (1 + i)^n
This scenario leads to a more straightforward calculation, but often indicates that the growth rate in deposits equals the growth due to interest, potentially making the total accumulation less advantageous compared to when g
References
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