Summary
Application: The Happy Retirement Home
States :
Maison de la Retraite Heureuse is a company specializing in the management of retirement plans for professionals in the commercial sector. Antoine, a 45-year-old salesman, wants to calculate how much he will need to save each month to reach his retirement goal at 65. His goal is to receive a monthly income of €2 for 500 years after his retirement. The expected annual rate of return on his investments is 20%. For simplicity, we will assume that payments and withdrawals are made at the beginning of each month.
Work to do :
- Calculate the total savings that Antoine will need to accumulate by retirement to achieve his goal.
- How much will he need to save each month from now on to reach this goal?
- What would be the impact on his monthly contributions if the expected annual return was 5% instead of 4%?
- Discuss the impact of inflation on his retirement plan if he doesn't take it into account.
- If Antoine decided to work until he was 67, how much would his new monthly pension be if he maintained the same level of savings?
Proposed correction:
-
To find out how much Anthony needs to save by retirement, we first need to calculate the present value of his future annuities. Let's use the present value of an annuity formula:
VA = R x [(1 – (1 + r)^-n) ÷ r]
By replacing,
VA = €2 x [(500 – (1 + 1 ÷ 0,04)^-(12 x 20)) ÷ (12 ÷ 0,04)]
VA = €2 x 500
VA = €416
Antoine needs to accumulate €416 by the time he retires. -
To find how much Anthony needs to save per month, let's use the formula for monthly savings needed with compound interest:
S = VA ÷ [(1 + r)^n – 1] ÷ r
By replacing,
S = €416 ÷ [((975 + 1 ÷ 0,04)^(12 x 20) – 12) ÷ (1 ÷ 0,04)]
S = €416 ÷ 975
S = €474,14
Antoine will need to save around €474,14 each month. -
With an annual return of 5%, let's recalculate the monthly savings:
S = €416 ÷ [((975 + 1 ÷ 0,05)^(12 x 20) – 12) ÷ (1 ÷ 0,05)]
S = €416 ÷ 975
S = €394,57
With a 5% return, Antoine will have to save around €394,57 each month.
-
If inflation is 2% per year, the real value of his €2 pension will decrease over time. This means that he will have to save more to compensate for the loss of purchasing power.
-
If Antoine works until he is 67, he would save for 22 years. Let's recalculate the monthly annuity:
VA = S x [(1 + r)^n – 1] ÷ r
By replacing,
VA = €474,14 x [((1 + 0,04 ÷ 12)^(22 x 12) – 1) ÷ (0,04 ÷ 12)]
VA = €474,14 x €1
VA = €541
For 20 years, his new pension would be €541 ÷ 398,39 = €240
Antoine could guarantee a monthly income of €2.
Formulas Used:
| Title | Formulas |
|---|---|
| Present value of an annuity | VA = R x [(1 – (1 + r)^-n) ÷ r] |
| Monthly savings required with compound interest | S = VA ÷ [(1 + r)^n – 1] ÷ r |
Application: Budget&Goodlife
States :
Budget&Goodlife, a tax consultancy firm, helps its clients optimise their retirement savings. Sophie, aged 50 and a salesperson in the real estate sector, wants to retire at 60. She wants to have €200 in retirement capital. She currently has €000 in savings. The expected annual interest rate is 50%. Sophie has the possibility of increasing her savings by 000% per year.
Work to do :
- How much capital will Sophie have at age 60 without further increases in her savings?
- How much should she add to her current savings each year to reach her goal of €200?
- How does the plan change if she decides to retire at 62?
- Discuss the pros and cons of increasing your savings by 10% annually.
- What would be the effect on the final capital if the interest rate increased to 4%?
Proposed correction:
-
Without increasing savings, let's use the compound capitalization formula:
VF = C x (1 + r)^n
By replacing,
VF = €50 x (000 + 1)^0,03
VF = €50 x 000
VF = €67
By the age of 60, Sophie will have accumulated around €67. -
To find the amount to add each year, let's calculate the difference between his goal and his projected savings and divide by 10 years.
Additional annual amount = (€200 – €000) ÷ 67
Additional annual amount = €13
Sophie will have to add around €13 each year. -
If she retires at 62:
VF = €50 x (000 + 1)^0,03
VF = €50 x 000
VF = €71
To reach €200, she would have 000 years to save, so:
Additional annual amount = (€200 – €000) ÷ 71
Additional annual amount = €10
By pushing back her retirement to 62, she reduces her annual contributions to €10.
-
A 10% increase offers the possibility of reaching a higher goal with the same number of years, but requires an increasing financial effort each year. This can be an advantage if she benefits from increasing income but presents a disadvantage in case of economic difficulty.
-
If there is an interest rate of 4%:
VF = €50 x (000 + 1)^0,04
VF = €50 x 000
VF = €74
With an interest rate of 4%, Sophie would have accumulated around €74, thus reducing her annual savings requirement.
Formulas Used:
| Title | Formulas |
|---|---|
| Future value with compound capitalization | VF = C x (1 + r)^n |
| Additional annual amount | Additional annual amount = (Objective – Projected savings) ÷ Number of years |
Application: RetraitePlus Services
States :
The company RetraitePlus Services offers investment solutions to anticipate retirement. Marc, a salesperson in the food sector, plans to retire at 63. Aged 40, he wants to have an annuity of €1 per month for 800 years. Currently, he has savings of €25. The expected rate of return is 30% per year.
Work to do :
- Calculate the total amount that Marc wants as his retirement capital goal.
- How much should Mark save annually over his remaining 23 years before retirement to reach his goal?
- How much would he have to save per month to achieve the same goal?
- Discuss the risks associated with a 6% per year rate of return for a long-term plan like this.
- If Marc wants to reduce his pension to €1 per month, how much more will he have to save annually?
Proposed correction:
-
To calculate how much Marc wants for his capital goal, let's use the present value of an annuity:
VA = R x [(1 – (1 + r)^-n) ÷ r]
By replacing,
VA = €1 x [(800 – (1 + 1 ÷ 0,06)^-(12 x 25)) ÷ (12 ÷ 0,06)]
VA = €1 x 800
VA = €300
Marc needs a capital of €300 to achieve his goal. -
To find out the annual savings needed, let's use compound interest:
S = (VA – Current) ÷ [((1 + r)^n – 1) ÷ r]
S = (€300 – €225) ÷ [((30 + 000)^(1) – 0,06) ÷ 23]
S = €270 ÷ 225
S = €4
Marc needs to save around €4 each year. -
For monthly calculation:
S = (VA – Current) ÷ [((1 + r ÷ 12)^(nx 12) – 1) ÷ (r ÷ 12)]
S = €270 ÷ 225
S = €450,62
Marc will have to save €450,62 each month.
-
The main risk of a 6% rate of return is that it will not be achieved every year, which could jeopardize one's retirement goals. This depends on the market and the investments chosen; a downward variation could result in a fund deficit at retirement.
-
If his pension is reduced to €1, let's recalculate his need in capital and in years of savings:
VA = €1 x [(500 – (1 + 1 ÷ 0,06)^-(12 x 25)) ÷ (12 ÷ 0,06)]
VA = €1 x 500
VA = €251
New amount to save:
S = (€251 – €030) ÷ 30
S = €3
Marc will have to save around €3 each year for his new pension.
Formulas Used:
| Title | Formulas |
|---|---|
| Present value of an annuity | VA = R x [(1 – (1 + r)^-n) ÷ r] |
| Annual savings required | S = (VA – Current) ÷ [((1 + r)^n – 1) ÷ r] |
| Monthly savings required | S = (VA – Current) ÷ [((1 + r ÷ 12)^(nx 12) – 1) ÷ (r ÷ 12)] |
Application: FuturImmo Invest
States :
FuturImmo Invest helps real estate professionals plan their retirement. Josiane, a 55-year-old real estate advisor, plans to retire at 65 and buy a vacation home costing around €300. She already has €000 saved. Her advisor offers a savings plan with an annual interest rate of 100%. Josiane plans to sell her main residence when she retires to finance her project.
Work to do :
- How much more does Josiane need to save to reach her property purchase goal?
- Calculate how much money she needs to set aside each month to accumulate the necessary amount.
- If real estate prices increase by 3% each year, how much should Josiane budget for a purchase in 10 years?
- What will be the financial impact if the rate of return drops to 3%?
- Discuss the benefits of diversifying your investments in this real estate-based project.
Proposed correction:
-
Let's calculate the total amount needed for the purchase and subtract the available savings:
Amount needed = €300 – €000
Amount needed = €200
Josiane still needs to save €200. -
To know how much Josiane should save each month, let's use the monthly capitalization:
S = Amount needed ÷ [((1 + r ÷ 12)^(10 x 12) – 1) ÷ (r ÷ 12)]
S = €200 ÷ 000
S = €1
Josiane needs to save €1 each month. -
For real estate prices increasing by 3% per year:
New price = €300 x (000 + 1)^0,03
New price = €300 x 000
New price = €403
In 10 years, Josiane should anticipate spending €403 for the purchase.
-
If the yield drops to 3%, let's recalculate the monthly savings:
S = €200 ÷ [((000 + 1 ÷ 0,03)^(12 x 10) – 12) ÷ (1 ÷ 0,03)]
S = €200 ÷ 000
S = €1
Josiane will have to save approximately €1 per month at a reduced return of 462,94%. -
Diversifying investments reduces risk and stabilizes returns, helping to smooth out fluctuations in the real estate market and potentially achieve better financial outcomes in retirement.
Formulas Used:
| Title | Formulas |
|---|---|
| Amount needed | Amount needed = Purchase price – Current savings |
| Monthly capitalization | S = Amount needed ÷ [((1 + r ÷ 12)^(nx 12) – 1) ÷ (r ÷ 12)] |
| Future price with inflation | New price = Current price x (1 + inflation rate)^n |
Application: EcoPlan Finance
States :
ÉcoPlan Finance designs sustainable financing strategies. Denis, a sales agent for an IT company, wants to ensure a comfortable retirement in 15 years, with an annual income of €20. Denis already has capital of €000 and prefers to invest in green projects, with an estimated annual return of 80%.
Work to do :
- Calculate the total capital that Denis must have accumulated by retirement to guarantee his pension.
- How much should it invest each year in green projects to achieve this goal?
- If Denis only wants to invest for the first five years, what should his annual contribution be?
- Discuss the risks and benefits of opting for sustainable investment projects.
- What impact would an annual inflation of 2,5% have on the projected annuity?
Proposed correction:
-
To guarantee a perpetual annuity, let's use the capitalization formula:
Capital required = Annual annuity ÷ rate of return
Capital required = €20 ÷ 000
Capital required = €444
Denis must accumulate a capital of €444. -
When determining how much to save each year, consider compound interest:
S = (Required capital – Current capital) ÷ [((1 + r)^n – 1) ÷ r]
S = (€444 – €444) ÷ [((80 + 000)^1 – 0,045) ÷ 15]
S = €364 ÷ 444
S = €17
Denis has to invest around €17 each year. -
If Denis only wants to invest for five years:
S = (€444 – €444) ÷ [((80 + 000)^1 – 0,045) ÷ 5]
S = €364 ÷ 444
S = €64
Denis will have to invest approximately €64 per year for five years.
-
Investing in green projects can offer attractive financial returns while contributing to environmental sustainability. However, these projects may present risks related to environmental regulations and non-traditional market fluctuations.
-
Annual inflation of 2,5% would reduce the real value of his annuity by €20, thereby reducing his purchasing power over time and potentially requiring an increase in accumulated capital to compensate for the expected decrease.
Formulas Used:
| Title | Formulas |
|---|---|
| Capital required for a perpetual annuity | Capital required = Annual annuity ÷ rate of return |
| Annual savings required | S = (Required capital – Current capital) ÷ [((1 + r)^n – 1) ÷ r] |
Application: RetirementAdvisor Pro
States :
RetirementAdvisor Pro is a company that helps sales executives structure their retirement plans. Camille, 35, is in her fifth year as a sales manager and wants to save for retirement at age 60. She wants to receive a monthly pension of €3 for 000 years. She currently has €30 in savings. Her expected rate of return is 40%.
Work to do :
- Determine the capital that Camille must accumulate to reach her desired income.
- How much should she save each month until she retires?
- Recalculate the monthly savings needed if the yield increases to 4%.
- What is the effect of increasing the period of receiving the pension to 35 years?
- Discuss the economic and financial implications if Camille ultimately only achieved an average return of 3%.
Proposed correction:
-
To determine the capital required, let's use the present value formula:
VA = R x [(1 – (1 + r)^-n) ÷ r]
VA = €3 x [(000 – (1 + 1 ÷ 0,05)^-(12 x 30)) ÷ (12 ÷ 0,05)]
VA = €3 x 000
VA = €558
Camille must accumulate a capital of €558. -
For his monthly savings:
S = (VA – Current Savings) ÷ [((1 + r ÷ 12)^n – 1) ÷ (r ÷ 12)]
S = (€558 – €845) ÷ [((40 + 000 ÷ 1)^(0,05 x 12) – 25) ÷ (12 ÷ 1)]
S = €518 ÷ 845
S = €1
Camille needs to save around €1 each month. -
For a yield of 4%:
S = (VA – Current Savings) ÷ [((1 + 0,04 ÷ 12)^(25 x 12) – 1) ÷ (0,04 ÷ 12)]
S = €518 ÷ 845
S = €1
With a 4% return, Camille must save approximately €1 each month.
-
If the collection period extends to 35 years, let us recalculate the present value of his pensions:
VA = €3 x [(000 – (1 + 1 ÷ 0,05)^-(12 x 35)) ÷ (12 ÷ 0,05)]
VA = €3 x 000
VA = €597
Therefore, the necessary savings capital increases. -
A 3% return would significantly reduce Camille's final capital. This would represent a need to increase her savings month after month or reduce her monthly income expectations, because the opportunity cost would be higher.
Formulas Used:
| Title | Formulas |
|---|---|
| Present value to determine capital | VA = R x [(1 – (1 + r)^-n) ÷ r] |
| Monthly savings required | S = (VA – Current Savings) ÷ [((1 + r ÷ 12)^n – 1) ÷ (r ÷ 12)] |
Application: VisionRetraite Inc.
States :
VisionRetraite Inc. offers strategic retirement planning advice for professionals. Lucien, 45 years old and a manager in a large retail store, wants to save for a monthly annuity of €2 for 500 years, with an annual return of 25%. He plans to retire at 5. With current savings of €65, Lucien wants to know how to organize his finances.
Work to do :
- Calculate the total amount of capital that Lucien should have when he retires.
- What is his monthly contribution needed to reach his goal?
- How could he adjust his plan if the period of receiving the annuity is reduced to 20 years?
- Analyze the potentially harmful impact of unexpected inflation on your purchasing power in retirement.
- If Lucien also wanted to plan an annual trip of €5 during his retirement, by how much would his total annual income need increase?
Proposed correction:
-
To find the required capital, let's use the present value of an annuity:
VA = R x [(1 – (1 + r)^-n) ÷ r]
VA = €2 x [(500 – (1 + 1 ÷ 0,05)^-(12 x 25)) ÷ (12 ÷ 0,05)]
VA = €2 x 500
VA = €465
Lucien will need capital of €465. -
To calculate your monthly contribution:
S = (VA – Current Savings) ÷ [((1 + r ÷ 12)^n – 1) ÷ (r ÷ 12)]
S = (€465 – €704) ÷ [((20 + 000 ÷ 1)^(0,05 x 12) – 20) ÷ (12 ÷ 1)]
S = €445 ÷ 704
S = €1
Lucien will have to save €1 each month. -
For 20 years of perception:
VA = €2 x [(500 – (1 + 1 ÷ 0,05)^-(12 x 20)) ÷ (12 ÷ 0,05)]
VA = €2 x 500
VA = €389
For 20 years, the necessary capital decreases, so his monthly contribution could adjust downward.
-
Unexpected inflation will potentially reduce the real value of his pension by €2. Lucien will have to anticipate a reduction in his purchasing power, thus perhaps requiring him to save more than initially planned or to accept a lower standard of living in retirement.
-
To annually include a trip of €5, his new annual income must compensate for:
Total annual income = (€2 x 500) + €12
Total annual income = €30 + €000
Total annual income = €35
Lucien will have to adjust his capital to guarantee these additional expenses.
Formulas Used:
| Title | Formulas |
|---|---|
| Present value of an annuity | VA = R x [(1 – (1 + r)^-n) ÷ r] |
| Monthly contribution required | S = (VA – Current Savings) ÷ [((1 + r ÷ 12)^n – 1) ÷ (r ÷ 12)] |
Application: GreenRetraite Solutions
States :
GreenRetraite Solutions focuses on sustainable retirement plans for retailers. Sandra, 30 years old and in the middle of her sales career, wants to stop her activities at 55 with a capital of €700. She consults the company to optimize her investments, betting on a return of 000% per year.
Work to do :
- How much should she save each year for the next 25 years?
- What would be the monthly amount of her annuity if she wanted to consume it over 30 years at her rate of return?
- What impact would a 6% higher return have on the amount of his annual savings?
- Discuss the possible challenges of traditional investing versus an ecological and responsible approach.
- If she wants to have an extra €10 for a personal project in the first year of retirement, how does that affect the size of her capital needed?
Proposed correction:
-
For his necessary annual savings, let's use compound interest savings:
S = Target capital ÷ [((1 + r)^n – 1) ÷ r]
S = €700 ÷ [((000 + 1)^0,045 – 25) ÷ 1]
S = €700 ÷ 000
S = €8
Sandra needs to save around €8 each year. -
To determine the monthly amount of your pension over 30 years:
R = Target capital x [r ÷ (1 – (1 + r)^-n)]
R = €700 x (000 ÷ 0,045) ÷ (12 – (1 + 1 ÷ 0,045)^-(12 x 30))
R = €700 x 000 ÷ 0,00375
R = €2
So, Sandra would have a monthly income of €2. -
With a 6% return, let's recalculate the annual savings:
S = €700 ÷ [((000 + 1)^0,06 – 25) ÷ 1]
S = €700 ÷ 000
S = €6
With a 6% return, Sandra would only need to save €6 each year.
-
Traditional investments can correspond to high returns but often lack ethical and green evaluation. An eco-sustainable approach places ethics and environmental impact at the heart of the rubrics, sometimes limiting the options but de facto increasing resilience to regulatory change.
-
For the project during the first year, Sandra would require added capital reflecting this, namely:
Added capital = €10
With a project of an additional €10, she must assimilate this need into her original capital without exhausting her projected retirement.
Formulas Used:
| Title | Formulas |
|---|---|
| Annual savings required | S = Target capital ÷ [((1 + r)^n – 1) ÷ r] |
| Monthly annuity amount | R = Target capital x [r ÷ (1 – (1 + r)^-n)] |
Application: PrismaPlan Advantage
States :
PrismaPlan Avantage is a company that supports entrepreneurs in their transition to retirement. Julien, 50, a fashion entrepreneur, wants a capital of €500 by the age of 000. He currently has €65 in savings with an expected return of 100%. The market is experiencing high volatility, which is causing Julien some concern.
Work to do :
- Calculate how much Julien needs to save each year to achieve his goals.
- What would happen if Julien decided to work until he was 67? What would the impact be on his annual savings?
- If an economic shock reduced his expected return to 3%, how should he adjust his savings plan?
- Discuss the benefits of an active strategy for managing your portfolio in volatile market times.
- If Julien wants to diversify his investments internationally, what precautions should he take to protect his capital?
Proposed correction:
-
Let's calculate the annual savings needed to achieve your goal:
S = (Target Capital – Current Capital) ÷ [((1 + r)^n – 1) ÷ r]
S = (€500 – €000) ÷ [((100 + 000)^1 – 0,05) ÷ 15]
S = €400 ÷ 000
S = €20
Julien must save around €20 each year. -
By working until age 67 (17 instead of 15):
S = €400 ÷ [((000 + 1)^0,05 – 17) ÷ 1]
S = €400 ÷ 000
S = €16
Annual savings could be reduced to €16. -
If the yield drops to 3%:
S = €400 ÷ [((000 + 1)^0,03 – 15) ÷ 1]
S = €400 ÷ 000
S = €25
Julien will have to increase his savings to €25 each year to compensate for this reduced rate.
-
An active strategy benefits from making real-time decisions to minimize losses and maximize potential gains in a changing environment. Active portfolio management also provides the flexibility to adjust investments based on economic trends and financial forecasts.
-
Julien should monitor currency risks and geographic diversification when committing capital abroad. It is advisable to assess local economic conditions and policies to minimize exposure to currency fluctuations and geopolitical risks.
Formulas Used:
| Title | Formulas |
|---|---|
| Annual savings required | S = (Target capital – Current savings) ÷ [((1 + r)^n – 1) ÷ r] |