MATH HUB Flashcards
What is the formula used to calculate the PRESENT VALUE of money?
future value ÷ (1 + interest rate) x number of years = present value
$5,000 ÷ (1 + 0.03)5 = $4,313.04
- The number of years is multiplied by itself (ex. 5 years = 3 x 3 x 3 x 3 x 3)
What is the formula used to calculate the FUTURE VALUE of money?
Present value x (1 + interest rate) x number of years = future value
$5,000 × (1 + 0.03)5 = 5,796.37
- The number of years is multiplied by itself (ex. 3 x 3 x 3 x 3 x 3)
Which bond has the higher rate of return?
- A GMC Bond purchased 3 years ago with an annual nominal return of 5.45%, sold last year when the inflation rate was 2.35%.
- A FPW Bond purchased 2 years ago when its annual nominal return was 5.55% is being this sold year, when the inflation rate is 2.45%.
They are equal:
GMC bond 5.45% - 2.35% = 3.1%
FPW bond 5.55% - 2.45% = 3.1%
BK would like to buy a T-Bill that gives him a real return rate of 4.32%. The rate of inflation is 3.02%.
What does the annual nominal rate need to be?
Calculation:
4.32% + 3.02% = 7.34% (can round down to 7.3)
What is Present Value ?
- Present value works backwards from a future date. It answers the question of how much is needed now to achieve a future savings goal.
- Used to help a person determine how much needs to be saved today to yield specified retirement savings at a future date.
- PV = FV ÷ (1 + i)n
Marko wants to go on a $3,000 trip in 3 years. He is starting to save now by investing his money in a guaranteed savings account with an annual interest rate of 2.4%.
How much money does Marko need to put aside today in order to reach his savings goal?
PV = 3,000 ÷ (1+0.024)3 = $2,793.97
Audrey invested her money in a very profitable company 7 years ago. The company has grown and she has managed to have a 13.4% interest rate annually. Audrey would like to withdraw her earnings of $55,670.
How much money did Audrey start with?
PV =55,670 ÷ (1+0.134)7 = $23,085
Frederick and Sally are preparing for the arrival of their fourth child. They want to buy a bigger house by the end of the year. Frederick and Sally have been saving for 6 years at an annual interest rate of 7.25%, and have a total of $70,000 for their move.
How much did they start with?
PV = 70,000 ÷ (1+0.0725)6 = $45,995.38
Spencer recently started a non-profit organization to support animal rescues. He needs to have $10,000 to cover the organization’s startup costs. He received $4,000 in donations and invested it in a high-risk fund for 5 years. The fund has an annual interest rate of 18%.
Will Spencer meet his goal?
No, he will not meet his goal.
FV = 4,000 × (1+0.18)5
= 4,000 × 2.287758
= $9,151.03
Donovan has three children aged 4, 6 and 7. He is opening a savings account for each of them to use for university when they turn 17. The accounts have interest rates of 5%, 7%, and 10% respectively. He will be investing $5,500, $6,500 and $7,500 respectively
How much will Donovan have saved when his children are ready for university?
Youngest Child:
4yrs old. $5,500 invested at 5% interest until age 17 (13 years).
FV = 5,500 × (1+0.05)13
= 5,500 × (1.05 × 1.05 × 1.05 × 1.05 × 1.05 × 1.05 × 1.05 × 1.05 × 1.05 × 1.05 × 1.05 × 1.05 × 1.05)
= 5,500 × 1.885649
= $10,371.07
Middle Child :
6yrs old. $6,500 invested at 7% interest until age 17 (11 years).
FV = 6,500 × (1+0.07)11
= 6,500 × (1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07 × 1.07)
= 6,500 × 2.104852
= $13,681.54
Oldest Child:
7yrs old. $7,500 invested at 10% interest until age 17 (10 years).
FV = 7,500 × (1+0.1)10
= 7,500 × (1.1 × 1.1 × 1.1× 1.1× 1.1 × 1.1× 1.1× 1.1× 1.1 × 1.1)
= 7,500 × 2.593742
= $19,453.07
Total = 10,371.07 + 13,681.54 + 19,453.07 = $43,505.68
Lachlan is 55 and would like to retire at age 65. He invested $120,000 in a low risk account with an annual interest rate of 4%. He also had another $25,000 in a high interest account with an annual interest rate of 8%.
How much will Lachlan have when he retires?
Investment 1:
FV = 120,000 × (1+ 0.04)10
= 120,000 × (1.04 × 1.04 × 1.04 × 1.04 × 1.04 × 1.04 × 1.04 × 1.04 × 1.04 × 1.04)
= 120,000 × (1.480244)
= $177,629.31
Investment 2:
FV = 25,000 × (1 + 0.08)10
= 25,000 × (1.08 × 1.08 × 1.08 × 1.08 × 1.08 × 1.08 × 1.08 × 1.08 × 1.08 × 1.08)
= 25,000 × 2.158925
= $53,973.12
Total = 177,629.31 + 53,973.12 = $231,602.43
Returns & Guarantees
Fatima purchased a segregated fund contract with a 75% maturity guarantee for $175,000. The investment has fluctuated over the past ten years and is currently valued at $156,800.
If she decides to withdraw her money, what amount will she receive?
[Ref. 1.3.1.4]
Fatima will receive $156,800.
175,000 × 75% = $131,250
$156,800 is greater, therefore Fatima will receive $156,800.
[Ref. 1.3.1.4]
Returns & Guarantees
Fatima purchased a segregated fund contract with a 75% maturity guarantee for $175,000. The investment has fluctuated over the past ten years and is currently valued at $156,800.
Instead of withdrawing her investment, Fatima decides to keep her segregated fund contract. She reset her contract at $185,000, with a 75% maturity guarantee for another 10 years. It is now valued at $195,000.
What is her new guarantee?
The new guarantee would be $138,750.
185,000 × 75% = $138,750
The new guarantee amount applies to the reset value, not the current value.
[Ref. 1.3.1.4]
Returns & Guarantees
Tania purchased a segregated fund with an 85% maturity guarantee for $65,000
If the value of the contract decreases, what is the maximum amount she could lose?
The maximum Tania could lose is $9,750.
65,000 - (65,000 × 85%)
= 65,000 - 55,250
= $9,750
[Ref. 1.3.1.4]
Returns & Guarantees
Arnold’s parents invested $35,000 in a segregated fund contract 10 years ago with an 85% maturity guarantee. The fund is now worth $80,000.
If Arnold keeps his money invested in the fund, resets the value, plus invests another $10,000, what is the new value of Arnold’s maturity guarantee?
The value of Arnold’s maturity guarantee after the reset and additional investment is $76,500.
(80,000 + 10,000) × 85% = $76,500
The value of Arnold’s maturity guarantee after the reset and additional investment is $76,500.
[Ref. 1.3.1.4]
Interest Rate
Chris is a new investor. He is a little anxious because of the constant changes in the market. He decides to play it safe and invests $14,000 in low risk bonds. He purchases a bond with a 3.2% coupon rate that has two payment dates.
How much interest will Chris receive per payment date?
[Ref. 1.3.4.4]
14,000 × 3.2%= $448 = $224 per payment date
[Ref. 1.3.4.4]
Interest Rate
Ruben plans to go on a 3-month vacation to St. Lucia. He needs $6,000 to pay for all the excursions he wants to go on.
A year ago, he invested $5,600 for his trip in a GIC with an interest rate of 4.67%.
Will he have enough for the trip ?
No, Ruben will not have enough. He will have $5,861.52
5,600 + (5,600 × 4.67%)
= 5,600 + 261.52
= $5,861.52
[Ref. 1.3.4.4]
Interest Rate
Amelia received a birthday gift of $500 from her uncle three years ago. She invested the money into a GIC.
For each of the last three years, the GIC rate has been 1.3%, 2.1% and 1.7%.
How much has Amelia earned in the last three years?
Amelia has earned $25.50
Year 1: 500 × 1.3% = $6.50
Year 2: 500 × 2.1% = $10.50
Year 3: 500 × 1.7% = $8.50
Total: 6.50 + 10.50 + 8.50 = $25.50
Mutual Fund Returns
Angela invested $3,000 in a TFSA last year. Over the year her investment grew through distributions and she ended the year with $3,600 in her TFSA.
What was Angela’s return for the year?
(3,600 - 3,000) ÷ 3,000 = 0.20
0.20 × 100 = 20%
REF. 1.3.7.4
Mutual Fund Returns
Desmond holds an annual sporting competition which usually makes a profit. Desmond invests this amount. Last year, Desmond earned $3,200 from the competition. By the end of the year, the invested earnings were valued at $4,050.
What was Desmond’s return?
(4,050 - 3,200) ÷ 3,200 = 0.2656
0.2656 × 100 = 26.56%
Mutual Fund Returns
Nina received some money as a gift and invested the money in a segregated fund. Over the past 5 years, Nina’s investment has grown from $580 to $700. She was hoping to get an interest rate of at least 20%.
Did Nina meet her interest rate goal?
Yes, Nina met her goal of 20%:
(700 - 580) ÷ 580 = 0.2068
0.2068 ×100 = 20.68%
Mutual Fund Returns
Luc has three investment statements:
Investment A: $5,890 invested; ended the year with $7,450
Investment B: $3,457 invested; ended the year with $5,670
Investment C: $6,770 invested; ended the year with $6,600
Which had the best interest rate?
Investment statement B had the best interest rate.
Investment A:
(7,450 - 5,890) ÷ 5,890 = 0.2648
0.2648 × 100 = 26.49%
Investment B:
(5,670 - 3,457) ÷ 3,457 = 0.64015
0.64015 × 100 = 64.02%
Investment C:
(6,600 - 6,770) ÷ 6,770 = -0.0251
0.0251 × 100 = -2.51% (represents a loss)
Capital Gains & Losses
Devohn invested $3,000 in a corporate class fund that reports unit increases as capital gains or losses. The account is non-registered and at the end of the year, her investment grew to $3,400. Her marginal tax rate is 20%.
How much tax does Devohn owe at the end of the year?
Devohn’s tax owing will be $40.
Capital gain = 3,400 - 3,000 = $400
50% taxable = 400 × 50% = $200
Marginal tax rate = 20%
Tax owing = 200 × 20% = $40
Capital Gains & Losses
Celia owns several investment accounts and just received her tax slip for the year.
The slip issued shows the $200 allocation as $125 interest and $75 capital gain.
Based on a marginal tax rate of 36%, how much tax is due?
Celia’s tax owing will be $58.50.
125 × 36% = $45
75 × 50% = $37.50
37.50 × 36% = $13.50
$45 + $13.50 = $58.50
