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
Capital Gains & Losses
Karl made three deposits into his segregated fund over the past 5 years. He opened the account with $4,080 and then deposited $341, $560 and $890. The value of his account is now $5,112.
What is the capital gain or loss that Karl will realize if he withdraws his money this year?
Karl will realize a loss of $759 if he withdraws his money this year.
5,112 - (4, 080 + 341 + 560 + 890) = -$759
Capital Gains & Losses
Conrad owns three segregated funds within his contract. Each fund has changed in value over the year.
If the funds only report changes in value as capital gains and losses, what will Conrad’s account statement show for this contract?
Conrad’s account statement will show $1,010 in capital gain.
(4,060 - $4,570) + (3,900 - 3,080) + (9,100 - 8,400)
= -510 + 820 + 700
= $1,010 in capital gain
Capital Gains & Losses
Conrad owns three segregated funds within his contract. Each fund has changed in value over the year.
Conrad’s marginal tax rate is 15%. How much tax does he have to pay?
Conrad will have to pay $75.75 in tax.
(1,010 × 50%) × 15% = $75.75 income tax
Front-End Sales charge
Alisha invests in a segregated fund. Her advisor explained to her that there is a 3% front-end load (FEL) sales charge on her investment. She deposits $3,500.
What is Alisha’s FEL sales charge?
Alisha’s FEL sales charge will be $105.
3,500 × 3% = $105
REF. 2.3.2.1
Front-End Sales charge
Andrew just learned that his new segregated fund contract will have a 2.5% front-end load (FEL) sales charge. He is planning to invest $7,800.
How much will go into his segregated fund?
$7,605 will go into Andrew’s segregated fund.
7,800 - (7,800 × 2.5%)
= 7,800 - 195
= $7,605
Front-End Sales charge
Gary invested $5,500 in a segregated fund contract with a 2.75% front-end load (FEL) sales charge for the initial and each subsequent deposit. Over the past 5 years, he has deposited another $690, $560, $780, $1,900 and $605 into the same account.
Calculate the total amount of FEL sales charges Gary has paid since he opened the account.
Gary’s FEK sales charges since he opened the account total $275.96.
(5,500 × 2.75%) + (690 × 2.75%) + (560 × 2.75%) + (780 × 2.75%) + (1,900 × 2.75%) + (605 × 2.75%)
= 151.25 + 18.975 + 15.4 + 21.45 + 52.25 + 16.6375
= $275.96
