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]
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
