B.12 Time value of money concepts and calculations

Question 1

Which of the following best defines the time value of money?

A. The principle that a dollar received today is worth more than a dollar received in the future
B. The principle that a dollar received in the future is worth more than a dollar received today
C. The principle that a dollar received today is worth the same as a dollar received in the future
D. The principle that a dollar received today is worth less than a dollar received in the future

Answer 1

A. The principle that a dollar received today is worth more than a dollar received in the future

The time value of money is the concept that money today is worth more than the same amount of money in the future because of its earning potential. This concept is based on the idea that money can be invested today to earn interest or other returns, so a dollar received today can be worth more than a dollar received in the future.

B.12 Time value of money concepts and calculations

Question 2

Which of the following is the formula for calculating the future value of an investment?

A. FV = PV × (1 + r)^n
B. FV = PV / (1 + r)^n
C. FV = PV + (PV × r × n)
D. FV = PV × (1 - r)^n

Answer 2

A. FV = PV × (1 + r)^n

The correct formula for calculating the future value (FV) of an investment is FV = PV × (1 + r)^n, where PV represents the present value of the investment, r is the annual interest rate, and n is the number of periods the money is invested for. This formula helps in determining how much an investment made today will be worth in the future, assuming a constant rate of return compounded over n periods.The future value (FV) of an investment is calculated using the formula FV = PV x (1 + i)^n, where PV is the present value of the investment, i is the interest rate, and n is the number of periods.

B.12 Time value of money concepts and calculations

Question 3

Which of the following is the formula for calculating the present value of an investment?

A. PV = FV / (1 + r)^n
B. PV = FV × (1 + r)^n
C. PV = FV + (r × n)
D. PV = FV - (r / n)

Answer 3

A. PV = FV / (1 + r)^n

The formula for calculating the present value (PV) of an investment is PV = FV / (1 + r)^n, where FV represents the future value of the investment, r is the discount rate or interest rate, and n is the number of periods until the payment or cash flow occurs. This formula helps determine how much a future sum of money is worth in today’s terms by discounting it at the given rate for the specified number of periods.

B.12 Time value of money concepts and calculations

Question 4

An investor is considering two investments, both of which will pay $1,000 in five years. One investment pays interest annually at a rate of 5%, while the other pays interest annually at a rate of 7%. Which of the two has the highest present value?

A. The investment that pays 5% interest annually
B. The investment that pays 7% interest annually
C. Both investments have the same present value
D. Insufficient information to determine

Answer 4

A. The investment that pays 5% interest annually.

The present value (PV) of an investment is calculated using the formula PV = FV / (1 + r)^n. Here, FV = $1,000, n = 5 years, and r is the interest rate. For the investment that pays interest at a rate of 5%, the present value is higher because the discount rate (r) is lower. When using a lower discount rate, the present value calculated will be higher compared to using a higher discount rate (7% in this case). Thus, the investment with the 5% interest rate has a higher present value, indicating it is more valuable today compared to the one offering a 7% rate.

B.12 Time value of money concepts and calculations

Question 5

Tom is considering an investment offer that quotes an annual interest rate of 8% compounded quarterly. He wants to compare this investment to another that offers a 7.9% interest rate compounded monthly. To make an accurate comparison, Tom needs to calculate the effective annual rate (EAR) for both investments. Which of the following best defines the effective annual rate (EAR)?

A. The annualized interest rate for a loan or investment that includes compound interest.
B. The annualized interest rate for a loan or investment that does not include compound interest.
C. The interest rate charged on a loan or investment on a monthly basis.
D. The interest rate charged on a loan or investment on a daily basis.

Answer 5

A. The annualized interest rate for a loan or investment that includes compound interest.

The effective annual rate (EAR) is the annual rate of interest that accounts for the effect of compounding over a given period. It provides a true reflection of the financial costs or earnings associated with a loan or investment, thereby allowing a more accurate comparison between different financial products with various compounding intervals. Option A correctly describes the EAR as it includes the compound interest effect, differentiating it from nominal rates that do not take compounding into consideration.

B.12 Time value of money concepts and calculations

Question 6

An investor deposits $5,000 today into a savings account that pays 3% interest annually. Approximately, much will the investor have in the account in 10 years?

A. $6,500
B. $6,700
C. $6,800
D. $6,900

Answer 6

B. $6,700

**Using the HP 10bII+ calculator, perform the following steps: **

Clear previous data: Clear previous calculation/data to avoide any errors.
Ensure Calculator is set to calculate 1 period per year: Press 1 > Gold Key > P/YR
Enter the number of periods: Press 10 then n to set the number of years to 10.
Enter the interest rate: Press 3 then i to set the annual interest rate to 3%.
Enter the present value: Press 5000 then PV to set the present value to $5,000 (the initial deposit).
Compute future value: Press FV to calculate the future value.

B.12 Time value of money concepts and calculations

Question 7

Which of the following is the formula for calculating the present value of an annuity?

A. PV = C / i
B. PV = C x (1 + i)n / i
C. PV = C x (1 - (1 + i)-n) / i
D. PV = C x (1 + i)n

Answer 7

C. PV = C x (1 - (1 + i)-n) / i

The present value (PV) of an annuity is calculated using the formula PV = C x (1 - (1 + i)-n) / i, where C is the periodic payment, i is the interest rate, and n is the number of periods.

B.12 Time value of money concepts and calculations

Question 8

An investor wants to have $1 million in 20 years for retirement. If the investor can earn 6% interest annually, approximately how much does the investor need to deposit annually to reach this goal?

A. $26,000
B. $27,000
C. $28,000
D. $29,000

Answer 8

$27,000

The amount of money an investor needs to deposit annually to reach a future value goal can be calculated using the formula PMT = FV x (i / ((1 + i)n - 1)), where PMT is the periodic payment, FV is the future value goal, i is the interest rate, and n is the number of periods.

To solve using the HP 10bII+ enter the following:
Press the Shift (orange key) and then Beg/End to make sure calclator is set to end
1,000,000 and then press FV
20 and then press N
6 and then press I/YR
press PMT

Answer = -27,000 (rounded)

B.12 Time value of money concepts and calculations

Question 9

Which of the following is the formula for calculating the future value of an annuity?

A. FV = C / i
B. FV = C x (1 + i)^n / i
C. FV = C x (1 - (1 + i)^-n) / i
D. FV = C x (1 + i)^n

Answer 9

B. FV = C x (1 + i)^n / i

The future value (FV) of an annuity is calculated using the formula FV = C x (1 + i)^n / i, where C is the periodic payment, i is the interest rate, and n is the number of periods.

B.12 Time value of money concepts and calculations

Question 10

An investor wants to have $500,000 in 10 years for a down payment on a house. If the investor can earn 8% interest annually, approximately how much does the investor need to deposit to reach this goal? - Choose the answer that is closest.

A. $34,000
B. $34,500
C. $35,000
D. $35,500

Answer 10

B. $34,500

To calculate how much the investor needs to deposit annually to accumulate $500,000 in 10 years with an annual interest rate of 8% using the HP 10bII+ financial calculator, follow these steps:

Clear the calculator: Press f and then CLX to clear any previous data.
Set the interest rate: Press 8 then i to enter the annual interest rate of 8%.
Set the number of periods: Press 10 then n to set the number of years to 10.
Enter the future value goal: Press 500000 then FV to set the future value as $500,000.
Compute the annual payment: Since you are solving for the annual payment (deposit), make sure the PV (present value) is set to 0 if not already. To do this, press 0 then PV. Now, calculate the payment by pressing PMT. This will show the amount that needs to be deposited annually.

B.12 Time value of money concepts and calculations

Question 11

Which of the following is the formula for calculating the present value of a perpetuity?

A. PV = C / i
B. PV = C x (1 + i)n / i
C. PV = C / (i - g)
D. PV = C x (1 - (1 + i)-n) / i

Answer 11

C. PV = C / (i - g)

The present value (PV) of a perpetuity is calculated using the formula PV = C / (i - g), where C is the periodic payment, i is the interest rate, and g is the growth rate.

B.12 Time value of money concepts and calculations

Question 12

An investor aims to have $50,000 saved in 5 years for a child’s college tuition. If the investor can earn an annual interest rate of 4%, how much does the investor need to deposit annually to achieve this goal?

A. $8,500
B. $9,000
C. $9,500
D. $10,000

Answer 12

C. $9,500

In this scenario, we are solving for the annuity given a future value. The formula for the future value of an annuity is FV = PMT × [(1 + r)^n - 1] / r, where PMT is the annuity payment, r is the interest rate per period, and n is the number of periods. We know the future value (FV) is $50,000, the interest rate (r) is 4% or 0.04, and the number of periods (n) is 5. We want to solve for PMT.

Rearranging the formula to solve for PMT gives us: PMT = FV × r / [(1 + r)^n - 1] = $50,000 × 0.04 / [(1 + 0.04)^5 - 1] ≈ $9,500.

B.12 Time value of money concepts and calculations

Question 13

An investor wants to have $1 million in 30 years for retirement. If the investor can earn 5% interest annually, how much does the investor need to deposit today to reach this goal?

A. $209,135.35
B. $220,347.88
C. $232,196.27
D. $244,715.82

Answer 13

A. $209,135.35

The amount of money an investor needs to deposit today to reach a future value goal can be calculated using the formula PV = FV / (1 + i)n, where PV is the present value, FV is the future value goal, i is the interest rate, and n is the number of periods. In this case, PV = $1,000,000 / (1 + 0.05)30 = $209,135.35.

B.12 Time value of money concepts and calculations

Question 14

Which of the following statements is true regarding the time value of money?

A. The present value of an investment increases as the interest rate increases
B. The future value of an investment increases as the number of compounding periods increases
C. The present value of an investment decreases as the number of compounding periods increases
D. The future value of an investment decreases as the interest rate decreases

Answer 14

B. The future value of an investment increases as the number of compounding periods increases

The time value of money concept states that the value of money changes over time due to the effects of inflation and interest. The future value of an investment increases as the number of compounding periods increases because each compounding period generates additional interest on the principal plus the previously accumulated interest. The present value of an investment decreases as the interest rate increases because a higher discount rate reduces the value of future cash flows.

B.12 Time value of money concepts and calculations

Question 15

An individual wants to have $100,000 in 10 years to purchase a house. Assuming an annual interest rate of 7% with quarterly compounding, approximately how much money does she need to invest today? Pick the answer that is closest to correct.

A. $48,600
B. $50,800
C. $51,200
D. $53,800

Answer 15

B. $50,800

To solve this problem using the HP 10bII+ financial calculator, follow these steps:

Clear the calculator: Press f and then CLX to clear any previous data.
Set the number of periods: Since the interest is compounded quarterly, multiply the number of years by the number of quarters per year. Press 40 (because 10 years * 4 quarters per year) then n.
Set the quarterly interest rate: Since the annual rate is 7%, the quarterly rate is 7% divided by 4. Press 1.75 then i (because 7% / 4 = 1.75%).
Enter the future value goal: Press 100000 then FV.
Calculate the present value: Press PV. The calculator will display the negative value of the present value required today to achieve the future value in the given time frame and rate. This is because cash outflows (investments) are conventionally entered as negative in financial calculators.

B.12 Time value of money concepts and calculations

Question 16

Sally is self-employed and has decided to save for her retirement in 3 years. She has a consistent Schedule C net income of $237,000, and she contributes $30,000 into a month purchase plan on the last day of each year. How much will his retirement account be worth at his retirement if he achieves 8% growth on his investments?

A. $95,312
B. $97,392
C. $100,182
D. $110,987

Answer 16

B. $97,392

This is a future value of an ordinary annuity problem since Susan contributes at the end of each year.

First, clear the TVM registers.
Press [orange shift key], then [C ALL].

Enter the number of periods (years).
Enter 3, then press [N].

Enter the interest rate.
Enter 8, then press [I/YR].

Enter the payment (annual contribution).
Enter 30000, then press [PMT].

Compute the future value.
Press [FV]. The calculator should display $97,392

B.12 Time value of money concepts and calculations

Question 17

Samantha transfers $600,000 to an irrevocable trust for the benefit of her niece. She also commits to adding $50,000 to the trust at the end of each year for 15 years. Given that the trust assets grow at a 7% after-tax rate, what will be the total amount in the trust at the end of 15 years following the last contribution, assuming no distributions have been made from the trust?

A. $1,200,000
B. $1,600,000
C. $2,100,000
D. $2,500,000

Answer 17

C. $2,100,000

HP 10BII+ Steps:

There are two components to solve for:

The future value of the initial lump sum amount of $600,000.

The future value of the annuity (the annual contribution of $50,000).

For the initial lump sum:

Using the HP 10BII+ calculator:

n = 15 (years)
i/y = 7 (annual growth rate in percentage)
PV = -600,000 (present value)
PMT = 0 (since there’s no additional payment for this calculation)
Compute FV which will give you the future value of the initial lump sum after 15 years.

For the annuity:

Using the HP 10BII+ calculator:

n = 15 (years)
i/y = 7 (annual growth rate in percentage)
PV = 0 (since we are only calculating the future value of the annuity)
PMT = -50,000 (annual contribution at the end of the year)
Compute FV which will give you the future value of the annuity after 15 years.

Add the future values of both components together to get the total amount in the trust at the end of 15 years.

The number will be close to $2,100,000

B.12 Time value of money concepts and calculations

Question 18

A client wants to buy a house in five years with a budget of $500,000. They currently have $50,000 saved and assume an annual return of 5%. How much does the client need to save annually to reach their goal?

A. $63,125
B. $72,151
C. $81,974
D. $92,865

Answer 18

B. $72,151

To determine how much the client needs to save annually, we need to use the Future Value formula. The formula calculates the future value of an investment with a certain rate of return.

Using the values given, we can solve for the annual savings amount required to reach the $500,000 goal:

FV = $500,000 - $50,000 = $450,000
r = 5% / 1 = 0.05
n = 5 x 1 = 5

PMT = FV / [(1 - (1 + r)^-n) / r]
PMT = $450,000 / [(1 - (1 + 0.05)^-5) / 0.05] = $72,151

B.12 Time value of money concepts and calculations

Question 19

A client wants to save $500,000 for their child’s college education in 18 years. Assume an annual return of 8%. How much does the client need to save** annually** to reach their goal?

A. $13,351
B. $18,724
C. $21,757
D. $25,179

Answer 19

A. $13,351

using HP 10bII+
Ensure compounding is set to 1
enter the following:
500,000 FV
18 N
8 I/YR
PMT

B.12 Time value of money concepts and calculations

Question 20

A client plans to retire in 20 years and aims to have an annual retirement income of $100,000 for 30 years after retirement. Assuming a yearly return of 5%, approximately how much does the client need to save annually starting today to meet this goal?

A. $20,000
B. $35,000
C. $46,000
D. $55,000

Answer 20

C. Approximately $46,000

Explanation: Retirement Phase: Determine the present value of the desired retirement annuities (30 years of $100,000 each at 5% per annum).

Savings Phase: Determine the annual savings needed to accumulate the amount from step 1 in 20 years at 5%.

Step 1: Retirement Phase

To determine the present value of the desired retirement annuities, we’ll use the Present Value of Ordinary Annuity formula:

Using the HP 10bII+ calculator for this:

1. Turn on the calculator.
2. Press [CLEAR ALL] to reset all settings, and ensure the calculator is set to END.
3. Enter 30, then press [N] (This sets the number of periods).
4. Enter 5, then press [I/Y] (This sets the interest rate).
5. Enter 100,000, then press [PMT] (This sets the periodic payment).
6. Press [PV] to compute the present value. This gives you the present value needed at the beginning of retirement. (should have approx $1,537,245)

Step 2: Savings Phase

Using the HP 10bII+ calculator:

1. Set (FV to 1,537,245 - The present value computed from step 1 is used the future value (since the amount you save now will be used to fund the retirement income).
2. Set ( N ) to 20 (for 20 years).
3. Set ( I/Y ) to 5 (for the annual return of 5%).

To solve the yearly payment (PMT):

1. Enter the value you got from step 1 (1,537,245), then press ([FV].
2. Enter 20, then press [N].
3. Enter 5, then press [I/Y].
4. Press [PMT] to compute the annual savings needed (should be approx 46,460)

B.12 Time value of money concepts and calculations

Question 21

A client plans to retire in 20 years and aims to have an annual retirement income of $100,000 for 30 years after retirement. Assuming a yearly return of 5%, approximately how much does the client need to save annually starting today to meet this goal?

A. $20,000
B. $35,000
C. $46,000
D. $55,000

Answer 21

C. Approximately $46,000

Retirement Phase: Determine the present value of the desired retirement annuities (30 years of $100,000 each at 5% per annum).

Savings Phase: Determine the annual savings needed to accumulate the amount from step 1 in 20 years at 5%.

Step 1: Retirement Phase

To determine the present value of the desired retirement annuities, we’ll use the Present Value of Ordinary Annuity formula:

Using the HP 10bII+ calculator for this:

1. Turn on the calculator.
2. Press [CLEAR ALL] to reset all settings, and ensure the calculator is set to END.
3. Enter 30, then press [N] (This sets the number of periods).
4. Enter 5, then press [I/Y] (This sets the interest rate).
5. Enter 100,000, then press [PMT] (This sets the periodic payment).
6. Press [PV] to compute the present value. This gives you the present value needed at the beginning of retirement. (should have approx $1,537,245)

Step 2: Savings Phase

Using the HP 10bII+ calculator:

1. Set (FV to 1,537,245 - The present value computed from step 1 is used the future value (since the amount you save now will be used to fund the retirement income).
2. Set ( N ) to 20 (for 20 years).
3. Set ( I/Y ) to 5 (for the annual return of 5%).

To solve the yearly payment (PMT):

1. Enter the value you got from step 1 (1,537,245), then press ([FV].
2. Enter 20, then press [N].
3. Enter 5, then press [I/Y].
4. Press [PMT] to compute the annual savings needed (should be approx 46,460)

B.12 Time value of money concepts and calculations