Introduction to Actuarial Science - Week 2

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Questions and Answers

How many years will it take for the business owner to save GH60,000 by contributing GH500 monthly at an 8% annual interest rate?

  • 7 years
  • 10 years (correct)
  • 15 years
  • 5 years

If Sandra contributes GH630 at the end of each month for 20 years at an 8% interest compounded semiannually, what will be the approximate future value of her retirement account?

  • GH150,000
  • GH300,000 (correct)
  • GH200,000
  • GH100,000

What is the formula to find the term of an ordinary annuity based on the future value?

  • n = ln(FV) + 1 / ln(1+r)
  • n = ln(FV * r) / ln(1+r)
  • n = (FV * r) / ln(1+r)
  • n = ln(FV) / (r * ln(1+r)) (correct)

How long will it take Rosemary to save $300,000 by contributing $10,000 monthly at a 15% interest rate compounded monthly?

<p>15 years (B)</p> Signup and view all the answers

What is the current value of an annuity of $7,500 paid at the end of each half-year for 10 years at a 9% interest rate compounded annually?

<p>$60,000 (A)</p> Signup and view all the answers

What is an annuity?

<p>A set of equal payments made at equal intervals of time. (B)</p> Signup and view all the answers

Which of the following describes a contingent annuity?

<p>The term begins but ends based on a specific event. (B)</p> Signup and view all the answers

In the formula for the future value of an annuity, which variable represents the total number of payment periods?

<p>t (D)</p> Signup and view all the answers

What happens to the future value of an ordinary annuity if the interest rate increases?

<p>The future value increases. (A)</p> Signup and view all the answers

If you invested $1,000 at an annual interest rate of 7% compounded quarterly for 3 years, what would be the accumulation considering quarterly compounding?

<p>$1,225.37 (B)</p> Signup and view all the answers

Which of the following is a characteristic of a perpetuity?

<p>Payments are continuous and last indefinitely. (B)</p> Signup and view all the answers

What does the term 'ordinary annuity' refer to?

<p>An annuity with payments at the end of each period. (D)</p> Signup and view all the answers

The formula for calculating future value of an ordinary annuity is derived from which principle?

<p>Compounding interest. (C)</p> Signup and view all the answers

What is the formula for the future value (FV) of an ordinary annuity with $n$ terms of payment?

<p>$\frac{A((1 + r)^n - 1)}{r}$ (C)</p> Signup and view all the answers

If Adam contributes $1200 yearly at an interest rate of 6% compounded annually, what will be his total FV by age 65?

<p>$185,714.36 (B)</p> Signup and view all the answers

What would be the quarterly interest rate if the annual interest rate is 4%?

<p>0.01 (C)</p> Signup and view all the answers

Which values should be used to determine the total amount paid by Eben until age 50 if he contributes $200 monthly at an interest rate of 3% compounded semiannually?

<p>A = 1200, r = 0.015, n = 50 (D)</p> Signup and view all the answers

In Kelvin's scenario, how large will the education fund be when his son starts college at age 18?

<p>$12,000 (C)</p> Signup and view all the answers

How is the future value formula for an ordinary annuity expressed using tables?

<p>FV = A·Sn¬r (B)</p> Signup and view all the answers

What is the correct method for obtaining the current value of an ordinary annuity?

<p>Reversing the amounts of the payments and discounting (A)</p> Signup and view all the answers

If an ordinary annuity has monthly contributions of $300, what would be the semiannual contribution amount?

<p>$1,800 (C)</p> Signup and view all the answers

What is the formula used to calculate the current value of an ordinary annuity?

<p>$A[1 - (1 + r)^{-n}] / r$ (D)</p> Signup and view all the answers

How would you determine the monthly interest rate if the annual interest rate is 6.5%?

<p>0.0054 (C)</p> Signup and view all the answers

If a man receives $500 a month for 10 years at an interest rate of 6.5%, what is the term in months for this annuity?

<p>120 (B)</p> Signup and view all the answers

For an annuity of $3,750 paid quarterly for 7 years at an interest rate of 8%, what is the total number of payments?

<p>28 (B)</p> Signup and view all the answers

What is the formula to find the payment amount (A) of an ordinary annuity when the current value (CV) is known?

<p>$A = CV imes r / (1 - (1 + r)^{-n})$ (D)</p> Signup and view all the answers

If Jim deposited $18,000 in an account with a 12% interest rate compounded semiannually for 6 years, what payment will his son receive?

<p>$1,800 (D)</p> Signup and view all the answers

What is the quarterly interest rate if the annual interest rate is 8%?

<p>0.02 (C)</p> Signup and view all the answers

In the context of future value calculations, what does the term $(1 + r)^{n} - 1$ relate to?

<p>The future value factor (D)</p> Signup and view all the answers

Flashcards

Annuity

A set of equal payments made at equal intervals of time.

Annuity Certain

An annuity where the start and end dates are known beforehand.

Term of an annuity

The time between the first payment and the last payment of an annuity.

Contingent Annuity

An annuity where the end date is unknown, and depends on an event.

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Perpetuity

An annuity that continues indefinitely.

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Future Value of an Annuity

The total value of all payments in an annuity at a specific future point in time.

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Ordinary Annuity

A type of annuity where payments are made at the end of each period.

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Role of Actuaries

The role of actuaries involves analyzing financial data and risks, especially to make informed decisions and manage uncertainties. They use mathematical and statistical models to evaluate financial risks, predict financial outcomes, and guide financial planning.

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Formula for Finding the Term of an Ordinary Annuity

The formula calculates the number of periods (n) needed for an ordinary annuity to reach a specific future value (FV) or current value (CV), given the interest rate (r) and regular payment (A).

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Present Value of an Annuity

The present worth of all future payments of an annuity when discounted to a point in time before the first payment.

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Future Value of an Ordinary Annuity

The total amount accumulated at the end of the annuity period, including all principal payments and accrued interest. This can be a powerful tool for long-term financial planning, such as retirement savings.

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Future Value of an Ordinary Annuity - First Quarter

The total amount accumulated in the first quarter of an ordinary annuity. It includes the initial payment plus the interest earned on it.

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Future Value of an Ordinary Annuity - Second Quarter

The total amount accumulated in the second quarter of an ordinary annuity. It includes the second payment plus the interest earned on both the first and second payments.

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Future Value Formula

The formula to calculate the future value of an ordinary annuity, where FV is the future value, A is the regular payment amount, r is the interest rate per period (annually, quarterly, etc.), and n is the total number of payment periods.

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Future Value Using Tables

A common way to understand and calculate future value. It involves finding the future value of a series of payments. The sum of these individual future values is the total, or future value, of the annuity.

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Present Value of an Ordinary Annuity

The present value of a future stream of cash flows. It represents the current amount of money that would be needed today to generate the same cash flows. This can be used to compare different investment opportunities.

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Total Mortgage Payment

The total amount of money you will have paid at the end of the mortgage, including both the principal payments and the accrued interest.

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Present Value (PV)

The present value (PV) of a future value (FV) is calculated by discounting the future value back to the present using the discount rate and the number of periods.

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Current Value (CV) of an Ordinary Annuity

The current value (CV) of an ordinary annuity is the present value of a series of equal payments made at the end of each period for a specified number of periods.

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CV Formula

The formula for calculating the current value (CV) of an ordinary annuity considers the payment amount (A), the interest rate (r), and the number of periods (n).

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Future Value (FV) of an Ordinary Annuity

The future value (FV) of an ordinary annuity is the total amount accumulated at the end of a specified period, including the principal and all accumulated interest.

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Finding Payment (A) from FV

To find the payment amount (A) of an ordinary annuity, we rearrange the Future Value (FV) formula to isolate A.

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Finding Payment (A) from CV

To find the payment amount (A) of an ordinary annuity, we rearrange the Current Value (CV) formula to isolate A.

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Using Tables for Annuity Calculations

Tables can be used as a shortcut to determine the present value of an annuity based on a given interest rate and number of periods.

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Applications of Annuities

Real-life examples of annuities include retirement plans, mortgages, and student loans.

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Study Notes

Introduction to Actuarial Science - Week 2

  • Actuarial science focuses on calculating and managing financial risks.
  • Actuaries apply mathematical and statistical principles to assess and manage risk in areas like insurance, pensions, and investments.
  • Core areas of actuarial application include definitions and scope, actuarial roles, simple and compound interest calculations, annuities, and future and present value calculations.

Simple and Compound Interest

  • Simple interest calculates interest only on the principal amount.
  • Compound interest calculates interest on both the principal and the accumulated interest.
  • Compound interest formulas determine accumulated amounts based on different compounding frequencies (annual, quarterly, monthly, weekly).

Annuities

  • An annuity is a series of equal payments or receipts made at fixed intervals of time.
  • Examples include periodic savings deposits, mortgages, life insurance premiums, and social security deductions.
  • Annuity types include term, annuity certain, contingent annuity, and perpetuity.

Future Value of an Ordinary Annuity

  • Future value is the accumulated amount a series of equal payments will reach after a specific time period.
  • The formula for future value (FV) is A[(1+r)^n - 1] / r, where A represents the periodic payment, r is the interest rate, and n is the number of payment periods.
  • Calculations for future value can be found with tables.

Current Value of an Ordinary Annuity

  • Current value is the present worth of a series of future payments.
  • The formula for the current value (CV) is A[1 - (1 + r)^-n] / r, where A represents the periodic payment, r is the interest rate, and n is the number of payment periods.
  • Calculations for current value can be found with tables.

Finding the Payment of an Ordinary Annuity (A)

  • Formulas for finding periodic payments (A) given the future value (FV) or current value (CV) of an ordinary annuity are provided.
  • The formulas use the respective future value and current value formulas.

Finding the Term of an Ordinary Annuity

  • Formulas for determining the number of payment periods (n) given the future or current value of an ordinary annuity are presented.
  • The formulas involve the use of logarithms (ln) and can be used with future value (FV) or current value (CV).

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