Perpetuities — Practice Quiz

A Financial Math cheat sheet for Perpetuities — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

The present value of a perpetuity, a stream of equal payments that continues indefinitely. \(PMT\) is the periodic payment and \(i = r/n\) the periodic rate.

$$PV = \dfrac{PMT}{i} \quad\text{or}\quad PV = \dfrac{PMT}{\,r/n\,}$$

Practice quiz

  1. A foundation wants to set up a scholarship that pays out $5,000$ annually in perpetuity. If the endowment can earn an annual interest rate of $4\%$, what is the present value needed to fund this scholarship?

    • $125,000$
    • $200,000$
    • $100,000$
    • $150,000$

    Answer: $125,000$

  2. An investor wishes to receive $1,200$ per month indefinitely. If the market interest rate is $6\%$ per annum, compounded monthly, what is the present value required to establish this perpetuity?

    • $240,000$
    • $120,000$
    • $200,000$
    • $1,200,000$

    Answer: $240,000$

  3. If a perpetuity has a present value of $500,000$ and makes annual payments of $20,000$, what is the implied annual interest rate?

    • $4\%$
    • $5\%$
    • $2.5\%$
    • $10\%$

    Answer: $4\%$

  4. Which of the following best describes a perpetuity?

    • A series of equal payments made over a fixed period.
    • A single lump sum payment made in the future.
    • A stream of equal payments that continues indefinitely.
    • A payment that increases by a fixed percentage each period.

    Answer: A stream of equal payments that continues indefinitely.

  5. A perpetuity currently pays $1,000$ annually and has a present value of $25,000$. If the interest rate decreases, what will happen to the present value of this perpetuity?

    • It will decrease.
    • It will increase.
    • It will remain the same.
    • It cannot be determined without the new interest rate.

    Answer: It will increase.

  6. A perpetuity has a present value of $1,000,000$ and an annual interest rate of $5\%$. If the periodic payment is doubled, what will be the new present value, assuming the interest rate remains constant?

    • $500,000$
    • $1,000,000$
    • $2,000,000$
    • $2,500,000$

    Answer: $2,000,000$

  7. A government bond promises to pay $300$ semi-annually forever. If the current market annual interest rate is $8\%$, compounded semi-annually, what is the present value of this bond?

    • $7,500$
    • $3,750$
    • $15,000$
    • $6,000$

    Answer: $7,500$

  8. What is the key difference between a perpetuity and an ordinary annuity?

    • Annuities have variable payments, while perpetuities have fixed payments.
    • Perpetuities have a finite number of payments, while annuities continue indefinitely.
    • Annuities have a finite number of payments, while perpetuities continue indefinitely.
    • Perpetuities are always larger in value than annuities.

    Answer: Annuities have a finite number of payments, while perpetuities continue indefinitely.

  9. In the formula $PV = \frac{PMT}{r/n}$, what does $n$ represent?

    • The total number of payments.
    • The annual interest rate.
    • The number of compounding periods per year.
    • The present value.

    Answer: The number of compounding periods per year.

  10. A charitable organization wants to establish a fund that provides $10,000$ at the end of each quarter indefinitely. If the fund can earn an annual interest rate of $4\%$, compounded quarterly, what is the minimum amount needed to establish this fund?

    • $1,000,000$
    • $400,000$
    • $250,000$
    • $10,000,000$

    Answer: $1,000,000$

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