Growing Annuity Formula

3 Key excerpts on "Growing Annuity Formula"

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  eBook - PDF

  Fundamentals of Corporate Finance

  • Robert Parrino, David S. Kidwell, Thomas Bates, Stuart L. Gillan(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  We then multiply the ordinary annuity value times (1 + i). Section 6.2 discusses the calculation of the present value and future value of an ordinary annuity and an annuity due. 3 Explain what a perpetuity is and where we see them in business, and calculate the value of a perpetuity. A perpetuity is like an annuity except that the cash flows are per- petual—they never end. British Treasury Department bonds, called consols, were the first widely used securities of this kind. The most common example of a perpetuity today is preferred stock. The issuer of preferred stock promises to pay fixed-rate dividends forever. The cash flows from corporations can also look like perpetuities. To calculate the present value of a perpetuity, we simply divide the constant cash flow (CF) by the interest rate (i). 4 Discuss growing annuities and perpetuities, as well as their application in business, and calculate their values. Financial managers often need to value cash flow streams that increase at a constant rate over time. These cash flow streams are called grow- ing annuities or growing perpetuities. An example of a growing annuity is a 10-year lease with an annual adjustment for the expected rate of inflation over the life of the contract. If the cash flows continue to grow at a constant rate indefinitely, this cash flow stream is called a growing perpetuity. Since a C-corporation has an indefinite life, when the cash flows from such a corporation are growing at a constant rate, they can be thought of as a growing perpetuity. The calculation of the value of a cash flow stream that grows at a constant rate is discussed in Section 6.4. 5 Discuss why the effective annual interest rate (EAR) is the appropriate way to annualize interest rates, and calculate the EAR. The EAR is the annual interest rate that takes compounding into account. Thus, the EAR is the true cost of borrowing or lending money.

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  eBook - PDF

  Corporate Finance

  • Peter Moles, Robert Parrino, David S. Kidwell(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  As you can see in this example, the PV of the annuity due is larger than the PV of the ordinary annuity. The reason is that the cash flows of the annuity due are shifted by one year and thus are discounted less. 222 PART 3 VALUATION OF FUTURE CASH FLOWS AND RISK over time. These cash flow streams are called grow- ing annuities or growing perpetuities. Growing Annuity Financial managers often need to compute the value of multiyear product or service contracts with cash flows that increase each year at a con- stant rate. These are called growing annuities. For example, you may want to value the cost of a 25-year lease that adjusts annually for the expected rate of inflation over the life of the con- tract. Equation (6.5) can be used to compute the present value of an annuity growing at a constant rate for a finite time period: PVA n ¼ CF 1 i  g  1  1 þ g 1 þ i   n   ð6:6Þ where: PVA n ¼ present value of a growing annuity with n periods CF 1 ¼ cash flow one period in the future (t ¼ 1) i ¼ interest rate, or discount rate g ¼ constant growth rate per period Growing annuity an annuity in which the cash flows increase at a constant rate You should be aware of several important points when applying Equation (6.6). First, the cash flow (CF 1 ) used is not the cash flow for the current period (CF 0 ), but is the cash flow to be received in the next period (t ¼ 1). The relation between these two cash flows is CF 1 ¼ CF 0  (1 þ g). Second, a necessary condition for using Equation (6.6) is that i > g. If this condition is not met, the calculations from the equation will be meaningless, as you will get a negative value for positive cash flows. A negative value essentially says that some- one would have to pay you money to get you to accept a positive cash flow. As an example of how Equation (6.6) is applied, suppose you work for a company that owns a number of coffee shops in the city and areas around Berlin.

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  eBook - ePub

  The Case Approach to Financial Planning: Bridging the Gap between Theory and Practice, Fifth Edition

  • John E. Grable, Ronald A Sages, Michelle E Kruger(Authors)
  • 2022(Publication Date)
  • The National Underwriter Company

    (Publisher)

  geometrically varying annuity formula must be used to determine a future value whenever a payment is expected to increase at a fixed geometric rate.

  Example : Assume that Jorge will make annual payments4 into a 401(k) for twenty years, earning an effective annual rate of 9 percent. He will begin with a $3,000 contribution, which is 5 percent of his income. Every year thereafter, he will increase his deposit by 3 percent to reflect the expected increase in his salary. Using this assumption, how much will Jorge accumulate at the end of twenty years?

  p041-1.jpg

  In this case, the fact that Jorge’s subsequent deposits into the 401(k) grow by 3 percent annually means that he will have approximately $189,915 in the account at the end of twenty years. When using growing annuities, it is often beneficial—or at least interesting—to compare the results of a growing annuity and a fixed annuity, keeping all of the variables besides the growth rate fixed.

  Solving the example, but without growing the payment by 3 percent per year, can be done using the same formula and entering a zero for the growth rate, or it can be solved using the fixed annuity equation as shown below:

  p041-2.jpg

  By removing the growth rate, the value in Jorge’s account at the end of twenty years is $36,435 ($189,915 – $153,480) less if he chooses not to increase his annual payment. As shown here, increasing payments over a long-term planning horizon can have a significant impact on meeting a client’s financial goals.

  It is also important to note that when using a growing annuity equation, the payment used in the calculation will be the first payment. Similarly, when using the equation to solve for the payment amount, the equation solves for the first payment (PMT1 ). It is important to understand that this distinction is relevant because the value increases or decreases across the series of payments in a growing annuity, whereas in a fixed annuity all of the payments are the same, so no differentiation is needed. All other payments must be derived by using PMT1 and then increasing payments by the growth rate. Therefore, PMT2 equals PMT1

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