Perpetuity It is a type of annuity whose payments continue forever.

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1 Perpetuity It is a type of annuity whose payments continue forever. Something to think about... How does an equal payment at an equal interval continue forever? Example: An individual might, for example donate a substantial amount of money to an educational institution to establish scholarship fund that is to pay out the fixed amount of scholarship awards each year forever. The funds would be invested by the institution and the perpetuity payments made with the investment earnings. Suppose, for example, that a certain amount donated is made to De La Salle-College of St. Benilde Antipolo Campus to endow a perpetual financial support fund. If the school invests the funds to earn an effective annual rate of 8%, the amount could be paid out in periodic payment on each anniversary of the investment. The principal amount would neither grow nor diminish in subsequent years.

2 Lesson 6.1: Computing for Ordinary Simple Perpetuity Ordinary Simple Perpetuity It is an annuity in which the payment interval equals the conversion period. Here, the payment of the perpetuity happens at the end of each rent period. A p. We will denote ordinary simple perpetuity using majuscule letter A and the subscript p or If the conversion period for the periodic interest rate is equals the payment interval, then it is considered ordinary simple perpetuity. Understanding Simple Perpetuity In an initial principal amount A p (present value) is invested at the periodic rate i, the interest earned in the first conversion period will be A p i. If this interest is then paid out instead of being added to the original principal, then the principal at the beginning of the second interval will again be A p, and it will again earned interest A p i in the second interval. In general, an amount A p earning a periodic interest rate i can sustain R (periodic payment) by the product of A p and i forever. Figure 6.1 Formula for Computing the Principal Amount/Present Value of Simple Perpetuity: A p = R p i p (Formula 6.1) Formula for Computing the Periodic Payment of Simple Perpetuity: R p = A p i p (Formula 6.2)

3 Formula for Computing the Interest Rate of Simple Perpetuity: i p = R p A p (Formula 6.3) Question: Is it possible to calculate the Final Value of a Simple Perpetuity? Why or Why not? Example 6.1: A commercial bank is considering the establishment in perpetuity of a Visiting Professorial Chair in Business Administration at a University. The cost would be Php100, per month. (a) If money can earn 12% compounded monthly in perpetuity, what endowment is required to fund the position? (b) What monthly compounded nominal rate would an endowment of P20million have to earn to fully fund the position? Answers: (a) The endowment required is P10,000, (b) The nominal rate is 6% compounded monthly.

4 Lesson 6.2: Computing for Perpetuities Due Perpetuity Due It is an annuity in which the payments are made at the beginning of each payment interval and continue forever. We will denote perpetuity due using majuscule letter A and the subscript p(due) or A p(due) A perpetuity due is equivalent to an immediate payment R followed by an ordinary simple perpetuity. Thus, the present value of a perpetuity due, A p(due), is R plus the present value of an ordinary simple annuity. Figure 6.2 Formula for Computing the Principal Amount/Present Value of Perpetuity Due: A p(due) = R i p(due) (1 + i p(due) )(Formula 6.4) Formula for Computing the Periodic Rent of Perpetuity Due: R p(due) = A p(due)i p(due) (1 + i p(due) ) (Formula 6.5) Formula for Computing the Interest Rate of Perpetuity Due: i p(due) = R p(due) A p(due) R p(due) (Formula 6.6) Example 6.2 A rural bank's preferred shares pay a dividend of Php60.00 per share every beginning of three months in perpetuity. The next quarter dividend is not to be paid. (a) At what purchase price will the shares provide an investor with a rate of return of 12% compounded quarterly?

5 (b) What will be the nominal rate of return if the investor is able to purchase to shares for Php3,060? Answers: (a) The purchase price of the dividend is Php2, (b) The nominal rate of return is 8% compounded quarterly.

6 Lesson 6.3: Computing the Deferred Perpetuities Deferred Perpetuities These are like deferred annuities. These are computed in two stages. The first stage is to compute the present value of an ordinary perpetuity, and the second stage is the computation of the discounted back to today's value, the same with an ordinary annuity. A p(def). We will denote deferred perpetuities using majuscule letter A and the subscript p(def) or Figure 6.3 Formula for Computing the Present Value of Deferred Perpetuities: R p(def) A p(def) = (Formula 6.7) d i p(def) (1 + i p(def) ) Formula for Computing the Periodic Payment of Deferred Perpetuities: R p(def) = A p(def) i p(def) (1 + i p(def) ) d (Formula 6.8) Example 6.3 Find the quarterly payments of a perpetuity whose present value is Php1,300, if money is worth 12.3% compounded quarterly; the first payment begins after 5 years. Answer: The perpetuity will have regular payments of Php24,

7 Let's Practice: Direction: Solve the following on one whole yellow paper. 1. If money is worth 7.8% compounded semiannually, what is the present value of an annuity of Php4, payments are made indefinitely at the end of every six months, if the first payment is made at the end of 3 years and 6 months? 2. Find the monthly payments of a perpetuity whose present value is Php47,200.00, if money is worth 9% compounded semiannually, payments are made at the beginning of every 6 months? 3. What payment at the end of each month forever can be provided by an endowment of Php380, invested at 12% compounded monthly?