Tiêu đề 93 Annuity and Gradient.pdf ✅

Similarly, the annuity can be computed from the future and present worth values by multiplying other factors, which are reciprocals of the first two. The new factors are called the sinking fund factor and capital recovery factor, respectively. sinking fund factor = A F = i (1 + i) n − 1 denoted by ( A F ,i,n) capital recovery factor = P A = i(1 +i) n (1 +i) n −1 denoted by ( A P ,i, n) When solving problems with annuities, the following format can be used as a calculator technique. V = ∑ (A)(1 + i) T − x end x = start where “T” is the time when the value “V” is being found, and “A” is how much the regular payment is, and i = r m . 2. Annuity Due and Deferred Annuity The annuity due at the start of each term is multiplied by a factor of (1 + i). Consider the cash flow diagram, The factor (1 + i) comes from the payments being shifted one term to the past. F = A(1 + i)[(1 + i) n − 1] i P = A[(1 + i) n − 1] i(1 + i) n−1 Sometimes, the payment series can start later. In this case, the annuity is said to be deferred. The future value is computed in the same way as an ordinary annuity. The present value is computed using compound interest from the future value. The calculator technique is still applicable using the appropriate starting and ending periods, both for annuities due at the start of the term and deferred annuities. 3. Perpetuity Consider an unending series of payments,
Since it is unending, its future value is infinite. However, its present value is not. For its present value, P = A[(1 + i) n − 1] i(1 + i) n P = A i [1 − 1 (1 + i) n ] Since the payment is unending, take the limit as n approaches infinity, P = A i Note that to use this equation, the first payment must be one term after when the present value is being evaluated. For example, the payment must start at year 1 when the present value is being evaluated at year 0. When skipping some compounding terms before the next payment (for example, the payment is every four years while the interest is annual), P = A (1 + i) n − 1 where n is how many terms of compounding occurred before the next payment. Note that to use this equation, the first payment must be n terms after when the present value is being evaluated. For example, the payment must start at year n when the present value is being evaluated at year 0. 4. Gradient A gradient is a series of regular payments that are not equal but are somehow related to time. An arithmetic gradient is distinguished by the payments having a common difference. A geometric gradient is distinguished by the payments having a common ratio. The future worth of an arithmetic gradient (evaluated at the time of last payment) can be computed with F = A[(1 + i) n − 1] i + G[(1 + i) n − in − 1] i 2 Here, A is the initial payment, and G is the constant increment. For a geometric gradient growing at a percent g, the present worth (beginning one term after the start) is P = A[(1 + i) n − (1 + g) n] (1 + i) n(i − g) Here, A is the initial payment. When i = g, P = An 1 + i
Solving problems with gradients using formulas is very complicated. The best way to solve these problems is to find a function of the payments in terms of time using regression in the calculator (“MODE-STAT-LIN” for arithmetic gradient and “MODE-STAT-ab exp” for geometric gradient). If the function of the payments is represented by A(x), V = ∑ [A(x)](1 + i) T − x end x = start where “T” is the time when the value “V” is being found, “A(x)” is the function representing the payment series, and i = r m . Future Value of Ordinary Annuity. F = A[(1 + i) n − 1] i Present Value of Ordinary Annuity. P = A[(1 + i) n − 1] i(1 + i) n Ordinary Annuity Factors. series compound amount factor = F A = (1 + i) n − 1 i denoted by ( F A ,i, n) series present worth factor = P A = (1 + i) n − 1 i(1 + i) n denoted by ( P A , i, n) Annuity Due at the Start of the Month. F = A(1 + i)[(1 + i) n − 1] i P = A[(1 + i) n − 1] i(1 + i) n−1 Calculator Technique for Annuities. V = ∑ (A)(1 + i) T − x end x = start Perpetuity. P = A i Skipping Perpetuity. P = A (1 + i) n − 1 Calculator Technique for Gradient. V = ∑ [A(x)](1 + i) T − x end x = start