Annuities present value

In effec t, in computing the average net annual cash flow per dollar invested, the value of f p of Eq. (9-46) has been obtained for this example. From tables of the annuity present-worth factor/ p the value of the interest rate is found to be = 0.25 when f p = 0.5124 with n = 3 years. 

The value of y will for most cases be less than 0.2 and with the right apphcation may well be less than 0.1. Values for the annuity present-worth factorwill in most cases be less than 0.15. 

The present value of an annuity that has been established to repay a loan is the amount of money borrowed. From the previous discussion... 

One special form of an annuity requires that payments be made at the beginning of each period instead of at the end of each period. This is known as an annuity due. An annuity in which the first payment is due after a definite number of years is called a deferred annuity. Determination of the periodic payments, amount of annuity, or present value for these two types of annuities can be accomplished by methods analogous to those used in the case of ordinary annuities. 

Equation 9-13 is defined as the annuity with I, the present value, and C, the equal payments, over n, years at interest, i. The I/C term is referred to as the payback period. If I and C are evaluated and the life of the project is known, i can be computed by trial and error. The value of i can be determined from Figure 9-3. 

In the accounting practice the operating and capital costs must be expressed on a common basis. The first are normally reported on yearly basis. With the capital the situation is different. We may consider that the capital is borrowed over a period of time, usually between five and ten years, the repayment will be done following a fixed sum called annuity. Suppose that the present value of the invested capital is P. If invested at a fixed interest rate i over n years, this will produce an equivalent amount F, as given by the equation (15.6). The reimbursement by yearly payments A can be... 

To ensure that all positive and negative cash flow items are included in the analysis and to better visualize these cash flows, it is useful to construct a cash flow diagram or taUe. A generalized illustration of a cash flow diagram for n periods of time is shown below. In the diagram, P is the present value of the first cost A is the value of an annuity, a imiform series of equal cash savings... 

This factor finds an annuity, or uniform series of payments, over n periods at /% interest per period that is equivalent to a present value, P. For example, what savings in annual manufacturing costs over an eight-year period would justify the purchase of a 120,000 machine if a firm s minimum attractive rate of return (MARR) were 20% ... 

To illustrate the process, let s value a 4-year, 6% coupon bond with a maturity value of 100. The coupon payments are 6 for the next four years. In addition, on the maturity date, the investor receives the repayment of principal ( 100). The value of a nonamortizing bond can be divided in two components (1) the present value of the coupon payments (i.e., an annuity) and (2) the present value of the maturity value (i.e., a lump sum). Therefore, when a single discount rate is employed, a bond s value can be thought of as the sum of two presents values—an annuity and a lump sum. 

We now have everything in place to value an annual coupon-paying bond. Recall, the present value of an annuity is equal to... 

Exhibit 3.7 illustrates the calculation. Once again for simplicity, we assume that 3-month LIBOR remains unchanged at 5% and there are 91 days in each coupon period. Since a risky floater can be thought of as par plus the differential risk annuity, all that is necessary is to take the present value of the annuity. Each annuity payment is computed as follows ... 

The quoted margin and required margin are in Columns (4) and (5), respectively. These cash flows are contained in Column (6). The discount factors are computed as described previously with the exception of the larger required margin. The discount factors appear in Column (7). The present value of the each cash flow is in Column (8) and is just the product of the cash flow (Column (6)) and its corresponding discount factor (Column (7)). The present value of the differential risk annuity is -0.1813 and is shown at the bottom of Column (8). 

Once the present valne of the differential risk annuity is determined, the price of our hypothetical 4-year floater is simply the sum of 100 (price of the floater per 100 of par value when the quoted margin and required margin are the same) and the present value of the differential risk annuity. In our example,... 

Current yield possesses a number of drawbacks as a potential return measure. Current yield considers only coupon interest and no other source of return that will affect an investor s yield. To see this, assume the current yield is a yield to perpetuity, the annual euro coupon payment is a perpetual annuity payment, and the security s price is the present value of the perpetual annuity. By rearranging terms such that the price equals the annual coupon payment divided by the current yield, we obtain the present value of a perpetual annuity formula as shown below... 

To see the significance of the second drawback, it is useful to partition the value of an option-free floater into two parts (1) the present value of the security s cash flows (i.e., coupon payments and matnrity value) if the discount margin equals the quoted margin and (2) the present value of an annuity which pays the difference between the quoted margin and the discount margin mnltiplied by 100 and divided by the number of periods per year. 

The present value of an annuity (PVA) will be used to reflect future amount of money that has been discounted to reflect its current value, as if it existed today, such as energy cost. The future cash flows of the annuity are discounted at the discount rate, and the higher the discount rate, the lower the present value of the annuity. 

PVA = Present Value of an Annuity PVF = Present Value of an annuity Factor n = number of years... 

During the 20 year life time the additional energy losses are 20.55 GWh. The present value of an annuity factor with 10 % discount rate over 20 years corresponds to 8.51. This gives the net present energy additional cost 612 kEUR for the Blair hoist alternative. 

Discount factors represent sinple ratios and can be multiplied or divided by each other to give additional discount factors. For exanple, assume that we need to know the present worth, P, of an aimuity, A—that is, the discount factor for P/A— but do not have the needed equatiom The only available formula containing the annuity term. A, is the one for F/A derived above. We can eliminate the future value, F, and introduce the present value, P, by multiplying by the ratio of P/F, from Equation 9.6. 

A present value of 1,019,000 is equivalent to a 20-year annuity of 100,000/yr when the effective interest rate is 9.5%. 

Operating Cost Methods. In the previous section, yearly savings were converted to an equivalent present value using the present value of an annuity, and this was measured against the capital cost. An alternative method is to convert all the investments to annual costs using the capital recovery factor and measure them against the yearly savings. 

In this chapter, PBP (= FCECF) did not consider the time value of money. In order to factor in the time value of money, future CFs have to be brought to the current value using the present worth annuity factor,/pA(r[Pg.336]

Table 17.7 Time Value of Money Annuity Factors—Uniform-Series Payments—Compound Interest (Details Presented in Section 17.5) ...

The concept of the time value of money is discussed. The following topics are presented sinple and conpound interest, effective and nominal interest rates, annuities, cash flow diagrams, and discount factors. In addition, the concepts of depreciation, inflation, and taxation are covered. 

The key to performing any economic analysis is the ability to evaluate and conpare equivalent investments. In order to understand that the equations presented in Table 9.1 provide a comparison of alternatives, it is suggested to replace the equal sign with the words is equivalent to. As an example, consider the equation given for the value of an annuity. A, needed to provide a specific future worth, F. From Table 9.1. Equation (9JT) can be expressed as... 

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