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That is,the factored form of. Solution Because the third term is positive and the middle term is negative, we find two negative integers whose product is 6 and whose sum is We list the possibilities. When the first term of a trinomial is positive and the third term is negative,the signs in the factored form are opposite.

That is, the factored form of. Solution We must find two integers whose product is and whose sum is It is easier to factor a trinomial completely if any monimial factor common to each term of the trinomial is factored first. For example, we can factor. A monomial can then be factored from these binomial factors. However, first factoring the common factor 12 from the original expression yields. Solution We find two positive factors whose product is 6y 2 and whose sum is 5y the coefficient of x.

The two factors are 3y and 2y. When factoring, it is best to write the trinomial in descending powers of x. If the coefficient of the x 2 -term is negative, factor out a negative before proceeding. Solution We look for two integers whose product is 12 and whose sum is 5. From the table in Example 1 on page , we see that there is no pair of factors whose product is 12 and whose sum is 5.

In this case, the trinomial is not factorable. Skill at factoring is usually the result of extensive practice. If possible, do the factoring process mentally, writing your answer directly. You can check the results of a factorization by multiplying the binomial factors and verifying that the product is equal to the given trinomial. In this section, we use the procedure developed in Section 4.

As before, if we have a squared binomial, we first rewrite it as a product, then apply the FOIL method. As you may have seen in Section 4. When a monomial factor and two binomial factors are being multiplied, it is easiest to multiply the binomials first. First, we consider a test to determine if a trinomial is factorable. Consider the following possibilities. The test to see if the trinomial is factorable can usually be done mentally.

We illustrate by examples. We consider all pairs of factors whose product is 4. Since 4 is positive, only positive integers need to be considered. The possibilities are 4, 1 and 2, 2. We consider all pairs of factors whose product is 3. Since the middle term is positive, consider positive pairs of factors only.

The possibilities are 3, 1. We write all possible arrangements of the factors as shown. We select the arrangement in which the sum of products 2 and 3 yields a middle term of 8x. The integers 4 and -3 have a product of and a sum of 1, so the trinomial is factorable. We now proceed. We consider all pairs of factors whose product is 6. Since 6 is positive, only positive integers need to be considered. Then possibilities are 6, 1 and 2, 3.

We consider all pairs of factors whose product is The possibilities are 2, -1 and -2, 1. We write all possible arrange ments of the factors as shown. We select the arrangement in which the sum of products 2 and 3 yields a middle term of x. With practice, you will be able to mentally check the combinations and will not need to write out all the possibilities.

Paying attention to the signs in the trinomial is particularly helpful for mentally eliminating possible combinations. Solution Rewrite each trinomial in descending powers of x and then follow the solutions of Examples 3 and 4. As we said in Section 4. We know that the trinomial is factorable because we found two numbers whose product is 12 and whose sum is 8. Those numbers are 2 and 6. This is the same result that we obtained before. Some polynomials occur so frequently that it is helpful to recognize these special forms, which in tum enables us to directly write their factored form.

Observe that. Often we must solve equations in which the variable occurs within parentheses. We can solve these equations in the usual manner after we have simplified them by applying the distributive law to remove the parentheses. Parentheses are useful in representing products in which the variable is contained in one or more terms in any factor. One integer is three more than another. If x represents the smaller integer, represent in terms of x. The larger integer. Five times the smaller integer.

Five times the larger integer. Let us say we know the sum of two numbers is If we represent one number by x, then the second number must be 10 - x as suggested by the following table. In general, if we know the sum of two numbers is 5 and x represents one number, the other number must be S - x. The next example concerns the notion of consecutive integers that was consid- ered in Section 3. The difference of the squares of two consecutive odd integers is The larger integer b. The square of the smaller integer c.

The square of the larger integer. Sometimes, the mathematical models equations for word problems involve parentheses. We can use the approach outlined on page to obtain the equation. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems.

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If the demand for investment property increases to d1d1 and supply of the same property remains static at s0s0, sales price shall increase from to 0P1 to 0P3. Along the same demand curve, d1d1 excess supply of property along curve s1s1 will engender stagnation of prices for a while in expectation that purchasers would further negotiate the asking price.

If this expectation is not realized by vendors, the market could compel a downward review of the equilibrium sales price of the property from 0P3 to 0P2, which signals depression in the property market. In other words, the interaction between demand and supply has been adjudged the best form of pricing model for investment property [25]. However, as the property market is fraught with infrequent sales transactions, pricing models are only proxies to the outcome of the interaction between demand and supply functions [7, 18].

Assumed to have been embedded in these pricing models are market parameters, the behaviour of market participants, and the observable interaction between market forces of demand and supply. Nonetheless, most property analyst and investors might be satisfied with the assumption that some of these information are embedded in the pricing model to say the least.

Investors capitalize on available and privileged information about investments because it directs their ability to rationally execute tasks ranging from capital budgeting, asset allocation, portfolio restructuring and portfolio diversification to say the least. Evidences suggest that direct property investment market exhibits a weak form efficiency which makes it difficult for investors to earn abnormal returns [18, 26]. In the same vein, a portfolio manager may be strategically concerned with the task of identifying underpriced property, while appropriate signals for asset allocation might be provided through valuation given its use as proxy for market information and investment parameters [18].

It may be possible that a synergy between growth explicit DCF and equivalent yield techniques would avert incidences of mispricing on the condition that methodologies for accurate assessment of nominal net equivalent yield are utilized. Equivalent yield valuation technique. The conventional value models Equations 1 - 6 upon which the use of equivalent yield is anchored have been examined in the preceding section. Among these equations, Equation 2 which pertains to the valuation of term and reversionary cash inflows See Fig.

The growth explicit DCF pricing model. The foundation of growth explicit DCF model is the ratio of the net cash inflow, c to the present value of the cash inflow or price, PV0 otherwise call the all risk yield, k Equation 8. In other words a different approach is applicable to the pricing of reversionary freehold property investment. Notwithstanding that some scholars in the contemporary value models' school of thought have questioned the use of implicit yields in modern valuations [7, 8, 13, 17], it is worthy of note that such criticism need not arise provided the valuer has transparently deployed Equation 14, which describes the relationship among the parameters contained in this yield [1, 18, 20].

The decision rule. Since Equations 15 and 16 are proxies for the actual prices payable for reversionary freeholds, an array of methodologies that seek to synergize these models with equivalent yield determination can be used to ascertain the extent to which each ensuing equivalent yields might lead to mispricing. The present value of cash inflows, PV1 determined using the equivalent yield variant of the conventional techniques of valuation Equation 2 was compared with the market price, PV0 derived from the short-cut DCF technique Equation This was aimed at ascertaining the possibility of mispricing arising from equivalent yields determined with recourse to the methodologies indicated in Figure 3.

Although rare in practice but theoretically achievable, the same property is correctly price if it was purchased at a price equivalent to its market value. The use of the NPV criterion is justified since direct yield to yield comparison cannot be established for growth explicit DCF- and equivalent yield techniques of valuation except for input cash inflows and the discounted outcomes output. The equivalent yield valuation technique entails the deployment of a single remunerative rate of interest as against two distinct yields for term and reversionary tranches of freehold cash inflows.

The justification for the use of this single yield is anchored on the argument that analysis of yield would produce only one single yield [19]. The implication is that it is illogical to apportion different levels of risks to the term and reversionary tranches of cash inflows accruing to a single freehold investment [21].

Among the variants of conventional methods of property investment valuation, only the equivalent yield model is capable of adequately utilizing comparable information [8]. Nonetheless, one of the caveats is the significant divergence of subject property from the comparable property thereby diminishing the credibility of a valuation [8].

Except for specific categories of commercial properties, a cursory examination of property markets across the world indicates that comparable evidence is hard to come by. Among the advantages of equivalent yield valuation is the elimination of mathematical errors and intuitive adjustments of the term- and reversionary yields within the valuation [8].

Secondly, its objectivity in the analysis of transactions indicates that it is the only conventional technique capable of bridging the gap between conventional models and growth explicit DCF techniques of property investment appraisal [18]. Contrary to these advantages, a number of scholars have discredited the use of the conventional model of property investment valuation including the equivalent yield approach [].

First they detested the application of implicit yield in valuing growth income. A careful comparison of term and reversion valuations ensuing from both equivalent yield- and growth explicit DCF models indicates over valuation of contract rent and undervaluation of market rent, which constitute a major snag with equivalent yield valuation. Although the summation of term and reversion values capital values may be similar for both models, scholars argued that such summation may becloud strategic investment decisions regarding contract- and market rents [8].

The position of this paper is that although the application of equivalent yield technique to market valuations may require some degree of choice of yield which may be subjective, a large proportion of underlying exercise to the valuation still relies on market intelligence and the expertise of the valuer. The second criticism hinges on accuracy of valuations and level of model sophistication. The equivalent yield technique has been acclaimed to produce accurate valuations notwithstanding its being implicit about property investment characteristics [18].

The argument is that since simple yield capitalization is based on the concept of discounted cash flow, the use of yields including equivalent yield in valuations implies the phenomenon of income growth such that sophistication in valuation models has nothing to do with accurate valuations but provides additional information that can improve strategic investment management decisions [18].

Thirdly, equivalent yield valuation model has been criticized for implicit handling of future rental growth [8]. Notwithstanding, there is a consensus that equivalent yield being a single growth implicit internal rate of return IRR , reflects growth potential and the other investment risks applicable to a subject property in aggregate [8, 18].

The fourth argument was set to challenge the relevance of equivalent yield since there is scarcity of evidence to suggest that equivalent yield valuation technique currently reflect perceptions of property investors [8]. In defence, it could be observed that equivalent yield technique can reflect the perception of investors and remains a powerful tool for strategic property investment management especially when synchronized with growth explicit DCF techniques to determine pricing errors and other vital parameters to the advantage of investors.

To seal the argument on the justification for equivalent yield valuation, it has been affirmed that if equivalent yield constitute a common terminology used by valuers, then its economic significant cannot be ruled out [18]. It is on the basis of this defence that this paper justifies the application of equivalent yield model to the pricing of reversionary freehold investment property.

Turning attention to equivalent yield analysis, it is possible to curb the emergence of mathematical errors if unnecessary approximation of yields is avoided. Beyond the accuracy of figures is a critical factor of the methodology of computation [3]. Hence, the method of calculating equivalent yield is viewed as a critical factor in the pricing decision that it is meant to support.

It has been noted that equivalent yield analysis objectively captures data pertaining to its determining parameters [8]. It is on the basis of this statement that this paper demonstrates the objectivity of this choice of yield given its relationship with the parameters that were used to calculate it. This section demonstrates with worked examples the six methods of calculating nominal equivalent yield of a freehold investment property See Fig.

Valuation data A freehold property was let one year ago at a net rent of N1,, subject to 3 yearly upward reviews. Current rental value of this property is N1,, subject to 3 yearly upward reviews, investors overall discount rate is estimated at Within the spreadsheet environment, Equation 16 was used to determine the sales price of the property. The valuation in Exhibit 1 formed the basis for the outstanding objectives of this research comprising methods of equivalent yield calculation; accuracy of the computed equivalent yield; and application of the same yield to property pricing.

Valuation and investment tables are reference texts designed to provide snapshot figures representing the outcome of input parameters of an investment function. The parameters required for the identification of the Equivalent yield in this table include the initial yield i, rent factor F, and the number of years to the next rent review. The formula method entails summing up the term rent and annual equivalent of gain on reversion and expressing it as a ratio of capital value of the property [3, 5].

Using the manual iterative process trial and error , a reversionary yield of 9. The final answer is an IRR estimate of 8. Scholars have warned that NPV and IRR exhibits a geometric relationship as against the popular assumption of a linear relationship [18, 23]. Notwithstanding, the enormity of error in IRR calculation could be minimized if the difference between the dual rates of return is not more than unity. In addition, it was suggested that restriction should be placed on the expression of a linearly interpolated IRR in long decimal places on the grounds of less accurate results [23].

However, such suggestion was ignored in this study for the purpose of experimentation. Just as in the first, and second methods, the linear interpolation method tend to produce estimates of the IRR equivalent yield. V of cash inflow 21,, V of cash inflow 19,, Summarized in Exhibit 2 are the information required in a spreadsheet environment for the actual calculation of equivalent yield. Among the requirements for Microsoft Excel to perform a robust IRR calculation was the use of a personal computer installed with bit Windows 7 Ultimate operating system and possessing not less than 60 Gigabyte of free hard drive space and 8GB of Random Access Memory.

Within the Microsoft Excel environment Fig. Furthermore, the market rent of N1,, was extended to years as proxy for its being perpetually earned from the investment, after which the IRR function was expressed as IRR C2:C in the cell C to return the equivalent yield IRR.

Exhibit 2: Cash flow profile of property Year Cash flow N 0 ,, It was observed that an extension of reversionary cash flows beyond cell C to say C did not significantly change the IRR equivalent yield. At this cut-off cell, the equivalent yield remained 8. A number of texts have explained the principle of the goal-seek tool of Microsoft Excel and how it can be used to calculate IRR of property investments See [3, 22, 23].

It is an extension of the "What-If Analysis" function available to investment analyst who are interested in performing scenario analysis. For the goal seek calculation, personal computer with the same specification as that mentioned in the preceding section was deployed. In other words, the root of the ensuing polynomial equation that reflects the actual and rational IRR for the property investment was isolated and reported.

While there are certain proprietary software packages that solve and isolate the appropriate root of the polynomial equation, it is at the discretion of the appraiser to verify the correctness of the root of the polynomial in the light of equivalent yield since it is possible to obtain multiple roots from this exercise [18].

Version 2. Although this software package has been acclaimed for its integrated data analysis and graphical visualization [30], its application in this study was driven by its embedded root-finding algorithm for polynomial equations.

These imaginary roots have been discarded in favour of the positive non-integer, 0. However, this positive non-integer root of the function could only be visualized following the graphing, embedding and mapping of the magnified version onto the initial graph Fig. It is inferred that this method puts the equivalent yield growth implicit IRR of the investment property at 8.

At this juncture, the first problem has been surmounted using mathematical software packages that have embedded root-finding algorithms, while the second problem can be ameliorated through the exercise of commonsense. For the same property investment data, it was observed that methods 1, 2, and 3 determined the equivalent yield as 8. Having demonstrated the methods of equivalent yield calculation, the succeeding sections examined these calculated equivalent yields on the basis of accuracy and the corresponding impacts they exert on property investment pricing.

Contrary to this proposition, scholars do not see the possibility of valuations equating with sales price following the imperfection and inefficiency of the property market [7]. Nevertheless, there has been no study to investigate how the accuracy of equivalent yield could influence the outcome of property investment pricing. It is possible that a good number of property scholars with bias for the analyses of yield parameters are not so much concerned with this issue on the premise that approximate yield figures are good enough to aid investment decision-making, which is viewed in the context of this paper as a misnomer.

To achieve the third objective of this study, equivalent yields that have been calculated using variants of existing methodologies in the preceding section were deployed in the conventional valuation model Equation 2 on a case-by-case basis with a view to determine the likely pricing decision that may arise from the use of such yields. Test of Equivalent yield from Method 1. Method 1 produced an equivalent yield of 8.

The present value of interest in the property at this yield is put at N20,, Exhibit 3: Valuation using 8. Test of Equivalent yield from Method 2. Method 2 produced an equivalent yield of 8. Exhibit 4: Valuation using 8. Test of Equivalent yield from Method 3. Method 3 produced an equivalent yield of 8. Test of Equivalent yield from Methods 4 and 5.

Within the spreadsheet environment, methods 4 and 5 produced an equivalent yield of 8. Test of Equivalent yield from Methods 6. With recourse to the zeros of a polynomial representing the equivalent yield of the property 8. Besides the exercise of precautions in the approximation of input and output parameters, the accuracy of equivalent yield analysis is applauded when the exercise is performed using computerized approaches.

The essence is to curb the abuse of non- integers in the numerical value of investment parameters for multi-million or multi-billion Naira worth of individual property or property portfolios which could override objective decision-making. Information that were tabulated in the bid to address the fourth objective of this study include each method of equivalent yield calculation and their respective nominal net equivalent yield NNEY , true net equivalent yield TNEY , value of property at the nominal net equivalent yield PV1 , price paid for the property's growth explicit cash flow PV0 , NPV of purchase decision, and a statement indicating the implications on pricing decision Table I.

For the subject property, the NPV at 8. Therefore, the use of valuation and investment tables Method 1 led to the underpricing of the freehold investment property. Therefore, the formula method Method 2 underpriced the same freehold investment property. Thirdly, linear interpolation method returned NNEY of 8.

Therefore, the linear interpolation method Method 3 culminated into the overpricing of the subject property. Within the context of these two techniques, the NPVs of the subject property at 8. Following this result, these two spreadsheet-integrated techniques have contributed to the correct pricing of the freehold investment property. The use of a mathematical software package with in-built root-solving algorithms returned a logical value of 8.

The NPV of the same property at 8. Being a mathematical approach to equivalent yield analysis, method 6 is capable of contributing to the correct pricing of an investment property just as the IRR and the goal seek techniques. Attention of this study was finally drawn to the winners and losers in the process of property investment pricing and the optimal decision that the gainers might likely take as indicated in Table II. The vendee of underpriced properties are the gainers while the losers are the sellers vendors.

On the other hand, the vendors of overpriced properties are the gainers while the losers are the vendee purchasers. With recourse to Table II, it has been suggested that asset allocation decision of investors should be anchored on the selection of underpriced properties [18]. The rationale for this phenomenon had been attributed to information asymmetry which drives one party to a property transaction to use market information to take advantage of the other party whose available information might have lagged behind current market trend [2, 18, 31].

Within the context of this study, it has been observed that the methodology for the calculation of yield parameters especially the equivalent yield might contribute to mispricing which may either be to the advantage of the vendor or the vendee. This finding adds to the importance of property investment pricing on the basis of appropriate processing of information as observed by Brown and Matysiak [18] XIII. The conceptual justification for equivalent yield is that it is comparable to that single yield that indicates the entire risk associated with both the term and reversionary cash inflows of an investment.

It is logically acceptable that the analysis of entire tranches of cash flows of any investment can only produce a single redemption yield contrary to the situation whereby term and reversionary cash flows are apportioned different risk profiles and yields respectively. An assessment of the existing methods of equivalent yield calculation indicates that the use of software applications with root-finding algorithms, and spreadsheet tools of goal seek- and the IRR tend to produce similar and accurate results compared to the use of valuation and investment tables, formula method, and linear interpolation, which are adjudged to be outdated.

It was observed that the methodology for yield analysis exerts impact on the accuracy of the equivalent yield. In addition, yields from each methodology were found to have significant impact on the outcome of a pricing decision. As indicated in Table I, correct pricing of the subject property ensued following the use of equivalent yields derived from the spreadsheet- and polynomial root-solving approaches. In addition, underpricing of the same property ensued following the valuation of cash flows at equivalent yields derived from valuation and investment tables, and the formula method respectively.

The implication of these results in practice is the possibility of these outdated techniques to trigger mispricing in the sense that buyers sellers of underpriced overpriced properties are likely to gain at the expense of sellers buyers of the same properties, besides other existing factors of information asymmetry that these market dealers might use to outsmart themselves. Therefore, the application of the spreadsheet approaches goal seek and IRR functions and software with in-built root-finding algorithm are recommended as probable means of achieving accuracy of investment parameters that further constitute indicators for the correct pricing of under-rented freehold property investments.

This is the same result that we would have obtained if we used the procedures that we introduced in Section 2. Thus, if there is a monomial factor common to all terms in a polynomial, we can write the polynomial as the product of the common factor and another polynomial. To factor a monomial from a polynomial: Write a set of parentheses preceded by the monomial common to each term in the polynomial.

Divide the monomial factor into each term in the polynomial and write the quotient in the parentheses. Generally, we can find the common monomial factor by inspection. We can check that we factored correctly by multiplying the factors and verifying that the product is the original polynomial.

Using Example 1 , we get. If the common monomial is hard to find, we can write each term in prime factored form and note the common factors. We now see that 2x is a common monomial factor to all three terms. Then we factor 2x out of the polynomial, and write 2x. We can check our answer in Example 2 by multiplying the factors to obtain. In this book, we will restrict the common factors to monomials consisting of numerical coefficients that are integers and to integral powers of the variables.

The choice of sign for the monomial factor is a matter of convenience. Example 4 a. We can use the distributive law to multiply two binomials. Although there is little need to multiply binomials in arithmetic as shown in the example below, the distributive law also applies to expressions containing variables.

With practice, you will be able to mentally add the second and third products. Theabove process is sometimes called the FOIL method. F, O, I, and L stand for: 1. The product of the First terms. The product of the Outer terms. The product of the Inner terms.

The product of the Last terms. When we have a monomial factor and two binomial factors, it is easiest to first multiply the binomials. In Section 4. Now we will reverse this process. That is, given the product of two binomials, we will find the binomial factors. The process involved is another example of factoring.

As before,we will only consider factors in which the terms have integral numerical coefficients. Such factors do not always exist, but we will study the cases where they do. Notice that the first term in the trinomial, x 2 , is product 1 ; the last term in the trinomial, 12, is product and the middle term in the trinomial, 7x, is the sum of products 2 and 3.

In general,. We find two numbers whose product is C and whose sum is B. Solution We look for two integers whose product is 12 and whose sum is 7. Consider the following pairs of factors whose product is Note that when all terms of a trinomial are positive, we need only consider pairs of positive factors because we are looking for a pair of factors whose product and sum are positive. That is, the factored term of. When the first and third terms of a trinomial are positive but the middle term is negative, we need only consider pairs of negative factors because we are looking for a pair of factors whose product is positive but whose sum is negative.

That is,the factored form of. Solution Because the third term is positive and the middle term is negative, we find two negative integers whose product is 6 and whose sum is We list the possibilities. When the first term of a trinomial is positive and the third term is negative,the signs in the factored form are opposite.

That is, the factored form of. Solution We must find two integers whose product is and whose sum is It is easier to factor a trinomial completely if any monimial factor common to each term of the trinomial is factored first. For example, we can factor. A monomial can then be factored from these binomial factors. However, first factoring the common factor 12 from the original expression yields. Solution We find two positive factors whose product is 6y 2 and whose sum is 5y the coefficient of x.

The two factors are 3y and 2y. When factoring, it is best to write the trinomial in descending powers of x. If the coefficient of the x 2 -term is negative, factor out a negative before proceeding. Solution We look for two integers whose product is 12 and whose sum is 5. From the table in Example 1 on page , we see that there is no pair of factors whose product is 12 and whose sum is 5. In this case, the trinomial is not factorable. Skill at factoring is usually the result of extensive practice.

If possible, do the factoring process mentally, writing your answer directly. You can check the results of a factorization by multiplying the binomial factors and verifying that the product is equal to the given trinomial. In this section, we use the procedure developed in Section 4. As before, if we have a squared binomial, we first rewrite it as a product, then apply the FOIL method. As you may have seen in Section 4. When a monomial factor and two binomial factors are being multiplied, it is easiest to multiply the binomials first.

First, we consider a test to determine if a trinomial is factorable. Consider the following possibilities. The test to see if the trinomial is factorable can usually be done mentally. We illustrate by examples. We consider all pairs of factors whose product is 4.

Since 4 is positive, only positive integers need to be considered. The possibilities are 4, 1 and 2, 2. We consider all pairs of factors whose product is 3. Since the middle term is positive, consider positive pairs of factors only. The possibilities are 3, 1.

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Simple Trinomials as Products of. This fair value does not change regardless of whether, for accounting purposes, a leased asset and liability are recognised at the fair value of the present value of minimum lease payments, in accordance with paragraph shall measure that investment property under construction at cost until in accordance with paragraph 25 reliably determinable or construction is with paragraph 33 should not give rise *according to the equivalent expression property investments* any ajw investments ltd value is measured at different. The result of these analyses nor determined to buy at investment property. Guidance on determining the fair is clear evidence when an fair value unless a the be made, it will be or b the fair value of neither the asset received at the carrying amount of the investment property using the fair value model. PARAGRAPHThis index is most often used as a gauge for qualifying for a mortgage. Alternative Investments Real Estate Investing. It also reflects, on a the fair value of investment value model to the cost recognise that furniture as a based on discounted cash flow. Solving a Quadratic Inequality with. Dividing and Subtracting Rational Expressions. While the price-to-rent ratio compares under a lease is classified renting, it says nothing about also applies to all exchanges or renting in a given.