Forms of Business Investment

ANNUITIES & SINKING FUNDS 8
Annuities & Sinking Funds
Introduction
When it comes to business investing and expansion, two tools in the toolbox that are often used include sinking funds and annuities. These tools may be a little hard to understand for the layperson. However, with a little consideration and review, the picture can be quite clear. The scenario posed for this assignment is fairly basic. There is a business that owns a vinyl fencing company. There are two purchases that are on the horizon. One is an annuity as is being offered by an insurance company. Regardless of the option chosen, the stated need in terms of savings is $2600 every six months. There is a choice of an ordinary annuity and an annuity due. The second matter centers on the need to purchase a machine. The machine costs $23,000 and it needs to be purchased within the next three years as that is when the existing machine will reach the end of its useful life. The idea is to use a sinking fund that compounds quarterly and that earns an annual rate of 7.2 percent. While one must be careful and diligent in making the necessary calculations, the choices to be made in the above scenarios are quite clear.









Purchase One - Annuity Decision
Ordinary Annuity
The scenario for the annuity makes is clear that the desired payment per six months is $2600. However, since the annuities in question both compound semi-annually, that makes the formula quite easy to render. The ordinary formula and variables would be as follows:

(UTEP, 2018).

· As can be seen by the contextual paragraph noted above, PMT would be $2600 because that is the desired payment per compounding period. Similarly, if a zero balance is acceptable at the end, then that means PV is what needs to be in the annuity right now.
· Also noted in the paragraph is what “i” would be. It would be r/m, where r is the annual rate and m is the amount of times compounded per year. Since the compounding is semi-annual, m would be 2. The rate, as given in the problem, is five percent, or 0.05. In other words, r/m would become 0.05 / 2, which is 0.025. Since i has already been calculated, most of the bottom is taken care of.
· Based on the above, we can figure out what -n would be. Over five years, there would be 10 periods of compounding (5 years * 2 times per year). Thus, -n would be -10. Given that, the bottom would become 1 – (1 + 0.025) -10. Parentheses and exponents have to be cleared first, in that order. Thus, one would first simplify that to 1 – (1.025) -10. 1.025 to the negative tenth power would be 0.7812, rounded to the fourth digit. This leaves 1 – 0.7812, or 0.2188. This leaves…
· 2600 = PV (0.025/0.2188)
· 2600 = PV (0.1143)
· 2600 / 0.1143 = PV
· PV = 22747.16















Annuity Due
Basically, the investment would need to be $22,747 (or more) for the ordinary annuity option. The formula for the annuity due version has the same variables. However, the structure is different, as shown below…

(Finance Formulas, 2018)
Of course, the payment would be 2600, as noted before. That would be the P in the formula above. R would remain 0.05 and the number of periods would remain 10.
· The top of the fraction would be wise to solve first…and it looks like this…
· 1 – (1 + 0.05) – (10 – 1)
· 1 – (1.05) – (9)
· 1 – 0.64
· 0.36


· One should then plug that into the rest of the formula
· 2600 + 2600 (0.36 / 0.05)
· 2600 + 2600 (7.2)
· 2600 + 18720
· 21320
Conclusion
In short, there is not a huge difference between the two methods. However, the annuity due method requires approximately $1000 less in investment. Thus, it is clearly the superior option. A thousand bucks may not seem like a lot in the grand scheme of things. However, that money could easily be spent on other things. Thus, the annuity due option is the way to go for the given scenario.














Sinking Fund
The details of the sinking fund are fairly basic. Essentially, the machine to be replaced will be obsolete in three years. Between now and then, the cost to replace the machine must be secured. With that in mind, the decision has been made to use as sinking fund that compounds quarterly at an annual rate of 7.2 percent. The total amount that must be raised is $23,000. The formula to figure out what must be invested is as follows:

(UTEP, 2018)
Here is the workflow for this formula:
· PMT would be the amount that must be paid per period to get the desired outcome. That is the figure we are not aware of as of yet.
· FV would be $23,000…since that is the amount that must be accumulated.
· The number of compounding periods per year…or m, would be four (quarterly)
· The annual rate would be 0.072, per the stated problem.
· i = r / m…so that would be 0.018. This is the interest per quarter.
· t would be 3, given that this is over three years
· n would be mt, per the above, so that would be 12…number of compounding periods

· As with prior formulas, getting the complex part of the fraction simplified is the first thing to do…
· (1 + 0.018)12 – 1
· (1.018) 12 – 1
· 1.24 – 1
· 0.24
· Thus, we get the following
· 23000 (0.018 / 0.24)
· 23000 (0.075)
· 1725
Thus, the monthly payment for the sinking fund would be $1725 if the business wants to be able to replace their equipment in three years. The upside of using a sinking fund is pretty clear. Indeed, the money that has to be sunk into the sinking fund is less on the front end and makes more than the annuity on the back end. In the end, if cash flow is more important for other things in the near-term, then a sinking fund is the way to go. However, if liquidity is good, then an annuity can be as safer option. Indeed, if the sinking fund payments are missed or otherwise bungled, that could lead to equipment not being replaced when it is worn out. Of the two scenarios posed in this assignment, the sinking fund/equipment purchase is clearly more important because the equipment is necessary to continue operations. Expanding to a new warehouse facility is not unimportant. However, even the existing business level with the existing warehouse will suffer if the post-maker is no longer usable and money is not there to replace it (ETG, 2018).
References
ETG. (2018). How to manage the risks of obsolete technologies - Eagle Technologies. Eagle
Technologies. Retrieved 16 February 2018, from https://eagletechnologies.com/how-to-
manage-the-risks-of-obsolete-technologies/
Finance Formulas. (2018). Present Value of Annuity Due - Formula and Calculator.
Financeformulas.net. Retrieved 16 February 2018, from http://financeformulas.net/
Present_Value_of_Annuity_Due.html
UTEP. (2018). Annuities and Sinking Funds. math.utep.edu. Retrieved 16 February 2018, from
http://www.math.utep.edu/Faculty/cmmundy/Math%201320/Worksheets/Sinking%20Fu
nds%20&%20Annuities.pdf