Question

Question 1

At the beginning of 1976 a relative migrated to Australia with $10,000 ‘spare cash’. The money could have been used to buy a block of land or invested in an ‘at-call’ savings account that paid interest at 8% p.a. compounded half-yearly. At the end of 2018, the land was valued by a local real estate agent who was keen to list the property on behalf of his agency, at a price of approximately $400,000.

Required:

1. Which of the alternative investments had the higher value at the end of 2018? Justify your response with appropriate calculations.

(Students should write no more than 50 words for this part of the question).

2. i)Assuming the half-yearly compounding of interest, what was the rate of growth in the land value over the total period expressed as a nominal annual interest rate?

2. What was the rate of growth in the land value over the total period expressed as an effective interest rate?

3. What was the rate of growth in the ‘at-call’ savings account over the total period expressed as an effective interest rate?

3. What investment in the savings account would have been necessary at the beginning

of 1976, to have the same value as the land was worth at the end of 2018? Briefly explain your response.

(Students should write no more than 50 words for this part of the question).

4. Recognising both your finance skills and ‘common sense’, one of your friends has asked whether your calculations above allow you to determine which of the investments would have been ‘better’ to make at the beginning of 1976, given the outcomes discussed above at the end of 2018. Provide a well-reasoned, complete response to this question taking into consideration various financial and non-financial issues.

(Students should write no more than 100 words for this part of the question).

e. i)You have now been provided further information that the investment in the land required the owner to make continuous annual payments of council rates over the total period held. These amounts are determined in accordance with Table 1 below. Assuming the land was sold at the end of 2018 (but ignoring the expected sale value), what is the adjusted present value at the beginning of 1976 of all the cash outflows relating to the acquisition and continued ownership of the land?

Note:     For the purposes of this question assume the following:

1. Rates are payable on the anniversary of each year of land ownership.

2. The annual amount of the rates are determined in accordance with the following formula;

Initial Purchase Cost ($) x Factor (times) x Relevant Percentage (%)

3. Rates are still payable for the 2018 year (for the full year).

Anniversary number	Factor	Relevant
of years land held	(times)	Percentage (%)
1 to 5 years	1.0	1.5
6 to 10 years	1.5	1.5
11 to 15 years	3.0	1.0
16 to 20 years	6.0	1.0
21 to 25 years	10.0	0.8
26 to 30 years	20.0	0.8
31 to 35 years	25.0	0.6
36 to 40 years	30.0	0.6
41 to 45 years	40.0	0.4

Table 1

2. Taking into consideration the calculations from part e) i) of this question, what is the rate of growth in the land value over the total period expressed as an effective interest rate?

(Students should write no more than 50 words for this part of the question).

Answer 1

Let us start the problem by listing down the data on hand;

Money at hand is the same as present value of money i.e $10,000

The nos of years the money is invested for is 2018-1976+1 = 43 years (From beginning of 1976 to end of 2018)

If we consider half yearly compounding, the number of periods is 2 x 43 = 86.

a) Now the easiest way to calculate the future value of the investment in at-call savings account would be to use Excel formula given by =FV(rate,nper,pmt,PV)

Here the rate is 8% divided by 2 since the compounding is done half yearly hence it is 4%, nper is the number of time periods which in this case is 2 x 43 = 86, Since there is no intermediate payment made the pmt value of 0 and finally the present value is -$10,000 since we are depositing the money at the beginning of the period into the bank.

Hence we calculate the future value of the investment as such =FV(4%,86,0,-10,000) and this gives us the value of $ 291,654.5.

The real estate agent had valued the land at $400,000 hence the investment in land is a better option

b) i) In this instance again we use excel to determine the Effective interest rate using the formula =Rate(nper,pmt,pv, FV)

Here we have PV = -$10000, nper = 86, FV = $400000, pmt = 0

Using the above formula we work thus =Rate(86,0,-10000,400000) and obtain the Effective rate as 4%.

Now using the formula of Nominal Annual Interest Rate r= m[(1+i)^{1/m _-1]}

Using m= 86, i =4% as obtained earlier, we calclulate r = 3.9223% hence the Nominal Annual Interest rate is 3.9223%

ii) The growth in the value of the land as expressed as Effective Annual interest rate would be

(400000/10000)^^1/43 -1 = 0.08957 or 8.96%

iii) Similarly for the at-call money deposited in bank the Effective interest rate is

(291654.5/10000)^{^1/43} -1 = 0.08159 or 8.16%

c) The problem here asks for the amount of money one would have to invest in bank at the rate mentioned in order to have the same value at the end of the period that is the same as the value of the land i.e $400,000

Again using excel we can easily determine the value using =PV(rate,nper,pmt,FV)

we can calculate using the following parameters rate = 8%/2, nper = 86,pmt=0 and FV = $400,000 (Value of land in 2018)

Hence PV(8%/2,86,0,400000) is equal to -13714.9 rounded to -13715. Hence an investment of $13715 is to be made in the beginning of 1976 to be able to have the same value as the value of the land at the end of 2018.

d) Based purely on finance calculations, land seemed to be a much better investment option compared to investment in at call savings account on the bank. But there are several other assumptions that have been made to arrive at this result which would be different if the underlying assumptions were not valid for some reason or the other.

1. First the interest rate of 8% compounded half yearly was assumed to be constant and over the duration of the 43 years. This is a very difficult assumption to make. The Interest rates change depending upon several factors one of which could be inflation and this could change the calculations.
2. If the interest rate could go up the money could be reinvested at higher interest rates thereby generating more returns
3. The real estate agent had estimated the value of the land at $400,000.00 however we do not know the basis of this estimation. Was there a similar market transaction that had happened which could be the basis of his estimation. We do not know for sure.
4. Finally during actual sale of the land, we do not know the state of the economy and this would impact the nos of buyers and hence also the actual sale price could be lower than the estimated price. This would depend upon whether the sale was a distress sate or not.
5. Also in this calculation there is no mention of the amount of tax or other costs associated with the possession of the land that would have to be paid is not explicitly mentioned. These taxes could effect the end result significantly

e)i ) This is a complex calculation consisting of several cash outflows during the course of the land holding and the best way to do the calculation is by calculating the NPV value using the NPV function in Excel

There are different cash outflows like $150 in the first 5 years and $ 225 from year 6 through to year 10 and if we lay out all these cashflows year wise and finally the Futire value of $400,000 at the end of the 43 year period and then use the NPV function then we have the adjuested present value of money that would be required at the beginning of 1976

ii) Lastly for the effective interest rate the same cash flow schedule as laid out in ei) would have to be used only here we use the excel function IRR instead of NPV and the effective rate would be determined