Jenny Walters, who owns a real estate agency, bought an old house to use as her
ID: 1218253 • Letter: J
Question
Jenny Walters, who owns a real estate agency, bought an old house to use as her business office. She found that the ceiling was poorly insulated and that the heat loss could be cut significantly if six inches of foam insulation were installed. She estimated that with the insulation, she could cut the heating bill by $80 per month and the air-conditioning cost by $70 per month. Assuming that the summer season is three months (June, July, and August) of the year and that the winter season is another three months (December, January, and February) of the year, how much can Jenny spend on insulation if she expects to keep the property for five years? Assume that neither heating nor air-conditioning would be required during the fall and spring seasons. If she decides to install the insulation, it will be done at the beginning of May. Jenny's interest rate is 6% compounded monthly.Explanation / Answer
The foam insulation is installed in the month of May, beginning from now. The interest rate is 6 % per annum or 0.5 % monthly. We need to calculate the accumulated value of present value of all the benefits that the insulation provides.
In summers, in the month of June, July and August, the benefit per month amounts to $70 accruing to Jenny till next five years. Now in the month of June, the second month after installation, the future benefit is $70. Similarly, the future benefit is $70 in the month of July, the third month after installation, and lastly, the fourth month August, also saves $70.
In this way, 14th, 15th and 16th months of next year will save a future value of $70 each month, and continuing this, the final year will have 50th, 51th and 52th month of summer that will save $70 for each of these months. Compute the present value of these future benefits at r = 0.5%
PV = 70 [ 1/(1.005)2 + 1/(1.005)3 + 1/(1.005)4 + 1/(1.005)14 + + 1/(1.005)15 + 1/(1.005)16 + 1/(1.005)26 + 1/(1.005)27 + 1/(1.005)28 + 1/(1.005)38 + 1/(1.005)39 + 1/(1.005)40 + 1/(1.005)50 + 1/(1.005)51 + 1/(1.005)52]
= $921
In winters, in the month of December, January and February, the benefit per month amounts to $80 accruing to Jenny till next five years. Now in the month of December, the eighth month after installation, the future benefit is $80. Similarly, the future benefit is $80 in the month of January, the ninth month after installation, and lastly, the tenth month of February also saves $80.
In this way, 20th, 21st and 22nd months of next year will save a future value of $80 each month, and continuing this, the final year will have 56th, 57th and 58th month of summer that will save $80 for each of these months. Compute the present value of these future benefits at r = 0.5%
PV = 80 [ 1/(1.005)8 + 1/(1.005)9 + 1/(1.005)10 + 1/(1.005)20 + + 1/(1.005)21 + 1/(1.005)22 + 1/(1.005)32 + 1/(1.005)33 + 1/(1.005)34 + 1/(1.005)44 + 1/(1.005)45 + 1/(1.005)46 + 1/(1.005)56 + 1/(1.005)57 + 1/(1.005)58]
= $1021.55
Hence the present value of total benefits is 1012.55 + 921 = $1942.544. This is the amount she can afford on insulation. The table summerizes this.
FV of Benefit Summers
Discount Factor
PV of benefit
FV of Benefit Benefits
Discount Factor
PV of benefit
70
0.990075
69.30522
80
0.960885
76.87082
70
0.985149
68.96041
80
0.956105
76.48837
70
0.980248
68.61733
80
0.951348
76.10784
70
0.932556
65.27895
80
0.905063
72.40503
70
0.927917
64.95418
80
0.90056
72.04481
70
0.9233
64.63103
80
0.89608
71.68638
70
0.87838
61.48659
80
0.852484
68.19869
70
0.87401
61.18069
80
0.848242
67.85939
70
0.869662
60.87631
80
0.844022
67.52178
70
0.827351
57.91455
80
0.802959
64.23671
70
0.823235
57.62642
80
0.798964
63.91712
70
0.819139
57.33972
80
0.794989
63.59913
70
0.779286
54.55002
80
0.756311
60.5049
70
0.775409
54.27863
80
0.752548
60.20388
70
0.771551
54.00859
80
0.748804
59.90436
13.15727
921.0086
12.76936
1021.549
FV of Benefit Summers
Discount Factor
PV of benefit
FV of Benefit Benefits
Discount Factor
PV of benefit
70
0.990075
69.30522
80
0.960885
76.87082
70
0.985149
68.96041
80
0.956105
76.48837
70
0.980248
68.61733
80
0.951348
76.10784
70
0.932556
65.27895
80
0.905063
72.40503
70
0.927917
64.95418
80
0.90056
72.04481
70
0.9233
64.63103
80
0.89608
71.68638
70
0.87838
61.48659
80
0.852484
68.19869
70
0.87401
61.18069
80
0.848242
67.85939
70
0.869662
60.87631
80
0.844022
67.52178
70
0.827351
57.91455
80
0.802959
64.23671
70
0.823235
57.62642
80
0.798964
63.91712
70
0.819139
57.33972
80
0.794989
63.59913
70
0.779286
54.55002
80
0.756311
60.5049
70
0.775409
54.27863
80
0.752548
60.20388
70
0.771551
54.00859
80
0.748804
59.90436
13.15727
921.0086
12.76936
1021.549
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