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Can you help with an Investment class (FINC 3231) question?

I am taking my last class for college graduation and am in need of help with this homework problem. I have done the first paragraph but I am baffled by the second paragraph. Please help if you can. The question is this:
 
1. An investor is looking at a $1.5 million parcel of land. She has two choices: 1) she can put 20% down and use a 15 year loan rate found in the WSJ. 2) She can put no money down and use a 30 year loan rate found in the WSJ but she will have to pay private mortgage insurance (PMI) of 0.1% a month on the balance of the loan until she has 20% equity in the property. (Look up PMI on the internet if you are not sure how this works.) The property will appreciate in value at a rate of 0.2% a month.

**Determine which option she should choose. Assume the opportunity cost of her down payment is 0.25% a month. Setup the cash flows in each case and then create a column that nets them out. Think about how to find when she no longer needs to pay the monthly PMI fee. Calculate the NPV using the 15 year loan rate as the discount rate.** 

Create a data table where the column input is the difference in the mortgage rates, go from 0% to 2% in increments of 0.1%. The NPV is the other variable.
Create another data table where the column input is the PMI rate; go from 0% to 0.2% in increments of 0.01%. Again the other variable is NPV.
Create another data table where the column input is the down payment amount; go from 0% to 20% in increments of 1%. Again the other variable is NPV.
Discuss the findings.
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1 Answer

Twenty percent down is $300,000.   
 
Option 1:  20% down.  She owes 1,200,000 on which she pays x% of the balance per year where x is the WSJ rate for 15 year.
 
Option 2:  0% down.   She owes 1,500,000 on which she pays (y + .1)% of the balance were y  is the WSJ rate for 30 year loans.  After her equity in the property is greater than or equal to 20%, then she pays y% of the balance per month.
 
The property appreciates at .2% per month compounded.
 
Since we don't know how much she pays and how often.  Without this information, it is not possible to choose.  For example, if her payment is $1,200,000, then the first option is best.   If she makes no payments then she will loos the property in any case and which one is best depends on which state she lives in - if she lives in California she can use the parcel for nothing until the bank can throw her off,  so the second option with no down payment is the best.  In these two cases the appreciation doesn't matter.
 
So it's impossible to answer the basic question asked here without more information.  The problem doesn't say that in the first case she will make 15*12 payments or in the second case she will make 30*12 payments - it just says she will get the corresponding loan rate on the balance.  
 
Let's suppose that she does get a 15 year loan in the first case and a 30 year loan in the second case.
 
She pays:

On a 15 year loan
3% payments of 10358.72 
4% payments of 11095.32
5% payments of 11861.90
=================================

On a 30 year loan
3% payments of 6324.06
4% payments of 7161.23
5% payments of 8144.25

Until 20% equity, then

3% payments of 6405.25
4% payments of 7247.88
5% payments of 8052.32
 
 
 
In 30 years the value of the property is 1,500,000 * (1.002)360  = 3,079,434.78 .
 
 
 
In 30 years she will have paid
 
On a 15 year loan
3% payments  1,864,569.60
4% payments  1,997,157.60
5% payments  2,135,142.00
=================================

On a 30 year loan
3% payments  2,276,661.60
4% payments  2,578, 042.80
5% payments  2,931,930.00

Until 20% equity, then

3% payments  2,305,890.00
4% payments  2,609,236.80
5% payments  2,898,835.20
 
 
 
The total payments on the 30 year option is an average of the payments before 20% equity and after.
 
$300,000 should be added to the 15 year payments.
 
So this amount should be added to the payments for 15 year loan:
 
3%  2,164,569.60
4%  2,297,157.60
5%  2,435,142.00
 
Note that this analysis does not take into account the future value of the payments or of the down payment.   It appears that the 15 year option is superior; however, The buyer is giving up the ability to invest the $300,000, presumably at the same rate that the property appreciates.  If she did this, it would subtract from the total payments for the 30 year option.  This would reduce the total value obtained from the property for the 15 year option to 
 
property value after 30 years minus lost opportunity
 
 3,079,434.78 - 315,886.96 = 2,763,547.82
 
So the profit after 30 years is:
 
 On a 15 year loan
3% 2,763,547.82 - 2,164,569.60 
4% 2,763,547.82 - 2,297,157.60
5% 2,763,547.82 - 2,435,142.00
=================================

On a 30 year loan
3% 2,763,547.82 - 2,276,661.60  to 2,763,547.82 - 2,305,890.00
4% 2,763,547.82 - 2,578,042.80 to 2,763,547.82 - 2,609,236.80
5% 2,763,547.82 - 2,931,930.00 to 2,763,547.82 - 2,898,835.20

===========================
 
The above calculations did not include taxes, which do not take inflation into account.  
 
The property taxes that would need to be paid would probably be about 1% of the value of the property. Maintenance would also be about the same.  So let's assume she pays 2% of the current value of the property for property taxes and maintenance.  That would be between  $30,000 and $61,588.70 per year in property taxes. This means that an average of $45,794.35 per year must be subtracted from the profit.  30 * 45,795.35 is $1,373,830.50.  
 
In addition, if she sells the property after 30 years, she gets a 250,000 exemption on profits.  This means 
she has to pay taxes on 
 
3,079,434.78  - 1,500,000 = 1,329,434.78
 
Taxes on long-term income are about 20% now, so she would pay $265,886.95 in taxes; in addition Obamacare adds a 3% tax on the total value of the property, so she would pay an additional $92,383.04 in taxes.
 
The total tax would be $358,269.99 and this needs to be subtracted from the value of the property.
 
So it's apparent that none of the options she has are very good. She doesn't make a lot of money under these assumptions. As her adviser, I would recommend that she do something else with the money.
 
However, if she insists on buying the property, I would recommend the 15 year loan options.  In addition if she selected the 15 year option she could sell the property in 15 years and take advantage of other opportunities.
 
 
 
 
 
 
 
 
 
 
 

Comments

Hi Kenneth G., I do apologize. The 15 year rate is 3.57% and the 30 year rate is 4.5%. All this is done in Excel. 
 
Thanks for your assistance.
Payments are made monthly. The state is Georgia, but the question is in general. Thanks again.