Y could lend to Z at an interest rate of 11<i<12 and then borrow from W with 8<i<11  to invest in its own project.

[NEAS: If so, W would lend to Z at an interest slightly below Y's but more than 11%.]

I think that the rate could be 11%. Y would bid the rate down to 11% but at that point they would be indifferent so the rate could stay right there.  I believe that the solution is 8<i<=11.

[NEAS: There are arguments for strict or non-strict inequality; don't worry about that aspect.]

Jacob: What does this homework assignment teach us?

Rachel: The homework assignment shows the value of borrowing and lending. For Part A, Firms Y and Z each has $10 million to invest, so they each pick their better project. Firm W invests in both projects. If firms can not borrow or lend, they ignore the projects that other firms have.

For Part B, the $40 million in cash is used for the four best projects. Firm W has extra cash; its second project yields only 8%. Firm Z has too little cash; its second project yields 12%. Firm W can not invest in Firm Z’s project. It must lend cash to Firm Z, which then invests in its second project.

Jacob: What is the lending rate (the interest rate)?

Rachel: If the rate were less than 8%, Firm W would prefer to invest in its second project, not to lend the cash to Firm Z. If it were more than 12%, Firm Z would not borrow the cash.

Jacob: So is the lending rate (the interest rate) between 8% and 12%?

Rachel: If the rate were more 11%, Firm Y would offer to lend the money at a lower rate but still higher than 11%. We conclude that the interest rate is between 8% and 11%.

Jacob: How do we know that the firms will borrow and lend?

Rachel: We assume firms are rational and seek to maximize their incomes. If the interest rate is between 8% and 11%, Firm W makes only 8% by investing in its second project but more than that by lending cash to Firm Z.

Jacob: That leaves a wide range.

Rachel: In real life, there are 100,000 firms and a million projects. The lending rate is the after-tax return on the marginal project.

Jacob: What other items should we know for this homework assignment?

Rachel: One can’t invest in a project twice. Suppose a firm is a supermarket chain. A project may be to build a supermarket in a certain town. Building one supermarket makes money. Building two stores in the same town loses money.

What about hte possibility of Z doing the 15% project, and Y lending his 10 mil to Z at a rate of betwean 11 and 12, and W investing in his 2 projects of 11 and 8...or do we assume that Y invests in his own and W invests in his 11 and lends the other 10 mil to Z at a rate of 8
I ask this becasue no one seems to have mentioned it. FOr W the latter is advantageous because he has a biger range of interst, byt for Y the first is more advantageous since he can potentially get more than the 11%

[NEAS: If so, Z would borrow from W at a rate between 8% and 11% instead of borrowing from Y at a rate between 11% and 12%.]

Company Z would invest in the project with the highest yield, 15%.

They as well would probably opt to borrow money from company W at a yield between 8% and 11% and invest it into the second project.

It would have to be at least 8% since otherwise company W would not lend their money.

It couldn't be more than 11% or else Y would get involved.

Profit for company Z would be $1,500,000 for their own money and their own project.

Profit for company Z’s second project would be calculated by:

Investment= $10,000,000*X where X is determined between company’s W and Z.

     Profit = (($10,000,000*1.12) - $10,000,000*(1+X) =(.12-X)*$10,000,000

     ROI = (.12-X)*$10,000,000/$10,000,000*(1+X).

                   = (.12-X)/(1+X)

Best case scenario for company Z is for X to be approximately 8% where the ROI for company Z’s second project would be approximately 3.7% or less depending on the value of X.

 

Worst case scenario for company Z is for X to be approximately 11% where the ROI for company Z’s second project would be approximately .9%

 

Provided that company Z’s ROI on project 2 is greater than their opportunity cost of capital they should take the deal.

 

 

This is what I come up with.  However, I am kind of unsure if this is adventageous for company Z.  Especially when I think to myself: if company Z waits a year and the project is still available they can make 12% on the investment rather that .9%-3.7%.  There should also be a risk associated with these projects.  The reason why they should borrow the money is that they are making money on money they don't really have.

 

Any other thoughts here?

 

[NEAS: The projects are available now.  A firm can't wait for a year.]

Peace.

This question neglects to mention the timing of the events (and it does matter).  The only thing that's holding W's interest rate to Z at 11% is the fact that if they go higher then Y can compete and possibly get a project that's slightly better than their current best option. 

If Y invests its money first before W and Z do anything then the rate range is between 8 and 12 non-inclusive.  It sounds picky but I'm sorry it makes a difference.

[NEAS: The firms are not competing. The projects are independent, but the first can borrow or lend from one another.]

please explain the reason of why interest rate cannot be greater than 11%.

I will try my best to help you.

y=a(x+2.4)(x-0.8) <-good job!

Now you know the parabola passes through a point(her coordinate is (1.2,–2.592) as you know).This is means even if we haven't known the letter "a" equals what,but we can conclude that where "x=1.2",it must exist"y=-2.592".So we can let x=1.2,y=-2.592 in equation to figure out the value of "a".

y=a(x+2.4)(x-0.8)
–2.592=a(1.2+2.4)(1.2-0.8)
So a=-1.8
then y=-1.8(x+2.4)(x-0.8)=3.456-2.88x-1.8x^2(… the factored form)
Please check the answer by yourself,it was calculated and expanded by mathematical software named"Mathematica".I am not responsible for its accuracy.

If you know a quadratic function just like this form:y=ax^2+bx+c (while the damned "a"doesn't equals zero;a,b or c are constants),you can know that:the x-coordinate of the vertex is -b/(2a),and the y-coordinate is (4ac-b^2)/4a.The conclusion can be inference by completing square.Every student have to recite it.

In this example“y=-1.8x^2-2.88x+3.456”,you can get it that"a=-1.8,b=-2.88,c=3.456".Then the x-coordinate of the vertex is -b/(2a)=-(-2.88)/(2×(-1.8))=-0.8,the y-coordinate of the vertax is (4ac-b^2)/4a=4.608(leave its process out for convinient).

According to "the point on the x-axis halfway between the two x-intercepts",you can also do "(-2.4+0.8)/2=-0.8" to calculated x-coordinate of the vertex.

vertex form:y=a(x+b/(2a))^2+(4ac-b^2)/4a
in your example is y=-1.8(x+0.8)^2+4.608

There is some calculated mistake in my software(maybe),please check the answers by yourself.

leasy lån