1, Midterm Econ 138
February 21, 2008
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Question 1 (NPV - 8 points)
Consider an infinite-horizon investment project that generates return Rt in period t 0. Denote the (constant) interest rate by r.
1.1
Part a (1 point)
P
-t t=0 Rt (1 + r)
Write down a basic formula for the NPV of this project. Solution: [1 point for correct formula, including specificed t = 0 and .]
1.2
Part b (2 points)
Suppose r = .05, and Rt = 1 for all t 0. What is the most you would pay for such this project, i.e., how much would you be willing to invest today, at t = 0, to get the returns to this project? Solution: Derivation of infinite geometric series (not required):
P
1 1+r 1 t 1 t=0 R( 1+r ) = R + R 1+r + R
1 t 1 t=0 R( 1+r ) = R 1+r + R r P 1 = 1+r R( 1+r )t = R t=0 P 1 t=0 R( 1+r )t = 1+r R r Here: $ 1.05 = $21. 0.05
P
1 1+r
1 1+r
2
2
+R +R
1 1+r
1 1+r
3
3
+ ... + ...
[1 point for correct formula, with or without numbers plugged in; 1 point for correct solution.]
1.3
Part c (2 points)
Now suppose instead that Rt = 5 t 0. Find an r such that this project is exactly as profitable as the one in the Part b, i.e., such that you would be willing to pay/invest exactly 1
the same amount for this project. Solution:
1+r r
[1 point for correct set-up; 1 point for correct solution.]
$5 = $21 $5 = r($21 - $5), so r =
5 16 .
1.4
Part d (3 points)
Let's return to the initial assumption that r = 0.05, and suppose Rt = 10 for all periods 0 t T and Rt = 0 for all periods thereafter (t > T ). Find the smallest value of T such that this asset is more valuable than the one in Part b. Hint 1: Derive the formula and then try out T = 0, T = 1, ... etc.
1 Hint 2: 1.05 = 0.952380952; 0.822702475
1 1.05
2
= 0.907029478;
1 1.05
3
= 0.863837599;
1 1.05
4
=
Solution: Derivation of finite geometric series (not required):
PT
1 t 1 t=0 R( 1+r ) = R + R 1+r + R
1 1+r
=
PT
1 t t=0 R( 1+r )
1 = R 1+r + R
r 1+r
Hence, the formula is
1 1.05
PT
1 t t=0 R( 1+r )
PT
1 t t=0 R( 1+r ) = R - R
1 1+r
=
1+r r R 1.05 0.05
1-
1 1+r
T = 0, 1 fail, but T = 2 works, i.e., the smallest T is T = 2.
T +1
10 1 -
0.9 (0.952380952)T +1
1 1+r
2
2
+R +R
1 1+r
T +1
1 1.05
1 1+r
1 1+r
3
3
+ ... + R + ... + R
1 1+r
1 1+r
T +1
T
T +1
T +1
21 1 -
1 1.05
T +1
1 10
9 10
[1 point for correct formula, with or without numbers plugged in; 1 point for solving it in terms of numbers; 1 point for finding the right T .]
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Question 2 (Basic Moral Hazard - 55 points)
Suppose a firm has an investment project costing I and cash C < I. The firm is run by a risk-neutral manager who, for simplicity, also owns the entire firm. The manager can either work hard, which gives no private benefit, or shirk, which yields a private benefit equal to $B. The project will either return R or 0 dollars, and the probability of a return of R is pH if the manager works hard, and pL if she shirks. As always, we assume that the market for external financing is competitive, that the interest rate is zero, and that the manager's liability is limited (i.e., she cannot be asked to pay money out of her private wealth.)
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2.1
Part a (2 points)
Suppose a lender were to provide the money to fill the financing gap I -C and to offer a contract that pays to the manager Rm when the project succeeds and 0 when it fails. Compute the manager's expected utility from both working hard and shirking (in terms of pH , pL , and B.) Solution: Expected utility from working hard: pH Rm ; expected utility from shirking: pL Rm + B. [1 point each for each piece]
2.2
Part b (1 point)
B p
Find the smallest Rm such that the manager will not shirk. Solution:
2.3
Part c (2 points)
Solution: If NPV negative, lenders lose money even for the lowest possible Rm , Rm = 0, [1 point for talking about `lowest possible Rm ] due to pL R being too small to recoup the financing outlay of the investors. [1 point for comparing to financing outlay]
Suppose that the NPV of the project is positive only if the manager works hard, and that pL R I - C. Explain why this means financing will not occur unless the manager works hard.
2.4
Part d (3 points)
Provide a definition of `pledgeable income,' the formula for pledgeable income, and the inequality for the financing condition. Solution: Pledgable income is largest amount the manager can offer to give lenders when the project succeeds while still finding it worthwhile to work hard. [1 point] Formula: (R -
B p ).
[1 point]
B p ).
Financing condition is I - C pH (R -
[1 point]
2.5
Part e (3 points)
Explain how changes in B, C and pH - pL affect whether or not the project can be financed. Give (correct) intuitive reasons for each of these comparative statics. Solution: Intuition: lower C means more financing outlay, potentially more than pledgeable income; higher B means higher temptation to shirk, hence higher incentive pay required to prevent manager from shirking, hence lower pledgeable income; lower difference in probability p 3 Financing is more difficult the lower is C, higher is B or lower is pH - pL .
means that a given amount of incentive pay Rm is less effective in convincing the manager not to shirk, hence Rm needs to be increased, hence the pledgeable income is lower. [overall 3 points, 1 for correct direction + correct explanation for each of the three variables; no point if either direction or explanation is wrong.]
2.6
Part f (1 point)
Suppose I = 100, R = 150, pH = .75, pL = .25, and B = 15. What is the smallest value of C for which the firm can find financing? Solution: 0.75(150 -
15 .5 )
= 100 - C C = 100 -
3 4
120 = 10
For the remaining parts of Question 2 assume the following: Going back to the general set-up (i.e., forget about the concrete numbers in part f), let's now assume that the manager also has illiquid assets A, for example, a machine used for production. Illiquid wealth cannot be used to finance a project (--the machine is needed for production!--), but can be sold after the project is over. Let's further assume pL R + A < I - C.
As before consider a financing contract that allocates an amount Rm to the manager in case of success and 0 otherwise. In addition, the contract specifies who gets A after the project is over (--Does the manager remain the owner of A? Or do the investors seize A?--) and how that depends on success and failure.
2.7
Part g (9 points)
Show that pledgeable income is maximized if A goes to the investors in case of failure (regardless of what happens with A in case of success.) In order to show this, first calculate the minimum Rm needed to convince the manager to work hard, separately for four cases: (1) the manager keeps A both in case of success and in case of failure, (2) she keeps A NOT in case of success BUT in case of failure, (3) she keeps A in case of success BUT NOT in case of failure, (4) she keeps A NEITHER in case of success NOR in case of failure. Then calculate the pledgeable income in those four cases. Solution: As before, the investors want the manager to work hard; otherwise they cannot recoup their financing outlay (pL R + A < I - C).
The manager works hard if his payoff in case of success is higher than in the case of failure. Hence, to induce the manager to work hard, the following inequality needs to hold 1. if she keeps A both in case of success and in case of failure: pH Rm + A pL Rm + B + A 4
[1 point]
Rm
B p
2. if she keeps A NOT in case of success but in case of failure: pH Rm + (1 - pH )A pL Rm + B + (1 - pL )A (pH - pL )Rm B + (pH - pL )A B Rm +A p [1 point] 3. if she keeps A in case of success BUT NOT in case of failure: pH (Rm + A) pL (Rm + A) + B (pH - pL )Rm B - (pH - pL )A B Rm -A p [1 point] 4. if she keeps A NEITHER in case of success NOR in case of failure: pH Rm pL Rm + B B Rm p
[1 point] The resulting pledgeable income is:
1. if the manager keeps A both in case of success and in case of failure: pH R - [1 point] 2. if she keeps A NOT in case of success but in case of failure: pH R + A - [1 point]
B p
B B - A = pH R - p p
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3. if she keeps A in case of success BUT NOT in case of failure: B B +A + A) + (1 - pH )A = pH R - pH (R - p p [1 point] 4. if she keeps A NEITHER in case of success NOR in case of failure: pH (R + A - [1 point] Hence, pledgeable income is maximized if A is allocated to the investors in case of failure (last two cases). [1 point] B B ) + (1 - pH )A = pH R - +A p p
2.8
Part h (4 points)
Your calculations in Part g have shown that we obtain the maximum pledgeable income if A goes to the investors in case of failure. Does it matter who gets A in case of success? Discuss in particular whether Rm can become negative. Solution: It does not matter [1 point for the correct conclusion] because:
B B B If p A [1 point for distinguishing the cases p A and p < A], then both the B minimum Rm in case A goes to the manager in case of success, p -A, and the minimum B Rm in case A goes to the investors in case of success, p , are positive; no problem here.
If
< A, then the both minimum Rm in case A goes to the manager in case of success, - A, is negative -- which may first seem to contradict limited liability [1 point for realizing that Rm < 0 might be a problem because of limited liability] -- but the manager can finance it by selling the asset; so again no problem! [1 point for realizing that problem solved with selling A.] And the minimum Rm in case A goes to the investors in case of B success, p , is again positive; no problem here.
B p
B p
2.9
Part i (2 points)
Based on your calculations above, we consider the contract that pays the manager Rm in case of success and allocates A to the investors regardless of success or failure. Derive the new minimum amount of cash the manager needs to have in order to obtain financing. Solution:
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Here, the minimum Rm is
B p .
Hence, the minimum cash is determined by pH (R -
B )+A=I -C p B ) C = I - A - pH (R - p [2 points]
2.10
Part j (2 points)
Going back to the numbers from Part f, i.e., I = 100, R = 150, pH = .75, pL = .25, and B = 15, assume that the manager also has a machine worth $20. What is the smallest value of C for which the firm can find financing? (Stick to the contract that pays the manager Rm in case of success and allocates A to the investors regardless of success or failure.) Solution: The minimum C = 100 - 20 - 3 120 = -10 [1 point], i.e., even if the entreprenur has no 4 cash at all she will receive financing. [1 point]
2.11
Part k (8 points)
Let's further assume C = 50, still assuming the numbers above and the contract that pays the manager Rm in case of success and allocates A to the investors regardless of success or failure. Under the assumption of competitive markets for outside financing, what is the net payoff the investors obtain, in case of failure and in case of success, and what is their expected payoff? What is the payoff the manager obtains, in case of failure and in case of success, and what is her expected payoff? (Hint: You first need to calculate the actual managerial rent Rm , which can be different from the minimum managerial rent that you calculated above.) Solution: The managerial rent is determined by pH (R - Rm ) + A = I -C
pH Rm = pH R + A - (I - C) 1 (A - (I - C)) Rm = R + pH 4 = Rm = 150 + (20 - 50) 3 Rm = 110 [2 points] Hence, investors obtain the following net payoff: 7
in case of success: R - Rm + A - (I - C) = 150 - 110 + 20 - 50 = 10 [1 point] in case of failure: A - (I - C) = 20 - 50 = -30 [1 point] expected payoff:
3 1 4 10 + 4 (-30)
= 0 [1 point]
The manager obtains the following net payoff: in case of success: Rm = 110 [1 point] in case of failure: 0 [1 point] expected payoff:
3 4 110
= 75 + 7.5 = 82.5 [1 point]
2.12
Part l (5 points)
Throughout this problem we have been assuming that the manager wants to implement the project. Using the concrete numbers from Part j, show that the manager is, indeed, better off implementing the project than not implementing the project, i.e. compare her net payoffs if she implements the project and if she doesn't. How does the difference in payoff relate to the NPV of the project? Solution: Her expected payoff if implementing is 3 110 = 82.5. [1 point] 4 Her payoff if not implementing is C + A = 70. [1 point] The difference is exactly equal to the NPV of the project pH R - I = 3 150 - 100 = 12.5. [2 4 points] That is she obtains the full expected surplus. Reason? Competitive markets for external financing. [1 point]
2.13
Part m (12 points)
Now assume that the machine is worth more in the hands of the manager than in the hands of the investors. One common reason for such a situation is that the investors can only sell it (they cannot `work' with it), while the manager can use it for more good investment projects. Concretely, assume that A = 40 if left to the manager and A = 20 if seized by the investors. All other values remain as above: i.e., I = 100, R = 150, pH = .75, pL = .25, and B = 15. What is now the best contract to maximize pledgeable income? Provide an intuition for your result. Hint: First show how the higher value of A for the manager (than for the investor) affect the minimum Rm and then how it affects the pledgeable income, i.e., go again over the four cases as in Part g. 8
Solution: The results for the minimum Rm are unaffected [1 point], though it is important to plug in 40 for A now: 1. Rm if she keeps A both in case of success and in case of failure: Rm [1 point] 2. if she keeps A NOT in case of success but in case of failure: Rm [1 point] 3. if she keeps A in case of success BUT NOT in case of failure: Rm [1 point] 4. if she keeps A NEITHER in case of success NOR in case of failure: Rm [1 point] To calculate the pledgeable income, we can use the formulas from Part g, plugging in 20 for A in the hands of the investors and 40 if in the hands of the manager: 1. if the manager keeps A both in case of success and in case of failure: pH R - [1 point] 2. if she keeps A NOT in case of success but in case of failure: pH [1 point] 9
B = 30 p
B + A = 30 + 40 = 70 p
B - A = 30 - 40 = -10 p
B = 30 p
B p
3 = (150 - 30) = 90 4
B B 3 - 40 = pH R - - 20 = (150 - 30 - 20) = 75 R + 20 - p p 4
3. if she keeps A in case of success BUT NOT in case of failure: B B 3 + 35 = (150 - 30) + 35 = 125 + 40) + (1 - pH )20 = pH R - pH (R - p p 4 [1 point] 4. if she keeps A NEITHER in case of success NOR in case of failure: pH (R + 20 - [1 point] Hence, pledgeable income is maximized if A is allocated to the investors in case of failure and to A in case of success. [1 point] Intuition: surplus is maximized if in the hands of entrepreneurs; incentives are maximized if in hands of investors in case of failure == hence in hands of manager in case of success and in hands of manager in case of failure. [2 points] B B ) + (1 - pH )20 = pH R - + 20 = 110 p p
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Question 3 (Empirics - 16 points)
Your calculations in Parts a-f relate to the empirical phenomenon of "Investment-Cash Flow Sensitivity" (discussed in class); those in Parts g-m to the empirical phenomenon of "CollateralValue Cash-Flow Sensitivity" (not discussed in class). Explain briefly: What does "Sensitivity of Investment to Cash Flow" mean? [1 points] Sensitivity of investment to cash flow means that there is a positive relationship between the amount of cash a firm has and the total investment of the firm. Firms have an easier time funding if they have more cash on hand. What does "Sensitivity of Investment to Collateral Value" mean? [2 points] Sensitivity of investment to collateral value means that there is a positive relationship between the amount of collateral (or asset value) a firm has and the total investment of the firm. Firms have an easier time funding if they have more assets. Why would we not expect to observe these two phenomena in an ideal Modigliani-Miller world. Your answer should include -- a statement of the Modigliani-Miller theorem, including all assumptions [2 points] Modigliani Miller states that the value of a project should not depend upon how you finance the project. It assumes: No taxes. No assymetric Information No transaction costs. Perfectly competitive markets -- an explanation of what investment should depend on and what it should not depend on (including cash flow and collateral value). [2 points] It should only depend on the NPV of the project. It should not depend up how much cash a firm has or assets. If the NPV is positive, you should do the project. If not, you should not do the project. How do we know that, in reality, investment is sensitive to cash flow? Explain how this relationship is tested empirically and what potential problems (confounds) with the 11
empirical test are. [3 points: 1 points for stating aggregate fact that most investment is financed out of internal funds; 2 additional points for describing the regression -- both points only if described carefully] Most investment is financed from internal funds for firms around the world. We would set up a regression like in problem set 2. We would need to add controls, like how good are their investment opportunities. How would you test whether, in reality, investment is sensitive to collateral value? List 3 suitable empirical proxies for (parts of ) `collateral value.' [up to 3 points] You would test it similarly to cash flows but instead of cash, you could use book value of assets, value of machinery, or value of inventory (among many others) Firms that use assets that are worth a lot to them but little if sold by investors (as assumed in Part m) will be suffering particularly much from "collateral-value cash-flow sensitivity." Can you think of some concrete examples of firms or industries or countries for whom that is the case? Try to list 3 examples. [up to 3 points] Here we are looking for assets that are more valuable to the firm than selling it. One case might be a pharmaceutical firm with patents. They already have the know how to use the patents and the set up for delivering the goods. It might not be worth as much to someone else.
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