Tirole, J. (2006). The Theory of Corporate Finance, chapter 4, p. 157-198 PDF

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ACF 2019 – Lecture 8-9

© Morten Hansen

9/26/2019

Determinants of Borrowing Capacity Source: Tirole, J. (2006). The Theory of Corporate Finance, chapter 4, p. 157-198 Lecture slides Notation:

A=¿ asset (cash) on hand A=¿ minimum necessary assets (cash) for investment in one project

A=¿ minimum necessary assets (cash) for investment in two projects I =¿ fixed investment costs

P=¿

Resale price of assets

Time line: t 0 , t 1 , t 2 t 0=¿ loan contract, investment I t 1=¿ moral hazard (behave/misbehave) t2 =¿ outcome, max(R , 0) pH =¿ probability of success if the borrower behaves pL =¿ probability of success if the borrower misbehaves pi=¿ probability of success pH + p L ≠1 the probability of success given the borrower’s behavior does not have to add to 1. ∆ p= p H − p L Sharing rule of the profit (in event of success): Rb=¿ Return (stake) to the borrower Rl=¿ Return (stake) to the lenders R=R b+ R l If the project is unsuccessful (fail) both sides receive nothing i.e. 0 y S denotes the probability that the borrower keeps the assets, A in case of success y F denotes the probability that the borrower keeps the assets, A in case of failure

Note the following terms are used interchangeable: 1. ‘borrower’ and ‘entrepreneur’ 2. ‘lenders’ and ‘investor’ 3. ‘shirking’ and ‘without/no/low effort’ 4. ‘hard work’ and ‘with/high effort’

Diversification By using diversification, the borrower can increase her probability of receiving funding. If the projects are uncorrelated (independent), it is possible to use the income from one good project as “collateral” for other projects. Two projects (cross-pledging) Consider 2 independent and identically distributed (i.i.d.) projects with: ***(i.i.d. if each random variable has the same probability distribution as the others and all are mutually independent)***

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Figure 1. Timeline for basic setting

The borrower has wealth

2 A and considers two types of financing;

(1) project financing (Tirole Chapter 3) – projects are considered on a stand-alone (separately) basis (2) cross-pledge financing – projects are considered on a cooperatively (jointly) basis In the cross-pledge financing setting, there is only two projects. If the firm wants to finance both projects, it then must ensure that the borrower work hard on both projects.

Figure 2. Outcomes and probabilities of two projects

From figure 2 it is possible to derive the following nine scenarios:

(1) The borrower is working hard on two projects – both are successful, hence 2 pH × p H = p H (2) The borrower is working hard on one project – she’s lucky and both are successful, hence pH × p L= p H p L and receive private benefit B for shirking on one project. (3) The borrower is working hard on none of the projects – she’s extremely lucky and both are successful, hence pL × p L = p 2L and receive private benefit 2 B for shirking on two projects. For ( 1)→ ( 3) successful.

scenarios above, the borrower will receive

R2

(2 R) as both projects are

(4 ) The borrower is working hard on two projects – she’s unlucky with one project as it’s a pH ( 1− p H ) and successful with the other project, hence pH . i.e. failure, hence 2 p H ( 1− p L) (5) The borrower is working hard on one project –one project as it’s a failure, the other project a success, hence pH ( 1− p L) and pL ( 1− pH ) , i.e. pH ( 1− p L) + p L ( 1−p H ) and receive private benefit B for shirking on one project.

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(6) The borrower is working hard on none of the projects – she’s lucky and one are successful, hence pL and the other project a failure pL ( 1− pL ) , i.e. 2 p L( 1− p L ) and receive private benefit 2 B for shirking on two projects. For ( 4 ) →(6) successful.

(7)

scenarios above, the borrower will receive

(R)

R1

as one project are

The borrower is working hard on two projects – she’s extremely unlucky and both are

failures, hence

(1− p H ) × (1− p H )= ( 1− p H )

2

(8) The borrower is working hard on one project – she’s unlucky and both are failures, hence (1− p H ) ( 1− p L) , but receives private benefit B for shirking on one project. (9) The borrower is working hard on none of the projects – both are failures, hence 2 (1− p L ) × ( 1− p L)= ( 1− p L) , but receives private benefit 2 B for shirking on two projects. For ( 7 ) →(9) scenarios above, the borrower will receive are successful.

R0

(0) as none of the projects

Putting it all together in table 1:

Table 1. Outcomes and probabilities of

{2,1,0 }

projects succeeding

The cross-pledge financing setting implies that the incentive constraints (I C b) for working hard on two projects, must be greater than working “hard” on only one or working “hard” on none of the projects. Hence, ***Notice here how each formula line is taken from the table – follow the line down, moral hazard line multiplied by borrower reward line***

Borrower must work hard on both projects over working hard on only one project,

p H p L R2 +[ p H ( 1− p L ) + p L( 1− p H ) ] R 1+ ( 1− p H )( 1−p L ) R 0 +B I C2b>1=⏟ p H R2 +2 pH (1− p H ) R 1+ ( 1− p H ) R0 ≥⏟ 2

2

Two projects

One project

Borrower must work hard on both projects over working hard on no projects,

I C2b>0=⏟ p L2 R 2+ 2 p L( 1− p L ) R1 + ( 1− p L) 2 R0 +2 B p H2 R 2+2 p H (1−p H ) R1 + ( 1− p H )2 R 0 ≥⏟ Two projects

No project

Borrower must work hard on one project over working hard on no projects,

I C1b>0=⏟ p L R2 +2 pL ( 1− p L) R1+ ( 1− p L ) R0 +2 B p H p L R2 + [ p H ( 1−p L ) + pL ( 1− p H ) ] R1 + ( 1−p H ) ( 1− p L ) R 0+ B ≥⏟ 2

2

No project

One project

Optimality of incentive scheme rewarding only if full success

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ACF 2019 – Lecture 8-9

If the payoff

R'2=

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( R2 , R 1 , R 0 )

(I C b2>1 )

satisfy

and

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(I C b2> 0)

, then so does

(R2' , 0,0 )

where

1 2 p R + 2 p H (1− p H ) R1 + ( 1− p H )2 R 0] 2 [ H 2 pH

Meaning that

(R2 , R 1 , R0 )

and

(R2' , 0,0 )

yield the same expected utility to the borrower.

The incentive constraints then become The (I C b )

constraint proof:

R1 and

Note for the next few lines,

I C2b>1= p H2 R2'

R0 are not set equal to 0

≥ p H p L R'2+ B

≥ p H p L R2 +( p H + p L −2 p H p L ) R 1+ ( 1− p H ) ( 1−p L ) R 0+ B

⏟ (1 )

(1 ) : [ p H ( 1− p L ) + p L( 1− p H ) ]⇒p H − p L p H + p L− p H p L ⇒p H +p L −2 p H pL '

Substitute in R2

R2 on right side

for

(

'

≥ p H p L R2 −2

(1− pH ) pH

R1 −

( 1− p H) p H2

)

R0 +( pH + p L −2 p H p L) R1 +( 1−p H )(1−

+¿ ¿ R0 pL 1− p L)− (⏟ ( 1−p H ) pH ⏟ ⏟ ¿0

¿ 1− p L

( ) { 0,1} ⏟ ¿

p H − p L R1 + ⏟ ≥ p H p L R'2+B+ ⏟ (1− p H ) ¿

(

Only consider contracts with scheme becomes

)

¿0

R1= R0 =0 , then the incentive compatibility constraints for this

I C2b>1= p H R2'

≥ p H p L R'2+ B

p H −PL ) R'2 pH (⏟

≥B

2

¿0

Δp

pH R2'

B Δp

≥

I C2b>0= p H2 R '2

(Eq. 4.2)

≥ p2LR2' +2 B

( p2H− p2L) R2'

≥2B

p H −P L ) R ( p H + p L )(⏟

' 2

≥2B

Δp

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( p H + p L ) R'2

≥2

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B (Eq. 4.1) Δp

Eq. (4.1) make good sense, because from project financing (single project case) the pay-off should be

B ∆p

greater than

which is the private benefit from shirking, and now with two projects the pay-

off must be 2 times that private benefit. Eq. (4.1) implies eq. (4.2)

pH R2' ≥

p H +p L ' 2 B R2 ≥ 2 Δp

2 pH R

The expected pledgeable income is equal to the expected income from the projects 2 H

2 H

2 p H R− p R2 . The

subtracted the expected payoff pay-off to the borrower p R2 , i.e. borrower has wealth 2 A and the investment costs is 2 I , hence The (I Rl )

constraint proof:

Remember: still only considering contracts with

R1=R0 =0 .

I Rl= p H ( 2 R−R2 ) +2 p H (1− p H ) R 2

≥ 2(I −A )

2

≥ 2(I −A )

2 p H R− p H R2 2 p H R−

2 H

2p B ( p H + pL ) Δ p

2 p H R−2 (1−d 2 )

pH B Δp

pL d 2= ( p H + p L) B R−( 1−d 2 ) Δp

where;

pH

[

≥ 2(I −A ) ≥ 2(I −A )

( 21 )

d2 ∈ 0 ;

]

≥ I − A (Eq. 4.3)

[

¿ I −p H R− (1−d 2) B Δp

´ A ≥ ´A

]

I.e. the minimum required cash the borrower needs when she’s using cross-pledging is smaller, hence two projects facilitate easier funding for the borrower, hence cross-pledging relaxes the

´ . However, it is important to note, that diversification does not financing constraint i.e. A´´ < A eliminate the private benefit. Summing up, the borrower can use cross-pledging for i.i.d projects and use the income from one project as collateral for another project and thereby reduces the amount of assets on hand needed. Thereby increasing the possibility for receiving funding. This is valid as long as her incentive constrains ensures that she will work hard on both projects.

Many projects

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The procedure for many projects is straight forward and almost identical with two projects in previous section. With many projects i.i.d projects, contracts on the form n∈N {Rn , 0 , … , 0 } should be considered. Hence,

pH R−I < B meaning that the pay-off form the project subtracted the investment cost is smaller than the private benefit, implying that the borrower needs some assets on hand.

(I C bn> 0 )

The

constraint becomes

pHn Rn ≥ pnL Rn +nB ⇔R n ≥

nB p − p nL n H

and then the project can only be financed if the expected income from the projects subtracted the cost is greater than the cost of financing, meaning that the incentive constraint (IRl ) becomes

n ( I −A ) ≤n p H R− pHn Rn

(I C bn> 0 )

when inserting

I −A ≤ p H R− pnH

B n p − pL

[

d n=

a project is only funded if

n H

I −A ≤ p H R−( 1−d n)

where;

( IRl )

in

B Δp

]

n−1 n−1 p L( pH − p L )

(pnH− p nL ) n , meaning that the "cost" becomes smaller for each n . And if pL converges to with is back to the normal financing constraint i.e. pH

This function increases with

n→∞

dn

pH R−B ≥ A − I With many projects the borrower can not shirk on some because of the law of large numbers.  

assuming pH R−I < B implies that there is credit rationing when A is too small for a given level of total net worth, the per project net worth tends to zero as n → ∞

Summing up, more projects is equal more pledgeable income with alleviates inventive constraint, but does not eliminate it, because of private benefit. So, even with an infinite n of projects, credit rationing still exists and therefore, net worth still matters.

Limits to diversification In the section above its assumed there is no correlation between the projects. This is a strong assumption and important fact of diversification. Below, is listed some problems with diversification Endogenous correlation

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If the projects are endogenous correlated, then there will be problems. Because the borrower would prefer to choose projects which is correlated as her utility increases

Ucb= p H R 2>U ib= p2H R 2 because pH I which makes sense, as the pay-off must exceed the investment costs for the NPV to be positive. This yields that

x ( p H R−I ) >( 1−x) ( I −P )=NPV which states that the expected profit must dominate the expected capital loss associated with distress. Hence, the project is financed if and only if

(

x p H R−

)

B + ( 1−x ) P ≥ I − A Δp

meaning that the threshold A´

´ =x p H A≥A

must be smaller than

A .

B −[ ( x p H R+ (1−x ) P )−I ] Δp

This make sense, because the assets in place is much easier to sell and convert to cash compared to if the assets were knowledge, i.e. the debt-holders is more secure. Intuitive effect of P and x ? ´ ↓ , hence less credit rationing When P↑ ⇒A When

B ∂ A´ B ´↑ ⇒A x↑⇒ − p H R + P>0 , i.e. if P> pH R− ⇒p H Δp Δp ∂x

(

)

if

x↑

Hence, reselling becomes less probably and thus more credit rationing. The cost of Assets collateralization: With this “quasi-cash”, assets which are sold, are up until now assumed to be costless (no cost of collateralization). However, this might not be true as assets may have a lower value for lenders than for the borrower Below are listed some possible deadweight losses attached to collateralization:       

ex-ante and ex-post transaction costs – simply just cost of selling the assets. borrower may have benefit of ownership (e.g. know-how) assets may be hard to sell differential prospects of future credit rationing for borrower and lenders – if the borrower have a possibility of being credit rationed, then she might add more value to “assets” borrower may be risk averse borrower may undertake suboptimal maintenance of assets assets may have an attached managerial rent (if one needs to replace the incumbent manager with a new manager after seizing collateral a new moral hazard problem appears)

Contingent pledges and strength of balance sheet It is now assumed that the borrower has no cash on hand, hence I is used to acquire some assets which is then used in the production with a value of A to the borrower and A ' ≤ A for the lender.

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As in the basic setting, the loan contract applies the sharing rule R=R b + R l and there is a contingent right for the lender to seize the assets, in case of the project is a failure. Then there’s is probability y S for the borrower to keeps the assets in case of success and probability y F for the borrower keeps the assets in case of failure. As always, the borrower wishes to maximize her utility:

U b= p H ( Rb + y S A )+( 1− p H )( 0+ y F A ) subject to the

(I C b )

and (I Rl ) constraints

yS

denotes the probability that the borrower keeps the assets, A in case of success

yF

denotes the probability that the borrower keeps the assets, A in case of failure

Figure 4. Outcome for contingent pledges

From figure 4 it is possible to derive the following scenarios:

pH ( Rb + y S A ) : The borrower works hard, and the project is successful, receives Rb and have probability y S of keeping the assets. (1):

(2): (1− p H )(0+ y F A ) : The borrower works hard and is unlucky and the project is a failure, receives nothing and have probability y F of keeping the assets. (3): p L ( Rb + y S A ) : The borrower shirks and is lucky and the project is successful, receives Rb and have probability y S of keeping the assets.

(4 ):( 1−p L )( 0+ y F A ) : The borrower shirks, and the project is a failure, receives nothing and have probability y F of keeping the assets. B : The borrower also receives a private benefit from shirking The

(I C b )

constraint is given by:

I C b= p H ( R b+ y S A )+(1− p H )(0+ y F A )

≥ p L( Rb + y S A ) + (1− p L )( 0+ y F A ) +B

Δ p ( Rb + y S A ) +(1−p L )( 0+ y F A )

≥B

Rb

≥

B −( y S− y F ) A Δp

Rb +( y S− y F) A

≥

B Δp

The

I Rl

(I Rl )

constraint is given by:

¿ p H [ (R−Rb ) +( 1− y S ) A' ]+ ( 1−p H ) [ 0+ (1− y F ) A ' ] ≥ I ¿ p H Rl +[ p H (1− y S ) + ( 1− pH )( 1− y F ) A ' ]≥ I

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Inserting lenders’ zero-NPV condition in the expected utility,

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Ub :

U b= p H R −I + A −[ p H (1− y S ) +( 1− p H ) ( 1− y F ) ] ( A−A ' )

⏟ DWL

Obtaining three possible regions: 1. Strong balance sheet:

{y S =1, y F =1, R b> 0 }

Within this region the borrower posts no

collateral, and the borrower always keeps the asset as i.e.

y S = y F =1 .

2. Intermediate (medium) balance sheet: { y S =1, y F ≤ 1, R b ≥ 0 } Within this region the borrower posts collateral only in case of failure, if the asset is pledged, it is better to pledge in case of failure because of incentive properties for this balance. 3. Weak balance sheet: { y S ≤ 1, y F =0, R b =0} Within this region the borrowers only compensation is a share of the asset, there is only some probability of keeping the asset in case of success. Summing up, the amount of collateral the borrower must pledge increases when the balance sheet becomes worse.

Pledging existing wealth if the borrower does have a very weak balance sheet, she can pledge her "personal wealth" e.g. her house etc. She can pledge the amount βC where β< 1

C ∈[ 0,C max ] , however the lender only value

C as

The borrowers net utility is then given by:

U b= p H R−I −(⏟ 1−p H ) ( 1−β ) C DWL

Which is the expected income subtracting the investment costs and the deadweight loss, denoted DWL=( 1− p H ) ( 1−β ) C . Hence, it is optimal to set C=0 , so the borrower should only pledge her personal wealth if she really must. Hence, if the financing constraint is violated, i.e. A < A´ , the amount will obviously be C>0 . The

(I C b )

(

pH R− The

constraint is given by:

)

B + p H C + ( 1− p H ) βC ≥ I − A Δp

(I Rl )

constraint is given by:

pH ( R−R b ) + (1− p H ) βC ≥ I − A Given the

(I Cb )

and

( I Rl )

constraint, the amount the borrower will pledge is

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ACF 2019 – Lecture 8-9

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