Option Approach to Risk-shifting Incentive Problem with Mutually Correlated Projects

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1 Option Approach to Risk-shifting Incentive Problem with Mutually Correlated Projects Hiroshi Inoue 1, Zhanwei Yang 1, Masatoshi Miyake 1 School of Management, T okyo University of Science, Kuki-shi Saitama , Japan Department of Industrial Administration, T okyo University of Science, Noda-shi Chiba , Japan Abstract A typical problem in financial contracting is the so-called risk-shifting problem. In this paper, we analyze the risk-shifting problem using a principalagent framework in which the principal lends money to the agent for a finite time of period. Extending the basic intuition of early models that convexity in the borrower s payoff is responsible for risk-shifting, a contract avoiding risk-shifting is developed. In particular, we use assets which are mutually correlated and hence become compliated than the case of a single asset[1], and the weighted sums of all assets are used with basket option approach. Keywords. Incentive problem, Option approach, Risk-shifting problem, Renegotiation-proof contract, Financing decision. 1 Introduction In financial contracting, two typical forms of moral hazard exist, each giving rise to specific incentive issues. These problems may be very closely related to each other. One is that when the behavior of agent is not observed by principal the agent may break contract and would have incentive to invest the project which gives higher return with risk after the agent borrowed the money. This problem is the so-called risk-shifting problem and was first studied by Jensen and Meckling[5]. The other is called observability problem for which the borrower is the only person that can observe project returns at no cost. His promised payment depends positively on realized project return and he might have an incentive to underestimate project return in order to reduce his payment to the lender. This problem usually arises in the context of a lender-borrower relationship since there exist asymmetry information between them. The borrower might try to influence inoue@ms.kuki.tus.ac.jp, y @yahoo.co.jp toshimiya1@yahoo.co.jp 1

2 the return distribution of the project to increase his payoff at the expense of the lender. These incentive problems inflict the principal a loss, who does not stand at advantage and produce social faults. In a serious situation principal would not have a contract with agent, causing financial activity and market to be withered. Lenders that are aware of the fact that borrowers have an incentive to increase the risk of their projects can use several ways to solve. Galai and Masulis[1] formalize Jensen and Meckling s idea in option pricing framework of Black and Scholes[13], who demonstrate the suboptimal risk-maximizing investment policy of shareholders in levered firms. In this paper, we analyze the risk-shifting problem using a principal-agent framework in which the principal lends money to the agent for a finite time of period. Extending the basic intuition of early models that convexity in the borrower s payoff is responsible for risk-shifting, a contract avoiding risk-shifting is developed. In particular, we use plural assets which are mutually correlated and the weighted sums of the all assets prices are used along with basket option approach, which extends Ziegler s results. This paper is organized as follows. Section describes a model for risk-shifting problem which exists between a lender and a borrower. A profit-sharing rule between the lender and the borrower is characterized with call and put options in section 3. In section 4, the properties of a contract to avoid risk-taking by the borrower are discussed, in particular, for the case of two assets. In section 5, the definition of renegotiation-proof incentive contract for n assets is provided and examples of a contract that are not renegotiation-proof are demonstrated. Also, the conditions of renegotiation-proof incentive contract are developed. In section 6 the feasible renegotiation-proof contract is discussed. Concluding remarks are provided in section 7. Risk-Shifting Problem Consider a financial intermediary that lends money to a borrower for investment in several projects that are available only to the borrower. Assume that the lender cannot observe the borrower s project choice and hence cannot assess the risk of the projects. At initial time, each of all projects has different prices S i i = 1,..., n with different risks. Assume also that the borrower can, at any time, change his mind and replace some projects with others at no cost. At any time, the agent can choose to invest all funds to some of a series of projects, i = 1,..., n, The value of the borrower s assets is assumed to follow the usual geometric Brownian motion ds i = µ i S i dt + σ i S i db i,t i = 1,..., n. 1

3 We assume that all projects have a finite life T and random terminal values, S i i = 1,..., n are observable by both the lender and the borrower. The lender pays an amount of D 0 to the borrower in exchange for a promise by the lender to pay him f n a i S at time T. a i i = 1,..., n indicate the weights, which imply the ratio for investing each asset and n a i = 1. Then, f n a i S is contingent payment for final sums of values, n a i S. c i = a i S i / a i S i i = 1,..., n. Assume that there exists some correlation ρ ij between ith asset and jth asset db i,t db j,t = ρ ij dt. After the financing contract is signed, the borrower chooses an investment project. At any time during the life of the contract, he may switch to another project involving higher of lower risk or to a newly weighted sums of several assets. For example, the borrower may sell ith asset, and with the money he could invest to higher or lower asset. In this context, the borrower can choose one or more assets with higher risk at any time t, which make the borrower bring some profit and at the same time make decrease the amount of payment to the lender. But, the lender usually can not observe the borrower s incentive behavior. In order to discuss the influence of the risk-shifting problem we assume that all assets have the same maturity period T, and the return on the project S i is observed by the lender and the borrower when the contract expires at time T. Also, the lender and the borrower agree on a single, end-of-period contingent payment to the principal, f n a i S. Suppose that lender and borrower have no other assets and limited liability. Then, the effective payment of the borrower to the lender at time T, whatever has been agreed upon, is given Π L [ ] a i Si = min a i Si, f a i Si. 3 On the other hand, the payoff to the borrower equals the difference between total project return and the amount paid out to the lender, Π B a i Si = a i Si a i Si Π L [ ] = max 0, a i Si f a i Si. 4 3

4 Thus, we can summarize the structure of this game as below. First, the financing contract is signed. The lender pays an amount of D 0 to the borrower in exchange for a promise by the lender to pay him f n a i S at the T. After receiving the money from the lender, the borrower invests in a projects and if he wishes, costlessly switch to projects involving more or less risk at any time. Finally, when the contract expires at time T, the return on the project n a i S is observed by both lender and borrower and the borrower pays min [ n a i S, f n a i S ] i to the lender. 3 Profit-Sharing Contract We look at profit-sharing rules avoiding risk-shifting. The lender and the borrower can agree on any payment, i.e. sharing rule. The sharing rule can be characterized as follows. A fixed payment from lender to borrower, D, α basket put options with strike price, X 1 and β basket call options with strike price, X. Thus, in our case, the underlying asset for option is described as the sum of n assets, [ Π L a i Si = D + α max X 1 ] [ ] a i Si, 0 + β max a i Si X, 0. 5 The sharing rule showing in Figure 1 attributes everything to the lender up to project returns of X 1, plus half of any returns in excess of X. Payoff n a i S i Share of the borrower f n a i S i β D α Share of the lender X 1 Final value of X n a i S i Figure 1: A profit-sharing rule between lender and borrower 4

5 Concerning Figure 1 there are three cases considered, depending on the size of the final value of n a i S and the strike prices. i n a i S < X 1 ; When the final total value is less than the strike price of put option the put option is exercised, and then the payoff to lender is described as D+α X 1 n a i S. ii X 1 < n a i S < X ; When the final total value exists between the strike price of put option and the strike price of call option none of the option is not exercised, so that the payoff to lender is a fixed payment, D. iii n a i S > X ; When the final total value is greater than the strike price of call option is exercised, and the payoff to lender is described as D + β n a i S X. 4 Developing an Incentive Contract We discuss the properties that avoid risk-shifting as shown in Figure 1. The first step is to determine the value of the payment to the principal and the agent using option pricing. From the structure of the contract above the current value of the payoff to the lender can be calculated as Π L = e rτ D + αv P X 1 + βv C X, 6 where r denotes the riskless interest rate, τ the remaining life of the loan, and V P and V C for the Black-Scholes put and call option values with an exercise price of X 1, X, respectively. The evaluation formulae are given by V P X 1 = V C X = 1 a i S i { exp τ a i S i { exp 1 τ σ x σ x c i σi N } d 1 σ x τ + K1 N d 1, c i σi N } d σ x τ + K N d. where d 1, d, K 1, K, σ x are defined below, d g = log K g + 1 τ n c iσ i σ x, K g = σ x = X g e rτ n a is i + exp 1 τ σx n c iσi ρ ij c i c j σ i σ j, j=1 1 g {1, }, and N denotes the cumulative standard normal distribution function. 7 5

6 On the other hand, the current value of the payoff to the borrower is described Π B = = a i S i Π L a i S i e rτ D αv P X 1 βv C X. 8 In order to avoid risk-shifting problem, the values of α, β, D, X 1 and X must be determined so that the borrower does not have any incentive to influence the risk of the projects. We note that arbitrage-free concept comes into our discussion and hence the borrower s payoff Π B should be independent of each of asset risks, σ i i = 1,..., n. Therefore, we have where V P X 1 σ i = V C X σ i = and A, B i, C i are described as Π B σ i = α V P X 1 σ i β V C X σ i = 0, 9 a i S i { AB i N d1 + σ x N d1 + K 1 e d 1 / C i }, a i S i { AB i N d + σ x N d + K e d / C i }, A = exp B i = τ c i C i = c i 1 τ σx c i σi, ρ ij c j σ j c i σ i, j=1 n j=1 ρ ijc j σ j πσx. Thus, 14 can be expressed as, for i = 1,..., n Π B = α V P X 1 β V C X σ i σ i σ i { { = a i S i ABi α N d1 + σ x τ N d1 +β N d + σ x N d } + C i α K 1 e d 1 / + β K e d /} = Let restrict our discussion of risk-shifting problem to the case of two assets. In 6

7 order for the borrower not to have risk-shifting incentive for respective asset risk σ 1, σ the following expressions must be satisfied. Π { B { = a 1 S 1 + a S A B 1 α N d1 + σ x τ N d1 σ 1 +β N d + σ x τ N d } + C 1 α K 1 e d 1 / + β K e d/} = 0, { { A B α N d1 + σ x τ N d1 Π B σ = a 1 S 1 + a S +β N d + σ x N d } + C where A, B i, C i i = 1, j = or i =, j = 1 are expressed as A 1 = exp τ σx c 1 σ1 c σ, α K 1 e d 1 / + β K e d /} = 0, 11 B i = τ c i σ i + ρ 1 c 1 c σ j c i σ i, c C i = i σ i + ρ 1 c 1 c σ j τ. 1 πσx Also, letting denote the common terms in 11 by α N d 1 + σ x N d1 + β N d + σ x N d = E 13 and we have α K 1 e d 1 / + β K e d / = F, 14 Π B = a 1 S 1 + a S σ 1 Π B = a 1 S 1 + a S σ A B 1E + C 1F = 0 A B E + C F = Concerning expressions of E and F there are four cases considered. 1 E 0 and F 0: From 15 the following relation is derived and substitution of 1 gives B 1 B = C 1, 16 C c 1 σ1 σ ρ 1 σ 1 + c ρ1 σ σ 1 σ =

8 In paticular, for the case of two assets, c 1 + c = 1 with 16, we have c 1 = ρ 1 σ σ 1σ ρ 1 σ 1 + ρ 1σ + σ 1σ c = ρ 1 σ 1 σ 1σ ρ 1 σ 1 + ρ 1σ + σ 1σ. 18 Thus, with 0 < c 1, c < 1 the following results are obtained ρ 1 < σ 1 σ for σ 1 < σ ρ 1 < σ σ 1 for σ 1 > σ. 19 Therefore, it will be concluded that in order for risk incentive not to occur the correlation coefficients of the assets, for which borrower would invest, needs to satisfy 19. E 0 & F = 0: For B 1 & B B 1 = τ c 1σ 1 + ρ 1 c 1 c σ c 1 σ 1 = 0, B = τ c σ + ρ 1 c 1 c σ 1 c σ = 0 0 hold. With c 1 + c = 1, σ 1 = σ, ρ 1 = 1 are obtained. In this particular case, the borrower invests one asset. 3 E = 0 & F 0: For C 1 & C C 1 = C = c 1 σ 1 + ρ 1 c 1 c σ τ = 0, πσx c σ + ρ 1 c 1 c σ 1 τ = 0 1 πσx hold, and with c 1 = σ σ 1 + σ, c = σ 1 σ 1 + σ, ρ 1 = 1 is obtained. In other words, the assets the borrower would invest satisfy negative perfect correlation as a necessary condition for which risk incentive does not occur. 4 E = 0 & F = 0: Both the common terms, 13 and 14 are zero. α N d 1 + σ x N d1 + β N d + σ x N d = 0 3 8

9 and α K 1 e d 1 / + β K e d / = 0. 4 From the discussion above we can summarize as follows; For 1 through 3 the borrower does not have incentive under the respective condition. But, if the conditions such as correlation coefficients are not satisfied there still exist risk incentive. In addition, the real values for σ 1, σ & ρ 1 are not observed by the lender. In other words, after the borrower received money from the lender, which of two assets among many assets are selected by borrower can not be observed by the lender. This is the premise for the problem of risk incentive. Therefore, the condition on 4 may be possible to avoid risk incentive. Namely, if the values satisfying 3 & 4 are provided, no matter which of two assets are initially selected by the borrower, risk incentive does not occur so that the profit of the lender is certainly guaranteed. Also, as long as lender has nothing to do with the risk σ i of the asset invested by borrower the conditions of 3 and 4 must be satisfied since risk incentive by borrower must be avoided. For 3 and 4 α, β, X 1 and X are free to be independently chosen there may exist an infinitive number of incentive compatible profit-sharing contracts at any point in time. To answer which one the lender and borrower should select we use renegotiation-proof incentive contract approach. 5 Renegotiation Proof Incentive Contract We use the same definition of renegotiation-proof incentive contract given by [1] and extend it to the case with n assets. A contract is renegotiation-proof if it does not give the agent risk-shifting incentives i at any point in time over the life of the contract, and ii for any value of any underlying asset S i i = 1,..., n We show some examples of a contract in Figure through Figure 4, that are not renegotiation-proof. Example1. The contract uses the parameters α=-0.5, β=0.5, X 1 =50, X =100, a 1 = a = 1, σ 1 =0.5, σ =0.5, τ=1, ρ=0.5, r=0.05. Thus, when investment is made with two assets, if the contract is not renegotiation-proof the risk-shifting incentive changes as shown in Figure through 4. Figure shows that the risk-shifting incentive changes for σ 1 and σ as S 1 varies when S =5 is fixed. Similarly, Figure 3 shows that S varies when S 1 =5 is fixed. In the both cases, when asset values are low the borrower makes project risk increase to obtain more profit since the borrower risk-shifting incentive Π B / σ i 9

10 is positive. In other words, the borrower has incentive to invest to the asset with more risk. Contrary to that, when the value of asset increases the borrower makes the asset risk decrease to obtain more profit since Π B / σ i is negative. In other words, it produces asset with less risk. Also, the value of asset goes up further, then Π B / σ i is going to approach 0. The risk incentive for the case, in which asset value S 1 is fixed, converges to 0 more rapidly than the case for S fixed. From this it is understood that as asset value boundlessly goes up a borrower can surely obtain profit, so that he is inclined to have less incentive by considering risk. Figure 4 shows changes of risk-shifting incentive as S 1 = S = 5 is fixed and the remaining period τ moves from 1 to 0 with 0.01 interval length. At that time Π B / σ i is positive through the entire time period and approaches 0 as the time comes closer to the maturity date. Namely, it is concluded that this contract can not avoid risk-shifting incentive of borrower as asset value and time change so that the contract is not renegotiation-proof. 6 4 risk-shifting incentive asset value S 1 Figure : Risk-shifting incentive for which S 1 varies from 0 to 500 with S =5 fixed 10

11 6 4 risk-shifting incentive asset value S Figure 3: Risk-shifting incentive for which S varies from 0 to 500 with S 1 =5 fixed risk-shifting incentive time t Figure 4: Risk-shifting incentive for which time varies from 0 to 1 with S 1 = S =5 fixed We will look at the changes of risk incentives for different correlations between assets as the asset value increases. Example. We use the same parameter values as shown in Example1. Fig.5 11

12 depicts risk-shifting incentive for which S 1 varies with S being fixed. Note that Π B / σ shows greately different variations while Π B / σ 1 behaves in similar ways with different correlation cofficient, ρ between two assets. In particular, the risk incentive Π B / σ shows for the values of S 1 beyond S 1 =50 the values with different signs, depending on the sign of ρ. For Figure 6 Π B / σ 1 shows roughly similar fluctuation except for the case of ρ=-1.0. This situation is remarkably observed for Π B / σ. risk-shifting incentive asset value S 1 asset value S 1 risk-shifting incentive Figure 5: Risk-shifting incentives for which S 1 varies from 0 to 500 with S =5 fixed risk-shifting incentive asset value S asset value S risk-shifting incentive Figure 6: Risk-shifting incentives for which S varies from 0 to 500 with S 1 =5 fixed Next, we discuss the conditions of renegotiation-proof incentive contract. When n assets exist if a contract is renegotiation-proof, the conditions of 18 must hold 1

13 for any values of τ and S i i = 1,..., n. Rewriting 3 and 4 α β = N d + σ x N d N d 1 + σ x N d1 5 α β = K e d / K 1 e d 1 / 6 If the contract is renegotiation-proof the values of α and β appearing in 5 & 6 must be independent of S i i = 1,..., n and τ. Substituting d g in 7 into 6 we obtain α d β = exp 1 d + log K / K 1 = exp log K1 K log K1 / K σxτ log K1 / K n c iσ i σ x σ x 7 Letting ψ = log K1 K log K1 / K σ xτ log K1 / K n c iσi σ x σx 8 the ψ must be constant for S i i = 1,..., n and τ. Taking derivative with respect to τ ψ τ τ = log K1 Q rk 1 / K 1 log K Q rk / K log K1 K log K1 / K σxτ Q rk1 / K 1 Q rk / K n c iσi σ x σx where K g and Q are defined as = 0, 9 K g = X g e rτ / Q = 1 σ x 9 can be expressed below a i S i c i σi 1 exp τ σx c i σi. Q rk 1 Q rk c i σi K 1 K σx τ log K 1 Q rk1 log K Q rk τ + log K1 K log K1 / K 1 K K =

14 When the first term is not zero the equation is a qurdratic equation with respect to τ. But, if the contract is renegotiation-proof 30 must hold for any value of τ, so that the terms of first, second and constant must be zero. Q rk 1 Q rk c i σi K 1 K σx = 0, log K1 Q rk 1 log K Q rk = 0, K 1 K log K1 K log K1 / K = From the above we have K 1 = K 3, and K 1 = K, which implies X 1 = X. 33 By substituting 3 and 33 into 7, we have α β = e0 = 1. Thus, α + β = Therefore, when there are n assets and if a contract is renegotiation-proof the above results X 1 = X and α + β = 0 must be satisfied. Similarly, letting the derivative of the expression with respect to S i be zero we obtain the same results. 6 The Feasible Renegotiation Proof Incentive Contract In finding the solution, the value of the constantpayment D and of the parameters α and β can now be determined using the conditions, X 1 = X and α + β = 0. First, since the agent cannot pay out more than the total final value n a i S i to the principal α must be negative and β = α is positive. Remark 1 If α were chosen to be positive the contract would call for the agent to make a positive payment to the agent when the project ends worthless, n a i S i = 0. That violates the feasible condition 4. Next, let X 1 = X = X for finding D since the strike prices for put and call options are equal. There are two cases of relationship between the total final value n a i S i and the strike price. We look at the payoff to the principal for each case. i n a i S i X ; Only put option is performed and the payoff to the 14

15 principal is given, Π L = D + α X a i Si = D β X a i Si = D + β a i Si X 35 ii n a i S i > X ; Only call option is performed with the payment to the principal, Π L = D + β a i Si X 36 Thus, the payoff to the principal is D + β n a i S i X for either case. With the feasible condition 4 the payoff to the principal must be below the final total value n a i S i, 0 D + β a i Si X a i Si 37 Rewriting it β a i Si D βx 1 β a i Si 38 The feasible condition 4 must holds for any value of n a i S i 0. Hence, must hold. obtained, D = βx 39 Substituting it into 50 or 51 the payment to the principal is Π L = f a i Si = β a i Si 40 where β is a positive constant. The only suitable renegotiation-proof is linear in the final total value n a i S i. In other words, the principal can receive 100β% of total profit of the total assets. This result is same as for that of a single asset. Substituting α = β, X 1 = X = X and D = βx into 5.9 D 0 = β a i S i, 41 which indicates the present value the principal receives. Therefore, if the principal can receive 100β% of final total value of asset n a i S i, going back to the beginning, he can agree to lend at most D 0 = β n a is i. n a is i is the necessary amount of initial total investment and S i is ith initial value of the asset. The borrower receives 1001 β % of final total value of the asset and hence he himself needs to finance at least 1001 β % of the total amount of investment at the initial time. 15

16 7 Concluding Remarks We used analysis of option to discuss risk-shifting problem on plural assets in which the sum of the all asset prices, where projects are mutually correlated, is considered the underlying assets and basket option approach was used, extending the result of a single asset of Ziegler. Then, in order to solve risk-shifting problem for the plural assets a profit-sharing contract model was developed. In other words, portfolio which consists of put option and call option as payout to lender and a fixed payment was used. In this case, the price of the underlying asset is the sum of prices of the plural assets. Then, profit-sharing contract can apply not only to a single asset but to the case of plural assets by renegotiation-proof incentive approach. In fact, it is not necessary for an investor to invest to a single asset when he has finance contract, so that incentive problem based on plural assets may be more realistic than the case of a single asset. In particular, we may find the conditions under which incentive contracts are renegotiation-proof. In this paper, though we use plural assets correlated mutually instead of a single one there still remain a problem that Black and Scholes model allows borrowers always to select infinite-volatility projects. Chesney and Ashner1999, 001 clarified this point, introducing a down side knock-out barrier for equity so that optimal volatility level of project become finite. This method may be a way to circumvent the limitation of the standard Black and Scholes and Merton models. Including the problem of circumventing the above situation, as a future work, we may develop more precise model which should be suited to the real circumstances. References [1] A.Ziegler, A Game Theory Analysis of Options, Springer, 004. [] D.Asaoka, Strategic Corporate Finance in Japanese, NTT Publishing Co, 006. [3] D.Gale and M.Hellwig, Incentive-Compatible Debt Contracts: The One- Period Problem, Review of Economics Studies, 5, , 1985 [4] F.Black and J.C.Cox, Valuing Corporate Securities: Some Effects of Bond Indenture Provisions, J. of Finance, 31, , [5] M.C.,Jensen and W.H. Meckling, Theory of the Firm, J. Financial Economics, [6] P.Wilmott and S.Hwsion and J.Dewynne, The Mathematics of Financial Derivatives, Cambridge University Press,

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