The Impact of Interlinked Insurance and Credit Contracts on Financial Market Deepening and Small Farm Productivity Michael Carter, Lan Cheng & Alexander Sarris June 2011 Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Summary Twin puzzles Ample evidence that uninsured risk depresses small holder productivity and the development of rural financial markets Yet, to date it has proven hard to sustain formal agricultural insurance despite apparent need (Gine/Yang; Sarris et al.) Explore prospects for resolving these twin puzzles with formal theory of the behavior of smallholder household and a competitive sector of rural lenders Demonstrate that Neither credit nor insurance markets likely to fully develop in isolation However, “interlinking” these markets and contracts is more likely to succeed How interlinkage works depends on collateral environment Insurance subsidies may be smart Account theoretically for some surprising empirical results Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Outline Small farm household model Technology choices & self-insurance Credit contracts & collateral environments Index insurance as ’mean preserving squeeze’ Competitive lender model Iso-expected profit contract locus (partial equilibrium) Interlinkage pivots contract locus Portfolio composition & supply price of risky ag credit Partial Equilibrium Analysis of Technology Choice (exogenous cost of capital to agriculture) No insurance baseline Independent & interlinked index insurance Numerical Simulation of Equilibrium Credit Market Agents differentiated by wealth & risk aversion Credit market equilibrium concept Technology choice with & without index insurance Are insurance subsidies ’smart’? Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Household Model Technology & self-insurance ’Safe,’ low yielding technology: y ℓ = θ g ℓ ; ρ ℓ = y ℓ where θ = ( θ c + θ s ) with support [ 0 , ¯ ¯ θ ] , pdf f ( θ ) , cdf F ( θ ) and E ( θ ) = 1. Capital-using, high returning technology: y h = θ g h ( K ) , where K is the amount of purchased inputs and we assume that g h > g ℓ . Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Household Model Technology & self-insurance To buy K need loan contract, denoted ℓ < K , r , χ > and returns to household are: θ g h ( K ) − ( 1 + r ) K , if θ > ˜ � θ ρ h = , − χ, otherwise where ˜ θ = ( 1 + r ) K − χ just permits full loan repayment g h ( K ) Consider case where high technology profitable for all: E [ ρ h ] > E [ ρ ℓ ] > 0 Implies that no one will be price-rationed out of credit market as always have a profitable project Consumption: c t = B + ρ t + W , t = h , ℓ , and lowest possible consumption under high technology is c = B + W − χ . Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Household Model Collateral & Returns Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Low Technology as Self Insurance Basis risk & actuarially unfair Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Index Insurance Express as mean preserving ’squeeze’ Index insurance contract, I t < ˆ θ c , z t , β > , pays off when covariant shock ( θ c ) is less than ˆ θ c the strike point; has actuarially fair premium z t (normalized by g t ); and a markup β . Payoff to insured producers given by: ( θ c + θ s ) g t + (ˆ θ c − θ c ) g t − z t g t − β = (ˆ � θ c + θ s − z t ) g t − β, if θ y I t = ( θ c + θ s ) g t − z t g t − β, = ( θ c + θ s − z t ) g t − β otherwise By defintion of actuarial fairness: z t g t = g t E [ 1 ( ˆ θ c > θ c )(ˆ θ c − θ c )] . Under insurance, gross farm income is determined by transformed random variable: θ I = θ + s ( θ ) where s ( θ ) = 1 ( ˆ θ c > θ c )( ˆ θ c − θ c ) − z t where E [ θ I ] = E [ θ ] = 1 and pdf [cdf], f I [ F I ] is a mean preserving squeeze of f : Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Mean Preserving Squeeze Key integral properties ˆ ¯ ˆ y ¯ θ [ F ( θ ) − F I ( θ )] d θ > 0 ∀ y < ¯ ¯ [ F ( θ ) − F I ( θ )] d θ = 0 ; θ 0 0 Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Intuition on Index Insurance Can it stochastically dominate self-insurance? Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Competitive Credit Market Model Three interest rates matter in the model: π is the exogeous (risk-free) opportunity cost of capital ¯ π a is the portfolio risk adjusted interested rate that a lender ¯ must earn on its agricultural loan portfolio r is the nominal interest charged to an individual borrower π a ) ≥ ¯ π a ( n a ) ≥ ¯ r ( χ | ¯ π Let’s look at each of these in turn Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Competitive Credit Market Model Iso-expected profit contract locus Under standard loan contract ℓ ( K , r , χ ) , lender profits are: rK , if θ > ˜ � θ π = χ + θ g h ( K ) − K , otherwise . Under this specification, lender profits are concave in the random variable θ and expected lender earnings are: ˜ ˜ θ ˆ ˜ E ( π ) = [ 1 − F ( θ )] rK + ( χ + θ g h ( K ) − K ) f ( θ ) d θ . 0 Iso-expected profit locus defined by the interest rate-collateral combinations that just yield expected returns equal to ¯ π a . − F (˜ θ ) ∂ r ∂χ = θ )) K < 0 1 − F (˜ Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Competive Credit Market Model Iso-expected profit locus & interlinkage Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Competitive Credit Market Model Insurance-Credit Interlinkage A credit contract is interlinked with insurance when the bank has first claim on insurance proceeds and thus treats its returns as driven by the insured probability functions f I and F I The insured iso-expected profit will lie below uninsured locus for all undercollateralized contracts (see figure again) Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Competitive Credit Market Model Portfolio composition and supply price of risky credit Assume lender has funds for n loans of size K that can divided between agricultural loans ( n a ) and non-agricultural loans ( n b ) that pay ¯ π for certain Lender’s gross rate of return on the portfolio of n loans will be given by: � n A i = 1 π ( θ i ) / K + n b ¯ π G = . n Because of reserve requirements and political economy risk, lender faces a penalty function, P ( G ) , that reduces net lender portfolio returns when G falls below a critical threshold level ˜ π . Net portfolio returns ( N ) are given by: � G if G > ˜ π N = ′′ ≤ 0 ′ , P G − P ( G ) otherwise , with P Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Competitive Credit Market Model Market Supply of Agricultural Credit To supply n a agricultural loans, the lender must fulfill a standard, zero profit participation constraint: E ( N ) ≥ ¯ π π a denote E ( π i ) , and F G ( f G ) denote the cdf (pdf) of Letting ¯ G, this condition can be rewriten as: ˜ π π + n a ˆ ¯ n ( ¯ π a − ¯ π ) − P ( G ) f G ( G ) dG ≥ ¯ π 0 This condition implicitly defines the market supply function, π a ( n a ) ¯ By elminating covariant risk, index insurance (which is a mean preserving squeeze of f G ) flattens this supply relationship Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Competitive Lender Model Portfolio composition and supply price of risky credit Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Techology Uptake Absent Formal Insurance Expected utility under low technology (self-insurance via income smoothing): ¯ ¯ θ ˆ V ℓ = u ( θ g ℓ + W + B ) f ( θ ) d θ 0 Expected utility under high technology (some implicit insurance if limited liability): ¯ ¯ θ ˆ V h = F (˜ θ ) u ( c ) + u ( θ g h − ( 1 + r ) K + W + B ) f ( θ ) d θ ˜ θ Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Technology Uptake Absent Formal Insurance Farmers’ decision on technology: ˜ θ ˆ F (˜ ∆ h ℓ = V h − V ℓ = θ ) u ( c ) − u ( θ g ℓ + W + B ) f ( θ ) d θ + 0 ¯ ¯ θ ˆ [ u ( θ g h − ( 1 + r ) K + W + B ) − u ( θ g ℓ + W + B )] f ( θ ) d θ ˜ θ First term is negative, second term is non-negative Under high collateral, c is low and risk averse may choose low technology(risk rationing) Under low collateral, lending is risky, r is high and risk averse may also choose low technology Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
Technology Uptake under Alternative Insurance Schemes Highly risk averse farmers (CRRA=3) & n a = N Michael Carter, Lan Cheng & Alexander Sarris Insurance-Credit Interlinkage
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