Introduction

Theory

Numerical example for Maize in a developing country

A Theory of Rational Demand for Indexed Insurance Daniel Clarke Department of Statistics, University of Oxford Centre for the Study of African Economies Fellow of the Institute of Actuaries S ECOND E UROPEAN R ESEARCH C ONFERENCE ON M ICROFINANCE , G RONINGEN 16 J UNE 2011

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Motivating question

When should consumers use financial contracts to hedge against a potentially material loss, and when should they not? (When should poor farmers purchase weather derivatives?)

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

An introduction to weather derivatives Motivation: Agriculture is an uncertain business, particularly for the poor (Dercon 2004, Collins et al. 2009) Traditional indemnity-based approaches to crop insurance were unsustainable (Hazell 1992, Skees et al. 1999). Weather derivatives can be fairly cheap whilst still offering protection against droughts (Hess et al. 2005).

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

An introduction to weather derivatives Motivation: Agriculture is an uncertain business, particularly for the poor (Dercon 2004, Collins et al. 2009) Traditional indemnity-based approaches to crop insurance were unsustainable (Hazell 1992, Skees et al. 1999). Weather derivatives can be fairly cheap whilst still offering protection against droughts (Hess et al. 2005). Stylised history of weather derivatives: 1996: First weather derivative (US). 2005: Weather derivatives piloted in India, Malawi, Ethiopia, Ukraine, Nicaragua. 2007: Govt of India pilots weather derivatives as an alternative to the NAIS, compulsory for loanee farmers.

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Two puzzles of weather derivatives for the poor 1

Demand for weather derivatives is lower than expected

2

Demand is particularly low for the most risk averse

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Two puzzles of weather derivatives for the poor 1

Demand for weather derivatives is lower than expected

2

Demand is particularly low for the most risk averse

Key empirical papers include: Giné et al. (2008) India (AP): 5% uptake Giné and Yang (2008) Malawi: 13% fewer people take up loan with weather derivative than loan without Cole et al. (2009) India (AP and Gujarat): 5-10% buy product, hedging only 2-5% of household agricultural income

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Two puzzles of weather derivatives for the poor 1

Demand for weather derivatives is lower than expected

2

Demand is particularly low for the most risk averse

Explanations include: ‘Insurance purchase is sensitive to price [...] credit constraints [...] trust’ (Cole et al. 2009) ‘The most likely explanation [for demand falling with risk aversion] is that it is uncertainty about the product itself (Is it reliable? How fast are pay-outs? How great is basis risk?) that drives down demand.’ (Karlan and Morduch 2009) ‘Poor farmers on the other hand are not sufficiently well insured and would benefit from purchase of insurance, but they are severely cash and credit constrained.’ (Binswanger-Mkhize 2011) Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Two puzzles of weather derivatives for the poor 1

Demand for weather derivatives is lower than expected

2

Demand is particularly low for the most risk averse

This paper can (partially) explain the puzzles as objective, rational responses to basis risk and actuarially unfair price: Basis risk:= the risk that the net income from the financial contract does not accurately reflect the incurred loss Actuarially unfair price:= E[Net transfer to insurer ] > 0

Mathematical framework encompasses all (perceived) risk of contractual nonperformance, including trust, exclusions, insurer default, etc: Schlesinger and Schulenburg 1987 and Doherty and Schlesinger 1990 Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

The 2 × 2 state model ˜ exposed to loss of L Initial background wealth w, Loss and index are imperfectly affiliated with joint probability structure: Loss = 0 Loss = L

Index = 0 1−q−r r 1−q

Index = I q+r −p p−r q

1−p p

Basis risk, r = P[Loss = L ∩ Index = 0] Positive basis risk: r > 0 Index and loss are affiliated: r < p(1 − q)

Can purchase indexed cover of αL at premium multiple of m: Premium of αqmL buys claim payment of αL if Index = I

Consumer is strictly risk averse expected utility maximiser Utility function u with u 0 > 0 and u 00 < 0 Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

The model: decision problem

Optimal cover α maximises expected utility: max EU(α) = (p − r )u(w − αqmL − (1 − α)L) + (q + r − p)u(w − αqmL + αL) α

+ (1 − q − r )u(w − αqmL) + ru(w − αqmL − L) s.t. α ≥ 0

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

What we talk about when we talk about insurance Result without basis risk (Indemnity insurance, r = 0) Shape of rational hedging: • More risk averse ⇒ buy more coverage • Infinitely risk averse ⇒ α = 1 • Fair price (m = 1) ⇒ α = 1 • Positive loading (m > 1) ⇒ buy less coverage • Insurance is inferior for DARA utility • Larger potential loss L ⇒ buy more coverage for DARA utility Level of rational hedging: • Positive loading (m > 1) ⇒ any level of cover α ∈ [0, 1] is optimal for some strictly risk averse, DARA utility function u

Daniel Clarke

Result with basis risk (Indexed cover, r > 0) ? ? ? ? ? ?

?

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Theorem 1: Infinitely risk averse consumer Probability mass functions Uninsured 100% indemnity insurance

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Theorem 1: Infinitely risk averse consumer Probability mass functions Uninsured 100% indemnity insurance

50% indexed cover

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Theorem 1: Infinitely risk averse consumer Probability mass functions Uninsured 100% indemnity insurance

50% indexed cover

For an infinitely risk averse, maximin, consumer: Indemnity insurance (r = 0): α = 1 Indexed cover (r > 0): α = 0

⇒ optimal purchase cannot be everywhere increasing in risk aversion. Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Theorem 2: CRRA and CARA

Optimal cover α

Figure: Rational hedging and risk aversion for CRRA and CARA 100%

100%

50%

50%

0%

0% 0

5

10

0

Optimal cover α

Coefficient of RRA

5

100%

50%

50%

0%

m m m m m

= 0.3 = 0.6 = 0.9 = 1.0 = 1.2

10

0% 5

= 0.75 = 1.00 = 1.25 = 1.50 = 1.75

Coefficient of ARA

100%

0

m m m m m

10

Coefficient of RRA Daniel Clarke

0

5

10

Coefficient of ARA A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Bounds for rational hedging Theorem (3) For any risk averse individual the optimal level of indexed cover is zero if E[i|l = L] ≤ mE[i]. Theorem (4) For any strictly risk averse individual with decreasing absolute risk aversion the optimal level of actuarially fair or unfair indexed coverage is zero if E[i|l = L] ≤ mE[i], or otherwise bounded above by the unique α ¯ that solves A¯ αα¯ (1 − α ¯ )1−α¯ = (A + BC − 1)α¯ × (B(1 − C))1−α¯   A + BC − 1 α ¯ ∈ 0, A+B−1 Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Numerical example from a developing country Suppose you are a financial advisor with the following data yij : Average maize yields (kg/ha) within subdistrict j in year i xij : Claim payment for weather index insurance product designed for maize that would have been made in subdistrict j in year i Nine years of data, i ∈ {1999, . . . , 2007} Yield and weather data and product details for 31 subdistricts j ∈ {1, . . . , 31} Total of n = 261 complete (xij , yij ) pairs How much weather index insurance would you advise your clients to purchase?

Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Data

Claim payment xij

Claim payment rate Xij

Figure: Unadjusted and adjusted joint empirical distribution of yields and claim payments

0

2,000 4,000 6,000 Yield yij (kg/ha)

Daniel Clarke

100%

50%

0% 0

2,000 4,000 6,000

Binned Yield Yij (kg/ha)

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Decision rule The financial adviser is to choose a level of coverage α ≥ 0, providing a maximum claim payment of αL, to maximise expected (objective) utility: 1X ¯ )) ˜ + Yij + αL(Xij − mX EU = u(w (1) n ij∈D

where ¯ denotes X

1 n

P

ij∈D

Xij

˜ is random initial background wealth (statistically independent w of the joint distribution of (X , Y ) m is the pricing multiple (premium / expected claim income) u is the utility function, assumed to satisfy u 0 > 0 and u 00 < 0 L is difference between maximum and minimum binned yield: 5, 381 − 831 = 4, 550 kg/ha Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Optimal purchase, for different insurance premiums Figure: Optimal purchase of index insurance for maize from decision makers with (indirect) CRRA utility function

m = 0.50 m = 0.75 m = 1.00 m = 1.25 m = 1.50

Optimal cover α

100%

50%

0% 0 10 20 Coefficient of RRA Daniel Clarke

A Theory of Rational Demand for Indexed Insurance

Introduction

Theory

Numerical example for Maize in a developing country

Upper bounds for financial advice Risk averse DARA upper bound for purchase of index insurance for maize

αDARA

10% 5% 0% 1

1.5 1.751

2

Pricing multiple m Also: No risk averse expected utility maximiser will optimally purchase any index insurance if m > 1.751. Cf.: Giné et al. (2007): Average premium multiple of 3.4 Cole et al. (2009): Premium multiples of seven products, ranging from 1.7 to 5.3 Daniel Clarke

A Theory of Rational Demand for Indexed Insurance