Musgrave Risk Model

The Domar-Musgrave risk measure was proposed by two very famous economists:-

Evsey David Domar was a Russian American economist, famous as co-author of the Harrod–Domar model. He was a Keynesian economist. He has made contributions in three main areas of economics: economic history, comparative economics and economic growth. His works in the field of economics is uncountable. The most renowned one is: proportional income taxation and risk-taking, with Richard Musgrave, 1944.

Richard Abel Musgrave (December 14, 1910 – January 15, 2007) was American economist of German heritage. His most cited work is the theory of public finance (1959), described as "the first English-language treatise in the field." and "a major contribution to public finance

An important contribution to the analysis of such effects was made in 1944 by Domar and Musgrave in their well-known article on "proportional income taxation and risk-taking" in which they advanced a new model for analyzing investor behavior.

Domar and Musgrave assume that for any given asset which the investor might consider, the returns are not calculated with absolute certainty, but there exists a probability distribution of returns as seen by the investor.

Following definitions was proposed by them:

“Risk” is defined as the sum of anticipated negative incomes weighted by their

Thus:

The indifference curves, then, are parallel straight lines. Even if we assume that b-b is positive, such indifference curves are incapable of generating the movement toward higher “risk taking” suggested by Domar and Musgrave. Under neither proportional income tax nor lump sum tax shifts would there be any change in the portfolio.

Domar – Musgrave Risk Index

Domar and Musgrave proposed a measure of risk that measures the expected value of returns less than the returns on a risk-free investment.

Letting ‘r’ denote the return on a risk – free asset, the domar – Musgrave risk index is given by:

In the case of a continuous return, or

In the discrete case.

In brief, the intuition behind the Domar-Musgrave result is that the imposition of a proportionate income tax narrows the potential gains and losses from a risky investment, because the government takes a portion of the gains and covers a portion of the losses. The government becomes in effect a partner in the risky venture, taking on part of the risk. This, in turn, can allow investors to take on more of that risky investment, possibly enough so that the increased potential returns offsets, at least somewhat, the imposition of the tax (and likewise, the increased potential loss is offset by the increased deduction for the