# Do bond interest rate risk premiums only compensate for the amount investors might lose?

Consider two different bonds: one riskless, and the other risky. Suppose the risky bond has a known probability of default. Suppose a market is made in these bonds, with investors trying to maximize return.

In fact, let's consider a specific scenario in which a riskless 1 year, $1,000 bond will return 5% in one year. Suppose a risky$1,000 bond will default (no interest, plus loss of principal) with probability .01. In this case, if the interest rate on the risky bond is 6.060606...%, then (speaking in simple terms), the bond will return $1060.060606... after one year 99 times out of 100, and$0 one time out of 100, for an average return of (99*1060.060606...+0*1060)/100=1050. In other words, a risk premium of .010606060... above and beyond the risk free rate .05 is enough to compensate for the .01 probability of default.

Assume we have a market with the two bonds described above. Assume it is perfectly efficient. If the risky bond sells for a rate higher than .060606..., there would be excess returns possible, and participants could buy down the interest rate to .060606..., at which point it matches the risk-adjusted return of the risk-free rate.

In this case, the risk premium of .01060606... is only enough to compensate for default risk. When investors buy the bond at this price, they are essentially buying a "risk free" bond, assuming they repeat this scenario indefinitely and can absorb the occasional loss over time.

My question is: in a perfectly efficient market, are all bonds priced in such a way so that they all return the same amount (on average), after accounting for risk? In other words, do risk premiums only compensate for the amount investors might lose?

Another way of saying it: In a perfectly efficient market, would all bond portfolios have an average return (after averaging out all defaults) equal to the risk free rate of return?

In answer to your last formulation, no. In a perfectly efficient market, different investors still have different risk tolerances (or utility functions). They're maximizing expected utility, not expected value.

The portfolios that maximize expected utility for different risk preferences are different, and thus generally have different expected values. (Look up mean-variance utility for a simple-ish example.)

Suppose you have log utility for money, u(x) = log(x), and your choice is to invest all of your money in either the risk-free bond or in the risky bond. In the risky bond, you have a positive probability of losing everything, achieving utility u(0) = -\infty. Your expected utility after purchase of the risky bond is:

Pr(default)*u(default) + (1-Pr(default))*u(nominalValue).

Since u(default)= -\infty, your expected utility is also negative infinity, and you would never make this investment. Instead you would purchase the risk-free bond.

But another person might have linear utility, u(x) = x, and he would be indifferent between the risk-free and risky bonds at the prices you mention above and might therefore purchase some. (In fact you probably would have bid up the price of the risk-free bond, so that the other investor strictly prefers the risky one.)

So two different investors' portfolios will have different expected returns, in general, because of their different risk preferences. Risk-averse investors get lower expected value. This should be very intuitive from portfolio theory in general: stocks have higher expected returns, but more variance. Risk-tolerant people can accept more stocks and more variance, risk-averse people purchase less stocks and more bonds.

The more general question about risk premia requires an equilibrium price analysis, which requires assumptions about the distribution of risk preferences among other things. Maybe John Cochrane's book would help with that---I don't know anything about financial economics. I would think that in the setup above, if you have positive quantities of these two investor types, the risk-free bond will become more expensive, so that the risky one offers a higher expected return. This is the general thing that happens in portfolio theory.

Anyway. I'm not a financial economist or anything.

Here's a standard introduction to expected utility theory: http://web.stanford.edu/~jdlevin/Econ%20202/Uncertainty.pdf

• An excellent answer. To generalise this further, consider the following axiom: For the expected rate of return to be equal for financial assets affected by uncertainty, it is necessary for the risk preferences of all investors to be neutral. Also, it's necessary but not sufficient. Even risk neutral investors may need to sell bonds, and therefore value liquidity, tenor, etc. – lilster Sep 22 '14 at 13:56

[...] are all bonds priced in such a way so that they all return the same amount (on average), after accounting for risk? In other words, do risk premiums ONLY compensate for the amount investors might lose?

No. GE might be able to issue a bond with lower yield than, say, a company from China with no previous records of its presence in the U.S. markets. A bond price not only contains the risk of default, but also encompasses the servicability of the bond by the issuer with a specific stream of income, location of main business, any specific terms and conditions in the prospectus, e.g.callable or not, insurances against default, etc.

Else for the same payoff, why would you take a higher risk? The payoff of a higher risk (not only default, but term structure, e.g. 5 years or 10 years, coupon payments) bond is more, to compensate for the extra risk it entails for the bondholder. The yield of a high risk bond will always be higher than a bond with lower risk.

If you travel back in time, to 2011-2012, you would see the yields on Greek bonds were in the range of 25-30%, to reflect the high risk of a Greek default. Some hedge funds made a killing by buying Greek bonds during the eurozone crisis.

If you go through the Efficient frontier theory, your argument is a counter statement to it. Same with individual bonds, or a portfolio of bonds. You always want to be compensated for the risk you take. The higher the risk, the higher the compensation, and vice versa.

When investors buy the bond at this price, they are essentially buying a "risk free" bond [...]

Logically yes, but no it isn't, and you shouldn't make that assumption.