If you own a €10 000 bond that has a 5% default probability and pays 10% p.a., you have a 95% probability to end up with €11 000 and a 5% probability of ending with nothing. (In actuality in case of default you would not lose everything, but this does not change the basic argument; so for simplicity's sake, I'll assume a default means losing 100%.) On average, €10 000 will turn into 95% × €11 000 + 5% × €0 = €10 450 — a return of 4.5% p.a. So odds are in your favor.
Odds may be in your favor but this is quite a gamble (after all, in Russian roulette too odds are in your favor). In such a situation the average is not meaningful. You will never make 4.5% a year: you either get 10% or lose everything.
If you instead spread your investment over 100 bonds, each with a default risk of 5% and returning 10% p.a., 95 of them will pay 110 € and 5 will default. Thus you always get a return of 4.5%. You receive €950 as interest and lose €500 from defaults, for a net gain of €450 — default is now merely a cost. (Of course other risks remain: interest rate and inflation risk cannot be diversified away, and there is the risk that the expected default rate of 5% was an underestimation — if 10% of bonds default, you end up with 90% × €11 000 + 10% × €0 = €9 900, a loss of 1%.)
A single bond is not an investment paying 4.5%, rather it pays 10% unless it defaults. With a portfolio, you got yourself a reliable investment with a return of 4.5%, net of default cost.
And life is much easier when you deal with costs rather than risks. Budgeting for instance is more efficient when you know what is going to happen (try budgeting if you make a living gambling).
Let us assume that every year you have a 1% probability of having a car accident, and that such an accident would cost you €20 000. On average accidents thus cost you 1% × €20 000 = €200. How would you feel about paying €220 a year for an insurance such that in case of an accident you have nothing to pay? On average accidents then cost you €20.
|cost (no insurance)||€0||€20 000|
Table 1: The cost of accidents, with and without insurance.
On the one hand, you pay more on average with an insurance than without. On the other hand, with an insurance you fork out only €220 in case of an accident. With an insurance, the possibility of having an accident is no longer a financial risk (of course other risks remain): it is a cost.
This is why insurance makes sense only when large sums of money and unlikely events are involved. Getting health insurance in case of hospitalization gets rid of the financial risk. But with health insurance for routine check-ups you trade a cost for another cost (and you pay for that).