Expected Return and Probability for Irish Prize Bonds

Irish Prize Bonds are a government-backed savings product where bondholders have the chance to win tax-free prizes in regular draws. The bonds do not pay interest; instead, bondholders are entered into a lottery for cash prizes.

The entire fund is approximately €4.5 billion. With each bond costing €6.25, the total number of bonds is $N = 7.2 \cdot 10^8$. All these bonds participate in the lottery.

There are 8,761 winning bonds per draw, and the total prize pool is €734,000. This means the expected value of a single prize is:

$$E_p = \frac{734,000}{8,761} \approx 83.78 \text{ euro}$$

The probability of a single bond winning in one draw is:

$$P_d = \frac{8,761}{7.2 \cdot 10^8}$$

The probability for a single bond to win in one year (52 draws) is:

$$P_a = P_d \times 52 \approx 0.0006327$$

For an investment of €25,000, a person would own 4,000 bonds. The expected number of wins for 4,000 bonds in one year is:

$$E_w = 4,000 \times P_a \approx 2.53$$

Thus, the expected annual return is:

$$E_r = E_w \times E_p \approx 2.53 \times 83.78 \approx 211.96 \text{ euro}$$

And the return on investment (ROI) is:

$$R = \frac{E_r}{25,000} \approx 0.00848 \text{ or } 0.85\%$$

Historical data analysis

A discussion on Boards.ie provides an opportunity to compare theoretical conclusions to practical results. One user reported an average annual return of €235 over 15 years on a €25,000 investment.

Another user shared a 20+ year history with total winnings of €3,550 since 1991. This gives an average annual return of approximately €162.7. Given their total investment of €84,950, this represents an average return of approximately 0.19% per year.

Conclusion

While some articles and reports criticize prize bonds, none make the mathematical disadvantage immediately obvious. The Irish Times states:

Of course, you can improve the odds of winning by buying more bonds.

While true, this statement overlooks an important fact: ==the return on investment does not depend on the number of bonds owned==. Whether you own 1 bond or 4,000, the expected return remains approximately 0.85% per year. You will win more frequently with more bonds, but you will lock away more capital for the same rate of return.

Update

The Irish Prize Bond website actually provides "Prize Fund rate", which is currently 1%. This is the same thing as the expected return calculated above. If you put €100 into prize bonds, you can expect to get back €1 on average per year in winnings.