I am buying a firm with an expected perpetual cash flow of $1,000 but am unsure of its risk. If I think the beta of the firm is .5, when in fact the beta is really 1, how much more will I offer for the firm than it is truly worth

Assuming that the risk free rate is 5%, you will pay $4, 849 more

Explanation:

The beta of a company or firm is a measure of the volatility, or systematic risk of a security, as it compares to the market. The beta of a frim or company is a measure of how the company’s equity market value changes with the changes in the overall market. It shows the sensitivity of the company’s equity to changes in the market. Systematic risk is the risk that cannot be diversified. This type od risk is due to changes in the market, and because of this, it cannot be avoided. This risk is caused by factors that are external to the firm.

Assume that the $1, 000 is a perpetuity. The risk- free rate is 5%

If beta is 5, the cash flow is discounted at 55%  

PV (beta = 5) = $1, 000 / .55 = $1, 818

If, however, beta is equal to 1, the investment will yield at 15%, and the price paid for the firm should be:

PV = $1, 000 / .15 = $6, 667

The difference $4, 849 [ $6, 667 - $1, 818], is the amount you will pay if you erroneously assumed that the beta is 5 rather 1.

If the cash flow lasts only one year:

PV (beta = 5) = $1, 000 / (1 + .55) = $645

PV (beta = 1) = $1, 000 / (1 + .15) = $869

With a difference of $224.

Incorrectly assuming the value of beta has substantial effects on the calculations of cash flows.