valuation multiples are essentially inverse discount rates — when the "safe" return goes up, the premium you pay for risky future earnings should compress. explain why?

Start with a concrete analogy, then build up to the math.

The sandwich shop analogy

Imagine a sandwich shop earns $1 profit per sandwich. You're considering buying the entire business for $20 — that's a 20x PE multiple. Your implied return is 1/20 = 5% per year.

Now imagine the government suddenly offers you a guaranteed 5% bond — zero risk, zero effort. Why would you pay $20 for a risky business when you can get the same 5% return by doing nothing? You wouldn't — unless the business gets cheaper. For the stock to be worth buying, the price has to fall, say to $14, so your implied return rises to 7% — enough above the risk-free 5% to compensate you for the risk of owning a business.

That gap between what a risky asset must return and what a safe asset returns is called the "equity risk premium." When the safe rate rises, the whole structure shifts upward — and multiples must compress to keep that premium intact.

The actual math

The formal version of this is the Gordon Growth Model. The fair value of any asset that produces earnings is:

Price = Earnings ÷ (Discount Rate − Growth Rate)

Rearranged: PE = 1 ÷ (Discount Rate − Growth Rate)

You can see immediately that PE is literally the inverse of the spread between your discount rate and growth. When the discount rate goes up and growth stays the same, the denominator gets bigger — and PE must shrink.