Risk-Free Rates in PE

Risk-free rates set the baseline return for everything else in finance. In private equity, they matter more than many people realize:

• They anchor the CAPM cost of equity. Every IRR hurdle you quote starts with $r_f$.
• They feed directly into the cost of debt. Lenders price spreads over Treasuries.
• They tug on exit multiples, because discount rates shape what future buyers are willing to pay.

This post is an interactive tour of those connections. One control panel at the top lets you set $r_f$ (and related assumptions), and every plot updates together. Each section zooms in on a different way $r_f$ sneaks into PE math:

1. WACC frontier. See how adding debt moves your discount rate, and how the entire curve shifts upward when $r_f$ rises. This is about target leverage and rate sensitivity.
2. DCF surface. Watch how valuation multiples explode or collapse as $(\text{WACC} - g)$ shrinks or widens. This makes duration risk from rates tangible.
3. Exit multiple vs $r_f$. Translate abstract rate moves into concrete “turns of EBITDA” on exit. This frames macro risk in underwriting language.
4. LBO engine. Simulate thousands of paths. $r_f$ hits both the interest bill and the exit multiple, reshaping the entire IRR distribution. Here you see rate risk in probabilistic terms.
5. Debt path. Trace how quickly deals de-risk. Higher $r_f$ slows deleveraging, keeping leverage higher for longer. This is the mechanical channel from rates into covenant pressure and IRR downside.

Taken together, the plots are not just math toys — they’re intuition pumps. They show how one simple number, the risk-free rate, flows through private equity from entry to exit, shaping valuation, leverage, and outcomes.

WACC frontier: why plot it, and what $r_f$ is telling you

Why do this at all? You want a quick, visual answer to three questions that matter in PE:

1. How much debt is “about right”? Look for the minimum of WACC across $D/V$. That is your “target leverage” starting point.
2. How sensitive is my valuation to rates? If the whole curve shifts up when $r_f$ rises, your DCF multiple falls. That’s your rate risk at a glance.
3. Which story about equity risk fits the deal? Constant-$\beta$ says debt stays cheap longer; Hamada says equity gets riskier as you lever, so WACC bends up sooner.

Two tiny formulas (that explain the picture).

• WACC: $$\text{WACC}(D/V) = \tfrac{E}{V}K_e + \tfrac{D}{V}K_d(1-\tau),\quad K_e = r_f + \beta\cdot\text{MRP},\quad K_d = r_f + \text{spread}.$$
• If $r_f$ changes by $\Delta r$: $$\Delta\text{WACC} = \Delta r\cdot\big(1 - \tau\cdot D/V\big).$$ Translation: when rates rise, the whole curve moves up (almost in parallel). Debt softens the blow a bit because of the tax shield.

How to use the plot.

• Bold curves are WACC (blue = constant-$\beta$, orange = Hamada).
• Dashed curves show $r_f + 100$ bps, so you can see the rate shock without touching any other inputs.
• Vertical dotted lines mark the min WACC for each curve (a practical leverage target); the gray dashed line is your current $D/V$.
• Components ($K_e$, $K_d$) are faint for context. We cap the y-axis so extreme Hamada tails don’t hide the useful part.

What value do you get?

• Faster debt sizing conversations. “At today’s $r_f$, the min WACC is near $D/V\approx X$; if $r_f$ goes up 100 bps, the min barely moves, but the rate level does.”
• Clear rate sensitivity for IC memos. “A +100 bps rate shock adds $\approx(1-\tau\cdot D/V)\cdot100$ bps to WACC, trimming EV/EBITDA by $\Delta \text{Multiple} \approx \frac{\Delta\text{WACC}}{\text{WACC}-g}$.”
• Reality check on leverage. If Hamada’s minimum sits at much lower $D/V$ than constant-$\beta$, the “debt is cheap” story is probably too generous for this deal.

Try this (60 seconds).

1. Set $g$ and spreads; toggle Hamada on.
2. Nudge $r_f$ up by 100 bps. Watch the dashed parallels rise.
3. Note the new WACC level at your current $D/V$ and the gap to the min. That gap is the price you pay (in discount rate) for your leverage choice.

DCF surface: how $r_f$ and growth shape valuation multiples

What this plot is showing. It’s the simplest possible DCF model:

where $FCF_1$ is next year’s free cash flow and $g$ is perpetual growth.

Divide $EV$ by EBITDA and you get an implied multiple. That’s what the colors show here.

• x-axis = shifts in the risk-free rate ($r_f$, in basis points).
• y-axis = growth rate $g$.
• color = $EV/EBITDA$ multiple from the one-stage DCF.

Where $r_f$ comes in. $WACC$ depends directly on $r_f$ (in both $K_e$ and $K_d$). Slide $r_f$ up by 100 bps and the denominator $(\text{WACC} - g)$ gets bigger. That shrinks $EV$, and the heatmap turns darker (lower multiple).

Why this is interesting.

• Duration intuition. When $WACC$ is close to $g$, the denominator is small. Even tiny changes in $r_f$ cause huge swings in the multiple. These are your “long duration” assets.
• Macro sensitivity. If you believe rates will stay high, you can see directly how much that compresses multiples at a given growth rate.
• Scenario planning. By pairing a growth guess with a rate guess, you get a quick “sanity band” for reasonable multiples.

How to use the chart.

• Pick a row (your view of $g$). Move along the x-axis to see what happens if $r_f$ is 3%, 4%, 5%…
• Or pick a column (a fixed $r_f$). Move up the y-axis to see how more growth justifies a higher multiple.
• Watch the bright zones near where $WACC \approx g$: that’s the danger zone where valuation can blow up.

What value this gives.

• Makes explicit how much of your multiple is just rates math versus how much is growth.
• Helps explain to IC or LPs: “This deal’s multiple isn’t magic — it’s what happens if you assume $g=3\%$ and $r_f=4\%$.”
• Provides a simple way to compare deals: which ones are fragile to rate changes, and which ones have enough growth cushion.

Exit multiple vs $r_f$: how discount rates pull on valuations

What this plot is showing. We assume a simple rule of thumb: when the risk-free rate $r_f$ rises by 100 bps, the exit multiple compresses by some number of turns (you set the sensitivity).

That gives us a line:

• x-axis = $r_f$ (in %),
• y-axis = exit multiple (in turns of EBITDA).

Where $r_f$ comes in. Think of exit multiples as today’s investors paying a price based on their discount rate. If $r_f$ goes up, both $K_e$ and $K_d$ rise, so the whole WACC is higher. A higher WACC means lower present value for a given stream of cash flows. One way the market expresses that is by lowering the multiple it is willing to pay at exit.

Why this is interesting.

• Quick sanity check. If you think $r_f$ might rise by 100 bps before exit, you can see directly how many turns that could shave off your underwriting.
• Scenario lens. It shows the sensitivity of your deal’s outcome to macro moves, not just company performance.
• Communication tool. It’s easier to say to IC or LPs: “Every +100 bps in $r_f$ costs us ~0.25x turns on exit.” That turns rate risk into something concrete.

How to use the chart.

• Look at your current $r_f$ (dotted vertical line). That’s your anchor.
• Shift left or right to see what happens if rates fall or rise.
• Adjust the “turns per 100 bps” sensitivity in the controls. This lets you test whether your view of market repricing is mild or severe.

What value this gives.

• It reframes rate debates. Instead of vague “higher rates are bad,” you can say “if 10y Treasuries go from 4% to 5%, our exit case drops from 10.0x to 9.75x.”
• It makes macro risk tangible inside your LBO math. Multiples don’t just fall from the sky; they move with capital costs, and $r_f$ is the cleanest knob for that.

LBO engine: IRR is a distribution, not a point

What’s happening here. We model a leveraged buyout across thousands of random trials:

• At entry: buy at an entry multiple, use debt + equity.
• During the hold: EBITDA grows stochastically; you pay interest at $K_d$ and use leftover cash to pay down debt.
• At exit: sell at an exit multiple that flexes with $r_f$.

The IRR is computed from equity in vs equity out. Instead of one neat number, you get a distribution of possible outcomes.

Where $r_f$ matters.

• Cost of debt ($K_d$). Higher $r_f$ makes interest heavier, slowing deleveraging.
• Exit multiple. If $r_f$ rises, the assumed exit multiple can fall, cutting terminal equity value.

So $r_f$ squeezes both ends: the path (debt paydown) and the final payoff (exit multiple).

This plot turns the simulation into a rate sensitivity curve: lines show p10, median, and p90 IRRs against $r_f$. The vertical dotted line marks the current $r_f$.

Why this is helpful.

• Reframes rate risk. Instead of saying “higher rates are bad,” you can quantify: +100 bps $r_f$ lowers median IRR from 18% to 15% and fattens the downside tail.
• Makes uncertainty visible. A histogram of possible IRRs is truer to PE than a single “deal IRR” number.
• Supports better IC/LP conversations. You can show exactly how sensitive the deal is to rates, and whether the downside is still tolerable.

Debt path: how quickly do you de-risk?

What this plot is showing. We track the debt balance through the hold across thousands of simulations. For each year, we plot percentiles:

• p90 (top line): slowest paydown, worst cases.
• Median: typical case.
• p10 (bottom line): fastest paydown, best cases.

Where $r_f$ comes in.

• Higher $r_f$ → higher $K_d$ → more cash goes to interest instead of principal.
• That slows the paydown path, pushing the curves higher (more debt left outstanding).

Why this is interesting.

• Deleveraging = risk reduction. The faster you pay down debt, the safer the equity becomes.
• Link to IRR. A deal that clears debt by year 3 is much more resilient than one still loaded at year 5. The shape of these lines explains the histogram you just saw: slower deleveraging fattens the left tail of IRR outcomes.
• Macro sensitivity. If you think rates will rise, this plot shows the mechanical effect: debt sticks around longer.

How to use the chart.

• Look at the median path: that’s your “base case” de-risking timeline.
• Watch the gap between p10 and p90: that’s the uncertainty in outcomes.
• Compare the curves under different $r_f$: if higher rates delay crossing below, say, 3× EBITDA, that’s a red flag for covenants and refinancing risk.

What value this gives.

• Turns abstract “rate risk” into something concrete: how many years until we’re safe?
• Helps explain to IC/LPs: “At today’s $r_f$, median debt is below 2× EBITDA by year 4. If $r_f$ rises 100 bps, it takes until year 6.”
• Reinforces that PE returns come from both entry/exit multiples and what happens in the middle — deleveraging is the bridge.