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VEE Accounting and Finance · Project and Divisional Discount Rates · Tutorial 686 of 996

Discount Rates Under Changing Leverage

Learn to update project equity beta and WACC as leverage changes, and distinguish a stable target ratio from a scheduled debt-paydown policy.

Advanced 12 min read

What You'll Learn

  • Distinguish debt-to-equity from debt-to-value leverage ratios.
  • Calculate equity beta under the stated tax-adjusted relevering assumptions.
  • Use CAPM consistently when debt is risk-free.
  • Determine how WACC changes as a scheduled debt balance changes.
  • Explain why a constant target leverage can support a constant WACC while scheduled paydown generally cannot.
  • Recognize when applying one initial WACC to every forecast period is inappropriate.

Why Discount Rates Change When Leverage Changes

A project's operating risk can remain unchanged while its financing mix evolves. If debt is paid down on a schedule, the equity claim generally becomes less exposed to financial leverage over time. Its beta and required return can therefore change, even when the project's asset beta does not. The project's WACC can change too, because debt's tax benefit depends on the debt weight.

The key distinction is between a constant target leverage policy, which maintains a chosen debt proportion, and a scheduled debt-paydown policy, which specifies debt repayments over time. A single WACC may be appropriate under stable target leverage and other stable assumptions. A scheduled repayment plan generally requires rates consistent with the changing leverage, or a valuation approach that handles financing effects separately.

Definition: At date \(t\), let \(D_t\) and \(E_t\) be the market values of project debt and equity, and let \(V_t=D_t+E_t\). The debt-to-equity ratio is \(D_t/E_t\); the debt-to-value ratio is \(w_{D,t}=D_t/V_t\). They are related by \(D_t/E_t=w_{D,t}/(1-w_{D,t})\), but are not interchangeable.

Assumptions Behind the Leverage Relations

The calculations below use a standard tax-adjusted relevering model. The project has constant operating risk, measured by asset beta \(\beta_A\); corporate tax rate \(T\) is constant and interest deductions can be used; debt is risk-free, so its beta is zero and its required return \(R_D\) equals the risk-free rate \(R_f\). Debt and equity are valued at market value. The standard relation also assumes the tax treatment of financing is represented by the tax-adjusted debt term shown below, with no distress costs or other financing effects.

Under those assumptions, the project's equity beta at a given leverage is

$$ \beta_{E,t}=\beta_A\left[1+(1-T)\frac{D_t}{E_t}\right]. $$

CAPM gives the required returns \(R_A=R_f+\beta_A(R_M-R_f)\) on the project's assets and \(R_{E,t}=R_f+\beta_{E,t}(R_M-R_f)\) on equity, where \(R_M-R_f\) is the market risk premium. Since debt is risk-free in this model, \(R_D=R_f\). Subtracting the asset CAPM equation from the equity CAPM equation gives a useful equivalent form:

$$ R_{E,t}=R_A+(1-T)\frac{D_t}{E_t}(R_A-R_D). $$

These equations are conditional results, not universal rules. They should not be used unchanged when debt is risky, tax shields cannot be used, or the financing policy changes the risk of the tax shields in a way the model does not capture. If debt is risky, its beta and required return must be modeled consistently rather than treating it as risk-free.

How the WACC Depends on Debt Weight

With tax-deductible interest, the period-\(t\) WACC is the market-value-weighted cost of equity plus the after-tax cost of debt:

$$ \mathrm{WACC}_t=(1-w_{D,t})R_{E,t}+w_{D,t}R_D(1-T). $$

In the model above, this can be simplified to a direct expression in the debt-to-value ratio. The simplification matters because it makes clear that, even if asset risk and the asset required return stay constant, WACC still depends on leverage.

Result: Under the stated assumptions, \(\mathrm{WACC}_t=R_A(1-Tw_{D,t})\). This expression is based on market-value weights and the tax-adjusted relevering relation; it is not a general formula for every debt policy or debt-risk model.

To see why, substitute the equity-return relation into the weighted WACC. Use \((1-w_{D,t})(D_t/E_t)=w_{D,t}\), which follows from \(V_t=D_t+E_t\):

$$ \begin{aligned} \mathrm{WACC}_t &=(1-w_{D,t})\left[R_A+(1-T)\frac{D_t}{E_t}(R_A-R_D)\right] +w_{D,t}R_D(1-T)\\ &=(1-w_{D,t})R_A+(1-T)w_{D,t}(R_A-R_D) +w_{D,t}R_D(1-T)\\ &=R_A\left[(1-w_{D,t})+(1-T)w_{D,t}\right]\\ &=R_A(1-Tw_{D,t}). \end{aligned} $$

The debt-return terms cancel in the second line: the negative debt-return component inside \(R_A-R_D\) is offset by the after-tax debt-cost term. Since \((1-w_{D,t})+(1-T)w_{D,t}=1-Tw_{D,t}\), the final expression follows.

Worked Example: Constant Target Leverage

Worked Example: Constant Target Leverage

A hypothetical project has asset beta \(\beta_A=0.80\), tax rate \(T=25\%\), and a constant target debt-to-equity ratio \(D/E=0.50\). Assume risk-free debt, annual risk-free rate \(R_f=3\%\), and annual market risk premium \(R_M-R_f=5\%\). Calculate the equity beta, required equity return, and WACC. The leverage target, asset risk, tax rate, and market inputs are assumed unchanged over the periods considered.

Method. First relever the asset beta using the target debt-to-equity ratio, then apply CAPM and calculate WACC using market-value weights. This avoids using \(D/E\) as though it were the WACC debt weight.

The tax-adjusted leverage factor is

$$ (1-T)\frac{D}{E}=(1-0.25)(0.50)=0.75(0.50)=0.375. $$

Thus the equity beta is

$$ \beta_E=0.80(1+0.375)=0.80(1.375)=1.10. $$

The asset required return is \(R_A=3\%+0.80(5\%)=7.00\%\). The equity required return is \(R_E=3\%+1.10(5\%)=8.50\%\). Convert the target ratio to a debt-to-value weight: \(D/V=0.50/(1+0.50)=1/3\), so \(E/V=2/3\). Therefore,

$$ \begin{aligned} \mathrm{WACC} &=\frac{2}{3}(8.50\%)+\frac{1}{3}(3.00\%)(1-0.25)\\ &=5.6667\%+0.7500\%=6.4167\%. \end{aligned} $$

As a check, \(R_A(1-TD/V)=7.00\%[1-0.25(1/3)]=6.4167\%\). The calculations agree. Rates are rounded to four decimal places where needed.

Scheduled Debt Paydown: Recalculate as Leverage Moves

With scheduled debt paydown, the debt balance at each date may be specified in advance, but the market value of equity can also change. Thus the leverage ratio is not determined by the debt balance alone: calculate \(D_t/E_t\) and \(D_t/V_t\) using the market values at each date. If the ratio falls, the tax-adjusted relevering model gives a lower equity beta. At the same time, the reduction in the debt weight reduces the modeled tax benefit in WACC, which can make WACC rise toward the asset required return.

Worked Example: Equity Beta and WACC During Scheduled Paydown

A hypothetical project has constant asset beta \(\beta_A=0.90\), tax rate \(T=25\%\), annual risk-free rate \(R_f=4\%\), and annual market risk premium \(5\%\). Its debt is risk-free, so \(R_D=4\%\). At three valuation dates, the market values, in millions of dollars, are: date 0, \(D_0=40\) and \(E_0=60\); date 1, \(D_1=30\) and \(E_1=70\); date 2, \(D_2=20\) and \(E_2=80\). Assume the operating risk and all other model assumptions remain unchanged. Find the equity beta and WACC at each date.

Method. At each date, calculate both leverage ratios from the given market values, then update beta and WACC. This avoids carrying the initial leverage ratio forward after repayments.

At date 0, \(D_0/E_0=40/60=2/3\), and \(D_0/V_0=40/(40+60)=0.40\). The tax-adjusted beta factor is \(1+0.75(2/3)=1.50\), so \(\beta_{E,0}=0.90(1.50)=1.35\). CAPM gives \(R_{E,0}=4\%+1.35(5\%)=10.75\%\). Then

$$ \mathrm{WACC}_0=8.50\%[1-0.25(0.40)]=8.50\%(0.90)=7.6500\%. $$

At date 1, \(D_1/E_1=30/70=3/7\) and \(D_1/V_1=30/100=0.30\). The beta factor is \(1+0.75(3/7)=1.3214286\), giving \(\beta_{E,1}=0.90(1.3214286)=1.1892857\). Therefore, \(R_{E,1}=4\%+1.1892857(5\%)=9.9464\%\), and

$$ \mathrm{WACC}_1=8.50\%[1-0.25(0.30)]=8.50\%(0.925)=7.8625\%. $$

At date 2, \(D_2/E_2=20/80=0.25\) and \(D_2/V_2=20/100=0.20\). The beta factor is \(1+0.75(0.25)=1.1875\), so \(\beta_{E,2}=0.90(1.1875)=1.06875\). Thus \(R_{E,2}=4\%+1.06875(5\%)=9.34375\%\), and

$$ \mathrm{WACC}_2=8.50\%[1-0.25(0.20)]=8.50\%(0.95)=8.0750\%. $$

The check is directional and numerical: equity beta falls from \(1.35\) to \(1.06875\) as leverage falls, while WACC rises from \(7.6500\%\) to \(8.0750\%\). The lower equity return does not imply a lower WACC here; the shrinking debt tax benefit more than offsets the change in the equity component.

Constant Target Versus Scheduled Paydown

A constant target debt ratio means leverage is maintained relative to market value as the project evolves. If the target, asset risk, tax rate, and market rates are constant, the equity beta and WACC from this model remain constant as well. The debt amount need not be constant: it may need to adjust to keep the target ratio.

A scheduled debt balance is different. Debt repayments are set by a schedule rather than by a continuing target ratio. Unless equity value changes in exactly the way required to maintain the same ratio, \(D_t/E_t\) and \(D_t/V_t\) change over time. Recompute the discount-rate inputs at the relevant dates instead of assuming that the initial WACC persists.

Worked Example: Same Initial Leverage, Different Financing Policies

Two hypothetical versions of the same project have \(\beta_A=0.80\), \(T=25\%\), risk-free debt, \(R_f=3\%\), and market risk premium \(5\%\). At date 0, both have \(D=40\) and \(E=80\), in millions of dollars. Later, Policy A maintains the same target \(D/E\) ratio. Policy B follows a debt-paydown schedule and has \(D=20\), \(E=80\) at date 1. Calculate date 0 and date 1 beta and WACC for each policy.

Method. Treat each date's market leverage separately. The common date-0 values provide a check that the two policies agree initially; the date-1 values reveal the effect of the policy difference.

At date 0, both projects have \(D/E=40/80=0.50\) and \(D/V=40/120=1/3\). Therefore, \(\beta_E=0.80[1+0.75(0.50)]=1.10\), and \(R_E=3\%+1.10(5\%)=8.50\%\). Their asset return is \(R_A=3\%+0.80(5\%)=7.00\%\). Each has

$$ \mathrm{WACC}_0=7.00\%[1-0.25(1/3)]=6.4167\%. $$

Under Policy A, date-1 target leverage remains \(D/E=0.50\), which corresponds to \(D/V=1/3\). The modeled date-1 equity beta is still \(1.10\), and WACC is still \(6.4167\%\). Under Policy B, date-1 leverage is \(D/E=20/80=0.25\), with \(D/V=20/(20+80)=0.20\). Its equity beta is \(0.80[1+0.75(0.25)]=0.95\), and its equity return is \(3\%+0.95(5\%)=7.75\%\). Its WACC is

$$ \mathrm{WACC}_1=7.00\%[1-0.25(0.20)]=6.6500\%. $$

The initial rates are identical because the initial leverage is identical. At date 1, the paydown policy has a lower equity beta but a higher WACC than the constant-target policy. A common initial WACC would miss this difference.

Discounting Cash Flows Consistently

A changing sequence of calculated WACCs is not automatically a license to discount every project's cash flows with an arbitrary sequence of rates. The cash flows and financing policy must match the valuation method. A WACC approach discounts operating cash flows using rates consistent with the forecast market-value weights and the project's risk. A method that values operating cash flows separately from financing effects can be more transparent when debt amounts and tax shields are explicitly scheduled.

In particular, do not use a constant WACC merely because the project began with a target ratio. A target that is maintained over time supports that assumption only if the other relevant inputs are also stable. Conversely, a debt balance that declines does not prove the debt ratio declines: equity market value may move too. Recheck the market-value weights before concluding how beta or WACC changes.

1
Identify the financing policy.
Decide whether the debt ratio is targeted through time or debt repayments are scheduled independently of market value.
2
Calculate market leverage at each relevant date.
Find \(D_t/E_t\) for relevering beta and \(D_t/V_t\) for WACC weights. Do not substitute one ratio for the other.
3
Apply a consistent risk model.
Use the tax-adjusted relation only under its assumptions, and make debt's beta and required return consistent with whether debt is risk-free or risky.
4
Match discounting to the policy.
Use a constant WACC only when the target leverage and other required inputs support it; otherwise account for changing financing consistently.

Common Exam Trap

A frequent error is to use the initial \(D/E\) ratio to relever beta at every date, or to use that ratio directly as the WACC debt weight. For example, if \(D/E=0.50\), then \(D/V=0.50/(1+0.50)=1/3\), not \(0.50\). The wrong substitution would produce \(\mathrm{WACC}=R_A[1-0.50T]\); the correct expression is \(R_A[1-T/3]\), under the assumptions stated above. A second error is to call debt risk-free while assigning it a cost different from \(R_f\) in a CAPM calculation. In the risk-free-debt model, set \(R_D=R_f\).

Key takeaway: With stable target leverage and stable risk and market inputs, equity beta and WACC can remain constant. With scheduled paydown, recalculate market leverage over time: equity beta can fall as debt is reduced, while WACC can rise as the debt tax benefit shrinks. Apply the resulting rates only in a valuation method consistent with the financing policy.

Check Your Understanding

Work each problem using the stated model assumptions, then check the final answer.

  1. A project has \(\beta_A=0.80\), \(T=25\%\), risk-free debt, and a target \(D/E=0.50\). Find its equity beta. (Answer: \(\beta_E=1.10\).)
  2. For a project, \(\beta_A=0.80\), \(T=20\%\), \(R_f=3\%\), and the market risk premium is \(5\%\). Debt is risk-free and the target debt-to-value ratio is \(0.25\). Find the equity beta, equity required return, and WACC. (Answer: \(\beta_E=1.0133\), \(R_E=8.0667\%\), and WACC \(=6.6500\%\).)
  3. A scheduled-paydown project has \(\beta_A=0.70\), \(T=30\%\), risk-free debt, \(R_f=2.5\%\), and a market risk premium of \(4.5\%\). At date 0, \(D=45\) and \(E=55\), in millions; at date 1, \(D=20\) and \(E=80\). Find each date's equity beta and WACC. (Answer: date 0, \(\beta_E=1.1009\) and WACC \(=4.8873\%\); date 1, \(\beta_E=0.8225\) and WACC \(=5.3110\%\).)
  4. A project has \(\beta_A=0.90\), \(T=25\%\), \(R_f=4\%\), and a market risk premium of \(5\%\); its debt is risk-free. Its debt-to-equity ratio falls from \(0.75\) to \(0.25\). Calculate the equity beta and WACC at both leverage levels. (Answer: at \(D/E=0.75\), \(\beta_E=1.40625\) and WACC \(=7.5893\%\); at \(D/E=0.25\), \(\beta_E=1.06875\) and WACC \(=8.0750\%\).)
  5. Two projects share \(\beta_A=0.80\), \(T=25\%\), \(R_f=3\%\), and a \(5\%\) market risk premium, with risk-free debt. Both start at \(D=40\), \(E=80\), in millions. One maintains its \(D/E\) target; the other moves to \(D=20\), \(E=80\) at the next date. Give the initial WACC and each project's next-date WACC. (Answer: initial WACC \(=6.4167\%\); next-date target-policy WACC \(=6.4167\%\); next-date scheduled-paydown WACC \(=6.6500\%\).)

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