Comparative Statics Of Equity And Debt Values

Why Comparative Statics Matter

In a structural model of corporate claims, equity is not simply the value of the firm’s assets less the face amount of its debt. The debt is risky: if assets are insufficient at maturity, debtholders may not receive the full promised amount. Equity and debt therefore have values that depend on the firm’s assets, the uncertainty in those assets, and the debt obligation. Comparative statics describe how those claim values respond when one input changes and the others are held fixed.

The key distinction is whether a change affects the total value of the claims or changes only how that value is divided. A rise in asset value raises the value of the firm’s claims in total. Greater asset volatility, under the model assumptions below, benefits equity at debt’s expense while leaving the total value unchanged. Increasing the debt face value also changes the allocation between the claims, not the total asset value.

The Structural Model

Consider a firm with current asset value $V$ and a single zero-coupon debt obligation with face value $F$, due at time $T$. At maturity, equity receives any assets remaining after debt is paid. Debtholders receive the lesser of the assets available and the promised face value. The model treats equity as a call option on firm assets and debt as the residual claim on those assets.

Model assumptions: The firm has one zero-coupon debt claim due at $T$; there are no interim payouts; the asset value follows geometric Brownian motion with constant volatility $\sigma$; the risk-free force of interest $\delta$ is constant; markets are frictionless; and default occurs only at maturity if assets are below $F$. Equity holders have limited liability. The model does not include taxes, bankruptcy costs, or renegotiation.

Let $N(z)$ denote the standard normal cumulative distribution function and $\phi(z)$ its density. Under these assumptions, the option-pricing formulas for equity and debt are

E=VN(d_1)-Fe^{-\delta T}N(d_2), \qquad D=V-E

where

d_1=\frac{\ln(V/F)+(\delta+\sigma^2/2)T}{\sigma\sqrt{T}}, \qquad d_2=d_1-\sigma\sqrt{T}.

Here $E$ is the current equity value and $D$ is the current market value of debt. The identity $D=V-E$ follows from the model’s claim structure: equity and debt together exhaust the firm’s assets, with no other claims or costs. Equivalently, debt is a risk-free zero-coupon bond less a put option on the assets. The put represents the shortfall debtholders bear when assets fall below the promised face value.

Asset Value: A Dollar Is Shared Between the Claims

Hold $F$, $T$, $\sigma$, and $\delta$ fixed. A higher asset value increases both equity and debt values, but not by a full dollar for each claim. The local sensitivities are

\frac{\partial E}{\partial V}=N(d_1), \qquad \frac{\partial D}{\partial V}=N(-d_1).

Because $N(d_1)+N(-d_1)=1$, the two sensitivities add to one. Thus, for a small increase in asset value, the increase in equity value plus the increase in debt value equals the increase in total claim value. These are marginal effects: they measure the change for a small change in $V$, holding the other inputs fixed, rather than guaranteeing that the same sensitivity applies over a large change.

Derivation: Since both $d_1$ and $d_2$ have derivative $1/(V\sigma\sqrt{T})$ with respect to $V$, differentiating the equity formula gives $$ \frac{\partial E}{\partial V} =N(d_1)+\frac{V\phi(d_1)-Fe^{-\delta T}\phi(d_2)}{V\sigma\sqrt{T}}. $$ The standard normal identity $V\phi(d_1)=Fe^{-\delta T}\phi(d_2)$ makes the second term zero. Hence $\partial E/\partial V=N(d_1)$. Since $D=V-E$, $\partial D/\partial V=1-N(d_1)=N(-d_1)$.

Worked Example: Allocating a Small Increase in Asset Value

A hypothetical firm has $V=\$150$, $F=\$120$, $T=1$ year, $\delta=0$, and $\sigma=0.30$. Estimate the local change in equity and debt values for a $\$1$ increase in asset value. Use the structural model and report sensitivities to four decimal places.

Method. Calculate $d_1$, then use the asset-value sensitivities. Since $T=1$ and $\delta=0$,

\[ d_1=\frac{\ln(150/120)+0.30^2/2}{0.30} =\frac{\ln(1.25)+0.045}{0.30} =\frac{0.2231436+0.045}{0.30} =0.8938. \]

The standard normal probabilities are $N(0.8938)\approx0.8143$ and $N(-0.8938)\approx0.1857$. Therefore, a small $\$1$ asset-value increase raises equity value locally by about $\$0.8143$ and debt value by about $\$0.1857$.

Check. The sensitivities sum to $0.8143+0.1857=1.0000$, as required because the increase in total claim value is one dollar. The common mistake is to assign the entire dollar to equity or to add a full dollar to both claims; the derivative shows why neither is correct.

Asset Volatility: Equity Gains at Debt’s Expense

Now hold $V$, $F$, $T$, and $\delta$ fixed and increase $\sigma$. Equity’s sensitivity to volatility is

\frac{\partial E}{\partial \sigma}=V\phi(d_1)\sqrt{T}>0.

This is the call option’s vega. It is positive because a call’s payoff is more valuable when the underlying asset value is more uncertain: equity participates in asset value above the debt obligation but cannot fall below zero. Since $D=V-E$ and $V$ is fixed,

\frac{\partial D}{\partial \sigma}=-V\phi(d_1)\sqrt{T}<0, \qquad \frac{\partial(E+D)}{\partial \sigma}=0.

This result is a redistribution in the model, not an increase in total claim value. Greater volatility raises the option value of equity and reduces the value of risky debt by exactly the same amount. This conclusion depends on the assumptions: if volatility changes operating cash flows, expected asset value, bankruptcy costs, or other claims, total value need not remain fixed.

Why the volatility formula holds: Differentiating the equity formula with respect to $\sigma$ produces terms from both $d_1$ and $d_2$. The terms multiplying their derivatives cancel by $V\phi(d_1)=Fe^{-\delta T}\phi(d_2)$. The remaining term is $V\phi(d_1)\sqrt{T}$, which is positive because $V$, $\phi(d_1)$, and $\sqrt{T}$ are positive.

Worked Example: Comparing Two Volatility Levels

A hypothetical firm has $V=F=\$100$, $T=1$ year, and $\delta=0$. Compare its equity and debt values when volatility is $0.20$ and $0.40$. Use the structural formulas, and round dollar values to the nearest cent.

At $\sigma=0.20$. Since $V=F$, $\ln(V/F)=0$. Thus $d_1=0.20/2=0.10$ and $d_2=0.10-0.20=-0.10$. Using $N(0.10)=0.5398278$ and $N(-0.10)=0.4601722$,

\[ E=100(0.5398278)-100(0.4601722)=\$7.9656\approx\$7.97. \]

Debt is $D=V-E=\$100-\$7.9656=\$92.0344\approx\$92.03$.

At $\sigma=0.40$. Now $d_1=0.40/2=0.20$ and $d_2=-0.20$. Using $N(0.20)=0.5792597$ and $N(-0.20)=0.4207403$,

\[ E=100(0.5792597)-100(0.4207403)=\$15.8519\approx\$15.85. \]

Debt is $D=\$100-\$15.8519=\$84.1481\approx\$84.15$. The equity increase is approximately $\$15.8519-\$7.9656=\$7.8863$; the debt decrease is approximately $\$92.0344-\$84.1481=\$7.8863$. In each case, equity and debt sum to $\$100$. The common mistake is to say that increased volatility raises the value of the firm’s total claims in this fixed-asset model; it raises equity value while lowering debt value by the same amount.

Debt Face Value: A Change in the Promised Claim

A change in $F$ changes the promised debt repayment, not the amount of assets currently in the firm. Holding $V$, $\sigma$, $T$, and $\delta$ fixed, the sensitivities are

\frac{\partial E}{\partial F}=-e^{-\delta T}N(d_2), \qquad \frac{\partial D}{\partial F}=e^{-\delta T}N(d_2).

The discounted probability weight $e^{-\delta T}N(d_2)$ is positive, so increasing face value lowers equity value and raises debt value locally. These changes offset: the sum of the sensitivities is zero. This does not mean a dollar of additional face value creates a dollar of debt market value; the debt is risky, and its market value responds by the amount specified by the sensitivity, not by one-for-one dollar-for-dollar face value.

The formulas also describe the effect of a marginal change. For a substantial change in $F$, $d_2$ changes too, so the original sensitivity cannot simply be multiplied by the full change and treated as an exact answer. Recalculate the model at the new face value, or integrate the changing sensitivity over the interval.

Worked Example: The Local Effect of Higher Debt Face Value

A hypothetical firm has $V=F=\$100$, $T=1$ year, $\delta=0$, and $\sigma=0.20$. Find the local changes in equity and debt values per additional dollar of face value. Then estimate the effect of a small $\$1$ increase. Report the sensitivities to four decimal places and dollar estimates to the nearest cent.

Method. First find $d_2$. With $V=F$, $\ln(V/F)=0$, so $d_1=0.10$ and $d_2=-0.10$. Since $e^{-\delta T}=1$, the sensitivities are

\[ \frac{\partial E}{\partial F}=-N(-0.10)=-0.4602, \qquad \frac{\partial D}{\partial F}=N(-0.10)=0.4602. \]

Thus, locally, a $\$1$ increase in face value reduces equity value by about $\$0.46$ and raises debt value by about $\$0.46$. The estimated changes sum to zero, consistent with fixed $V$ and $E+D=V$. This is a marginal estimate, not an exact repricing of any large face-value increase. The common mistake is to assume debt market value rises by the full additional dollar of promised repayment.

Reading the Effects Without Confusing Value and Redistribution

The main comparative-static results can be summarized by identifying what is held fixed in each derivative:

Change in input	Equity effect	Debt effect	Total claim value
Increase $V$, with other inputs fixed	Increases by $N(d_1)$ per unit	Increases by $N(-d_1)$ per unit	Increases by one unit per unit increase in $V$
Increase $\sigma$, with $V$ fixed	Increases by $V\phi(d_1)\sqrt{T}$ per unit	Decreases by the same amount	Unchanged
Increase $F$, with $V$ fixed	Decreases by $e^{-\delta T}N(d_2)$ per unit	Increases by the same amount	Unchanged

These are model-based conclusions, not universal statements about every financing decision. For example, issuing additional debt may bring cash into the firm, alter investment incentives, or introduce distress costs. Those effects change more than $F$ and fall outside this one-period comparative-static calculation. To use the results correctly, state which input changes, which inputs are held fixed, and whether the question asks for a marginal sensitivity or a full repricing.

The analysis is aligned with the SOA VEE Accounting and Finance syllabus. Its most useful exam discipline is to separate the two questions: What happens to total asset value? and How is that value divided between equity and debt? The identity $E+D=V$ answers the first, while the sensitivities explain the second.

Check Your Understanding

For each problem, state the relevant model effect and check whether the changes in equity and debt are consistent with the total-value identity.

A hypothetical firm has $V=F=\$80$, $T=1$, $\delta=0$, and $\sigma=0.30$. Calculate $d_1$ and the local sensitivities of equity and debt to asset value, to four decimal places. (Answer: $d_1=0.1500$; $\partial E/\partial V\approx0.5596$, $\partial D/\partial V\approx0.4404$.)
A hypothetical firm has $V=F=\$200$, $T=1$, $\delta=0$, and $\sigma=0.20$. Calculate equity and debt values, rounding to the nearest cent. (Answer: $E\approx\$15.93$, $D\approx\$184.07$.)
At $V=F=\$120$, $T=1$, $\delta=0$, and $\sigma=0.40$, find the local equity and debt sensitivities to face value, to four decimal places. (Answer: $\partial E/\partial F\approx-0.4207$, $\partial D/\partial F\approx0.4207$.)
A firm has $T=1$, $\delta=0$, $\sigma=0.20$, and $V/F=e^{-0.02}$. Find $d_1$, then state the local effects of a one-unit asset-value increase on equity and debt. (Answer: $d_1=0$; equity increases by $0.5000$ and debt by $0.5000$ per unit of asset value.)
With $V$, $F$, $T$, and $\delta$ fixed, asset volatility rises in the structural model. State the sign of each claim’s value change and the change in their total. (Answer: Equity value increases, debt value decreases by the same amount, and total claim value is unchanged.)

Change in input	Equity effect	Debt effect	Total claim value
Increase \(V\), with other inputs fixed	Increases by \(N(d_1)\) per unit	Increases by \(N(-d_1)\) per unit	Increases by one unit per unit increase in \(V\)
Increase \(\sigma\), with \(V\) fixed	Increases by \(V\phi(d_1)\sqrt{T}\) per unit	Decreases by the same amount	Unchanged
Increase \(F\), with \(V\) fixed	Decreases by \(e^{-\delta T}N(d_2)\) per unit	Increases by the same amount	Unchanged

What You'll Learn