ACCAP4：Merton's Models (1974)

ACCA P4考试：Merton's Models (1974)

1 Valuing Equity as an Option

Robert Merton (who worked with Black and Scholes on their original model) viewed equity investors as having a European call option over a firm's underlying assets, with the exercise price being the amount required to pay off debt.

If the value of assets rises above the level of outstanding debt, the call option moves into the money and equity investors can exercise their right to pay off the debt and take a gain (i.e. the net assets). If, however, the value of assets falls below the level of debt, shareholders can allow their option to lapse and walk away under the protection of limited liability.

Merton values equity using the Black-Scholes model with the ollowing inputs:

market value of assets;

volatility of asset value;

redemption value of debt;

time to redemption of debt; and

risk free rate.

Assuming that the firm's debt is long term in nature, then any hort-term liabilities (e.g. trade payables) should be deducted from the market value of assets (i.e. the value of assets is usually input as total assets less current liabilities).

For simplicity, Merton assumed that the firm's debt is a single discounted bond with zero coupon and redemption at face value, the time to redemption sets the period to expiry of the shareholders' call option.

However, in the real world, a firm's debt is unlikely to be a single discounted bond. Firms may issue coupon-paying debts with various maturities. There are two main methods of recalibrating a firm's actual debt before it is input into the Merton model:

Estimate the redemption value of a hypothetical zero coupon bond with the same fair value, yield and average time to maturity as the firm's actual debt.

Estimate the "Macaulay duration" of the firm's actual debt by multiplying the year of each payment of coupon or principal by the proportion of total present value received in that year and then summing. Duration is then used as the time to expiry of the call option, and the exercise price is set as the face value of the firm's actual debt plus the cumulative coupons to be paid.

2 Merton's Structural Debt Model

Having valued the equity, Merton then uses Modigliani and Miller's assertion that the market value of equity plus the market value of debt equals the total value of the firm's assets:

Value of debt = value of assets – value of equity

By comparing the theoretical market value of the debt to its face value, the yield can be easily implied and then compared to the risk-free rate to estimate the credit spread on the bond and hence the firm's default risk/credit risk. Hence Merton provides us with a "structural debt model" built on stronger foundations than ratio-based models such as Altman's Z-score.

As the yield is not known in advance (it is the output of the model), a variation on Macaulay duration must be used when calibrating the firm's debt to set the strike price of the call option. In this case, find the "nominal duration" of the firm's debt by multiplying the year of each payment of coupon or principal by the proportion of total payment and then summing (i.e. as per Macaulay duration but calculated using undiscounted cash flows).

Unfortunately, two of the inputs to Merton's model are not easily observed in practice—the value and volatility of assets.

However, for a quoted company, the output of the model is known (i.e. the equity value). Via some complex maths (which would not be required in the exam), the value and volatility of assets can be implied and hence the value and risk of debt.