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1、,Part 4 Chapter 11 Yulin Y Department of Finance, Xiamen University,Main line: 1 A partial-equilibrium one-period model 2 A general intertemporal equilibrium model of the asset market, includes three models(model 1 is based on a constant interest rate assumption, model 2 is a no-riskless-asset case,
2、 model 3 is the general model)., A partial-equilibrium one-period model,We follows the warrant pricing approach used in Chapter 7, that is, investors choose among three assets: the warrant, the stock of the firm and a riskless asset to form optimal portfolios which maximize their expected utility.,C
3、onsider an economy made up of only one firm with current value , and there exists a “representative man” acts so as to maximize the expected utility of wealth at the end of a period of length, that is, Define a random variable Z by and assume its probability distribution is known at present, more im
4、portantly,is independent of the particular structure of the firm, this is consistent with the MM(Modigliani-Miller) theorem. Define as the current value of the i th type of security issued by the firm. The different types of securities are distinguishable by their terminal value . For a debt issue(i
5、=1), ,Because each of the securities appears separately in the market, so: and Define , so we can rewrite as a maximization under constraint: ,The corresponding first-order conditions are: This can be rewritten in terms of util-prob distributions Q as: ,Where and is a new multiplier related to . dQ
6、is independent of the functions by the assumption that the value of the firm is independent of its capital structure, so is a set of integral equations linear in the , and we can rewrite as,* Suppose the firm issues just one type of security-equity, then Substituting in , we have,From , we can see t
7、hat the expected return on all securities in util-prob space must be equated. If U was linear, then dQ=dP and would imply the result for the risk-neutral case. Hence, the util-prob distribution is the distribution of returns adjusted for risk.,Some examples,Example 1: Firm issues two types of securi
8、ties, debt and equity with current value and respectively. From and , we have :,Suppose or for then as . In the limit, the debt becomes riskless, so will be replaced by r. Another useful form of is Since in equilibrium,So, . This is identical to the warrant pricing equation derived in Chapter 7. Thi
9、s equation can also be derived directly from the terminal value of equity in the same way as debt.,Example 2: Firms capital structure made up from three types of securities: debt, equity(N shares outstanding with current price per share of S, i.e. ), warrants (exercise price is ). Assume there are n
10、 warrants outstanding with current market value per warrant of W,i.e. . Because the warrant is a junior security to the debt, the current value of the debt will be the same as in the first example. The current value of the equity will be ,Where . Rewrite as ,In equilibrium, . So from we have If we d
11、efine normalized price of the firm as,And define the normalized price of a warrant as , then can be rewritten as which is of the same form as equation (7.24).,Example 3: Firms capital structure contains two securities:convertible bonds with a total terminal claim on the firm of either B dollars or a
12、lternatively the bonds can be exchanged for a total of n shares of equity; and N shares of equity with current price per share of S dollars.,So, , and Where is determined to be .,By inspection of this equation, we have the well-known result that the value of a convertible bond is equal to its value
13、as a straight bond plus a warrant with exercise price .,Example 4: A “dual” fund: it issues two types of securities to finance its assets: namely, capital shares(equity) which are entitled to all the accumulated capital gains(in excess of the fixed terminal payment); and income-shares(a type of bond
14、) which are entitled to all the ordinary,income in addition to a fixed terminal payment. Let be the instantaneous fixed proportion of total asset value earned as ordinary income, V be the current asset value of the fund and Z the total return on the fund.,Let be the current value of the income share
15、s with terminal claim on the fund of B dollars plus all interest and dividends earned, be the current value of the capital shares. So, from definition, we have,And Where . The current value of the capital shares can be less than the current net asset value of the capital shares, defined to,be V-B, because If , that is, then, ., A general intertemporal equilibrium model,Consider an economy with K consumers investors and n firms with current value .Each consumer acts so as to Define , where is the number of shares and is the price per share at time t.,Assume that e
