Investment timing with fixed and proportional costs of external financing

Abstract

We develop a dynamic model in which a firm exercises an option to expand production with a cash balance and costly external funds. By proving the properties of the option exercise regions, we reveal how the firm's financing and investment policy depends on its cash flow and balance. In the presence of only a proportional cost of external financing, the firm with more cash balance invests earlier; however, the presence of both proportional and fixed costs leads to a non-monotonic relation between the investment time and the cash balance. The firm with more cash balance invests later to save a fixed cost, particularly when the cash balance is close to the investment cost. Our results can potentially account for a variety of empirical results concerning the relation between investment and internal funds.

Keywords::

• real options
• optimal stopping
• investment timing
• costly external financing

JEL Classification::

• G13
• G31
• G32

Notes

^1. An incomplete list includes Bolton, Chen, and Wang (2011), Cleary (1999), Fazzari, Hubbard, and Petersen (1988), Gomes (2001), Hennessy and Whited (2005), and Kaplan and Zingales (1997).

^2. For example, the literature examined the effects of liquidity constraints (Boyle & Guthrie, 2003), shareholder–debtholder conflicts (Mauer & Sarkar, 2005; Sundaresan & Wang, 2007), asymmetric information (Grenadier & Malenko, 2011; Morellec & Schürhoff, 2011; Shibata & Nishihara, 2010), and debt capacities (Shibata & Nishihara, 2012).

^3. Milne and Robertson (1996) showed that the investment level increases with cash holdings in a dynamic dividend and investment model. In the real options literature, Hirth and Uhrig-Homburg (2010a) and Nishihara and Shibata (2010) also showed that the investment threshold decreases with internal funds.

^4. Several papers, including Bolton, Chen, & Wang (2011), Décamps, Mariotti, Rochet, and Villeneuve (2011), and Hugonnier, Malamud, and Morellec (2011), assume that the cash reserves process does not monotonically increase. In Section 4.2, we check numerically that our results do not qualitatively change even if Y(t) is not always increasing.

^5. In all figures in this paper, we set the axes in the same way as Boyle and Guthrie (2003) and Hirth and Uhrig-Homburg (2010b) for comparison.

^6. Nishihara (2012, 2013) showed the properties of multiple real options using similar techniques.

^7. In a static model, Cleary, Povel, and Raith (2007) and Guariglia (2008) showed both theoretically and empirically a U-shaped relation between investment level and internal funds.

^8. For simplicity, we do not consider an entry and exit problem but a liquidation problem.

^9. We set dt = 1/200 year and a lattice model with 401 × 3501 grids.

^10. In the computation, we used about 4000 iterations to ensure the convergence of value functions. Then, roughly speaking, it took 4000 × 401 × 3501 steps to compute the value function and exercise region for each parameter set. Owing to computational cost considerations, it is hard to discretize time and state space into a lattice with more grids.

Additional information

Funding

This work was supported by KAKENHI (Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science), [grant numbers 22710146, 22710142, 23310103, 21241040, 26350424 and 26285071].