Underwriting Limits and Optimal Leverage in Commercial Real Estate

Abstract

Risk-averse commercial mortgage lenders follow an underwriting policy with strict limits based on the property’s value and cash flow. A borrower then chooses the initial loan amount and amortization that fit into these requirements and maximizes the investment’s net present value. For an underwriting policy based on typical mortgage ratios, this optimization problem has a closed-form solution. Applying the formula to loan business data from life insurance companies, fluctuations in market parameters and cash flow-based policy limits can explain the major part of the observed variability in initial leverage. This analysis gives further support to observations that initial leverage is endogenous to the underwriting process, while cash flow-based and forward-looking measures are of primary importance in commercial loan risk management.

Appendices

Appendix 1: Proof that $L_{0}^{\times } < L_{T}^{max}$ Implies $L_{0}^{max} < L_{0}^{\times }$

Proof

If $L_{0}^{\times } < L_{T}^{max}$, then by inserting definition (21), we get especially

$$\begin{array}{@{}rcl@{}} \frac{\frac{T \cdot \bar{C}}{DSCR^{min}} + L_{T}^{max}}{1+T \cdot r} &<& L_{T}^{max} \\ \Leftrightarrow \quad L_{T}^{max} + \frac{T \cdot \bar{C}}{DSCR^{min}} &<& L_{T}^{max} + r \cdot T \cdot L_{T}^{max} \\ \Leftrightarrow \quad \frac{\bar{C}}{r \cdot DSCR^{min}} &<& L_{T}^{max} \end{array} $$

(30)

With ICR^min ≥ DSCR^min by basic assumption, this implies

$$\begin{array}{@{}rcl@{}} \frac{\bar{C}}{r \cdot ICR^{min}} \cdot (1+r \cdot T) &=& \frac{\bar{C}}{r \cdot ICR^{min}} + \frac{T \cdot \bar{C}}{ICR^{min}} \\ &\le& \frac{\bar{C}}{r \cdot DSCR^{min}} + \frac{T \cdot \bar{C}}{DSCR^{min}} \\ &<& L_{T}^{max} + \frac{T \cdot \bar{C}}{DSCR^{min}} \\ \Leftrightarrow \quad \frac{\bar{C}}{r \cdot DSCR^{min}} &<& \frac{L_{T}^{max} + \frac{T \cdot \bar{C}}{DSCR^{min}}}{1 + r \cdot T} \end{array} $$

(31)

From this we finally get

$$\begin{array}{@{}rcl@{}} L_{0}^{max} &=& \min \left\{ V_{0} \cdot LTV^{max} , \frac{\bar{C}}{r \cdot ICR^{min}} \right\} \\ &\le& \frac{\bar{C}}{r \cdot ICR^{min}} \\ &<& \frac{\frac{T \cdot \bar{C}}{DSCR^{min}} + L_{T}^{max}}{1 + r \cdot T} = L_{0}^{\times} \end{array} $$

(32)

□

Appendix 2: Proof that Inclination of NPV Contour Line is Higher than Inclination of Line Defined by $L_{T}^{max}$ Constraint

Proof

We first note, because f(r) = (1 + r)^−T is convex for r > − 1 and hence f(r) ≥ f(0) + f^′(0) ⋅ r, that

$$\begin{array}{@{}rcl@{}} \frac{1}{(1+r)^{T}} &\ge& 1 - T \cdot r \\ \Leftrightarrow \quad 1 &\ge& (1+r)^{T} - r \cdot T \cdot (1+r)^{T} \end{array} $$

(33)

This results in

$$ \frac{(1+r)^{T} - 1 - r \cdot T}{(1+r)^{T} \cdot r \cdot T} \le \frac{(1+r)^{T} - 1}{(1+r)^{T}} $$

(34)

Let us now take any two initial loan amounts $L_{0}^{\prime } \ge L_{0}^{\prime \prime }$ with their corresponding amortization amounts

$$ A^{\prime} := \frac{L_{0}^{\prime} - L_{T}^{max}}{T} \quad \text{and} \quad A^{\prime\prime} := \frac{L_{0}^{\prime\prime} - L_{T}^{max}}{T} $$

(35)

on the line defined by the $L_{T}^{max}$ constraint. Then with Eq. 34 we get

$$\begin{array}{@{}rcl@{}} A^{\prime} \cdot NPV_{A} - A^{\prime\prime} \cdot NPV_{A} &=& (L_{0}^{\prime} - L_{0}^{\prime\prime}) \cdot \frac{(1+r)^{T} - 1 - r \cdot T}{(1+r)^{T} \cdot r \cdot T} \\ &\le& (L_{0}^{\prime} - L_{0}^{\prime\prime}) \cdot \frac{(1+r)^{T} - 1}{(1+r)^{T}} = L_{0}^{\prime} \cdot NPV_{L_{0}} \\&&- L_{0}^{\prime\prime} \cdot NPV_{L_{0}}\\ \end{array} $$

(36)

Rearranged, this gives us

$$ L_{0}^{\prime} \cdot NPV_{L_{0}} - A^{\prime} \cdot NPV_{A} \ge L_{0}^{\prime\prime} \cdot NPV_{L_{0}} - A^{\prime\prime} \cdot NPV_{A} $$

(37)

This holds especially in cases c) and d) for the vertices defined by $L_{0}^{\prime } = L_{0}^{\times }$ and $L_{0}^{\prime \prime } = L_{T}^{max}$. □

Appendix 3: Modification if Mortgage Constant is Assumed

We assume that the lender annually pays a constant amount M := r ⋅ L₀ + A, where A is the amortization amount in the first year. In later years, the amortization stands higher at M − r ⋅ L_t− 1. This only gets relevant at the maturity of the loan where the loan amount due is given by

$$ L_{T} = L_{0} \cdot (1+r)^{T} - M \cdot \frac{(1+r)^{T} - 1}{r} = L_{0} - A \cdot \frac{(1+r)^{T} - 1}{r} $$

(38)

using the annuity accumulation formula. Hence, the double constraint in Eq. 19 becomes

$$ 0 \le L_{0} - A \cdot \frac{(1+r)^{T} - 1}{r} \le L_{T}^{max} $$

(39)

So, wherever T is used in connection with the policy constraints, it is substituted by ((1 + r)^T − 1)/r. The central example is Eq. 21 as the intersection of two constraints, so that $L_{0}^{\times }$ becomes

$$ \widetilde{L}_{0}^{\times} = \frac{(1+r)^{T} - 1}{r \cdot (1+r)^{T}} \cdot \frac{\bar{C}}{DSCR^{min}} + \frac{L_{T}^{max}}{(1+r)^{T}} $$

(40)

by rearrangement of terms. Similarly, the bivariate linear regression without constant considers

$$ LTV_{0} - \underbrace{\frac{(1+r)^{T} - 1}{r \cdot (1+r)^{T}} \cdot \frac{\bar{C}}{DSCR^{min}}}_{=: \widetilde{LTV}^{*}} = \frac{1}{(1+r)^{T}} \cdot \alpha + \frac{DSCR}{(1+r)^{T}} \cdot \beta $$

(41)

instead of Eq. 28. Table 4 reports the respective results, showing similar goodness-of-fit indicators as Table 3 for the original model.

Table 4 Parameter fitting results in the restricted model without YoD and ICR constraint; correlations compare observed LTV₀ to $\widetilde {LTV}^{*}$ as defined in Eq. 41 resp. to $\widetilde {L}_{0}^{\times } / V$ using Eq. 40 together with Eq. 27, the later computed using α and β from the bivariate linear regression without constant term; standard errors are given below each coefficient estimate