Real Estate­ Value: Let or Sell and Rent or Buy

It is important that a distinction is made between value and price in a real estate valuation. Unfortunately, these two words are often confused. The decision value is subjective (dependent on the individual) and takes into account the individual decision field (e.g., taxes, loss carryforwards, alternative investments, withdrawal preferences, etc.) and is the most significant value that characterizes the limit of advantageousness.

The benefit of the future cash flow is discounted to the present using a calculation interest rate (endogenous marginal interest rate).

In addition to the decision value, there is an arbitration value (umpire value), which an independent appraiser can calculate from the decision values of the parties involved. Argumentation values are values used to influence negotiations, which can be based on more or less plausible procedures.

Two other values should be mentioned. The liquidation value (sale of all assets) is considered the lower value limit in business valuation. The reconstruction value is more significant in the context of real estate valuation and corresponds to the new construction of the property, including land acquisition. In business valuation, it corresponds to a new foundation.

The price, on the other hand, is the amount paid in a transaction. It regularly deviates from the decision value.

The well-known capitalized earnings method according to ImmoWertV does not actually calculate values but market prices based on market data. The transformation factor is the property interest rate. Roughly speaking, it works like this:

• Properties a, b, and c are sold.
• An expert committee calculates a property interest rate from the sales.
• This property interest rate is used (adjusted for the specific property) for the value calculation (better price calculation) of property d.

However, the property interest rate is a kind of "black box" that implies a lot of data and should not be used for investment decisions. Why? It implies the overall development of relevant factors (yield curve, inflation, population, economy, etc.).

In a property bubble, the property interest rate tends to fall further and further. A low interest rate causes property prices and values to rise massively. Since the property interest rate is derived from market data, the capitalized earnings method according to ImmoWertV justifies inappropriately high prices during property bubbles. The situation is different with the functional valuation theory and the closely related original capitalized earnings method (Hering, 2017; Hering, 2021; Matschke & Brösel, 2013; Toll, 2011; Walochnik, 2021).

Explanations of the capitalized earnings method in various forms, including according to ImmoWertV, can be found on the linked page on the capitalized earnings method. The capitalized earnings method according to ImmoWertV has its justification within the legal scope of application.

Below you can see a graph that shows how strongly the interest rate influences the real estate value. If a perpetual annuity is discounted at an interest rate of 10%, the value is 120,000 monetary units (MU). At an interest rate of 5%, it is already 240,000 MU, and at an interest rate of 1%, it is 1,200,000 MU.

Figure 1: Discounting a Perpetual Annuity

Source: own presentation.

This section explores the question of when it is worthwhile to rent and when to acquire real estate. The examples are abstracted from real-life circumstances so that they are easy to understand. Complex calculations (progressive taxes, loss carryforwards, etc.) are not considered in these explanations.

Mr. Immanuel Biele lives alone and has an annual gross salary of 50,000 MU. This is taxed at a rate of 30%. The necessary living expenses amount to 10,000 MU. In addition, he can either rent a property for 12,000 MU per year or acquire one for a currently unknown price. Specifically, he can either rent one side of a semi-detached house or acquire the other side.

In the base case, there is no inflation; in the variation, there is growth (inflation) of 2%. Salary, living costs, and rent develop in line with inflation (indexation).

Investments at the bank can be made at 5%. Investments are taxed at 25%, so the net interest rate is 0.05*(1-0.25) = 3.75%. (Formulas for net interest rates can be found in Schneeloch et al. (2020).)

There are further case distinctions with loans. In the first case group, unlimited loans can be taken out at 10%. In the other case group, an installment loan (annuity loan) over 10 years can be taken out at 10% and is tied to the acquisition price (mathematically through a restriction in a linear program). The loan can be represented as a cash flow as follows:

Without inflation: [1, -0.1627, -0.1627, -0.1627, -0.1627, -0.1627, -0.1627, -0.1627, -0.1627, -0.1627, -0.1627]

With inflation: [1, -0.1509, -0.154, -0.157, -0.1602, -0.1634, -0.1666, -0.17, -0.1734, -0.1768, -0.1804]

Mr. Biele is faced with the question of whether to rent or acquire a property. To do this, he seeks his decision value. This is the value at which the advantageousness reverses.

Renting Real Estate

In the case of renting without growth (inflation), Mr. Biele has a salary of 50,000 MU. He pays 15,000 MU in taxes. The necessary living costs amount to 10,000 MU, and the rent is 12,000 MU per year. Net, he has 50,000 - 15,000 - 10,000 - 12,000 = 13,000 MU at his disposal (withdrawal). He uses this particularly for leisure activities and vacations.

Table 5: Renting Real Estate without Growth

The case is similar to the above case with the addition that his salary, living expenses, and rent are adjusted for inflation.

Table 6: Renting Real Estate with Growth

Acquiring Real Estate

In the case of acquiring real estate, two groups of cases will be presented. This is followed by a few variations in tables.

In the first case group (extreme case), only the loan interest (no annuity loan) and no repayment are paid.

In the second case group, an annuity loan is taken out in the amount of the acquisition price. Further loans are avoided by Mr. Biele.

Note: The following explanations do not apply to a real estate acquisition for investment purposes, i.e., for letting. In that case, taxes on the rental income would need to be considered.

Acquiring Real Estate - Interest Payment Only

In the extreme case, Mr. Biele only pays the interest. An expert in the valuation of real estate and companies tells him that at a value (decision value) of a maximum of 120,000 MU, he can achieve the same withdrawal stream as when renting. To illustrate, he provides the following financial plan. It becomes clear that he only pays the interest amounting to 12,000 MU without ever repaying.

The calculation interest rate (endogenous marginal interest rate) is 0.1. Calculated as a perpetual annuity, this results in 120,000/0.1 = 120,000 MU.

Table 7: Acquiring Real Estate without Growth (Extreme Case Only Interest Payment)

In the variation, the appraiser takes into account inflation and thus price increases. The decision value, i.e., the value at which the advantageousness between purchase and rent turns, is 150,000 MU. In this case too, the appraiser provides a financial plan for the valuation of real estate and companies. Only the interest is paid, and the loan increases annually so that the interest can just be serviced. (After all, purchasing power and the saved rent increase.)

Table 8: Acquiring Real Estate with Growth (Extreme Case Only Interest Payment)

Acquiring Real Estate - Annuity Loan

Mr. Immanuel Biele takes out a loan from his bank in the amount of the acquisition price. The loan (annuity loan) has an interest rate of 10% and is repaid over 10 years. After that, he wants to remain debt-free.

An expert in the valuation of real estate and companies tells him that with a maximum value (decision value) of 73,734.81 MU, he can achieve the same withdrawal stream as when renting. He provides him with the following financial plan for clarification.

The calculation interest rate (endogenous marginal interest rate) is 0.1.

The value of 73,734.81 MU can also be calculated using the formula for the present value factor bwf(i,n) = ((1+i)^n - 1)/((1+i)^n * i). This results in 12,000 * 6.14 = 73,734.81 MU. If Mr. Biele were to allow unlimited indebtedness, as in the extreme case, he could discount the saved rental payments after paying off the loan. The endogenous marginal interest rate is the loan interest rate of 0.1. The value is (12,000/0.1) * (1+0.1)^(-10) = 46,265.19 MU. Together this results in 46,265.19 + 73,734.81 = 120,000 MU. This is the value from the extreme case.

(For mathematicians: The marginal price was calculated using a linear program in two ways. Once by taking out an annuity loan, which was tied to the acquisition price, and once through a restriction in year 10, which prevents further debt. In the first case, the dual value of the liquidity restriction in t=1 corresponds to the present value factor of 6.14. In the second case, the endogenous marginal interest rate of 0.1 can be calculated using the dual values of the liquidity restriction.)

Table 9: Acquiring Real Estate without Growth (Annuity Loan 10 Years)

In the following case (annuity loan), the appraiser takes into account inflation and thus the price increase. The decision value, i.e., the value at which the advantageousness between purchase and rent turns, is 79,503.73 MU. In this case too, the appraiser provides a financial plan for the valuation of real estate and companies. The annuity loan is repaid.

The calculation interest rate (endogenous marginal interest rate) is 0.1.

The value of 79,503.73 MU can also be calculated using the formula for the present value factor with growth bwfg(i,g,n) = (1 - (1+g)^n * (1+i)^(-n)) / (i - g). This results in 12,000 * 6.63 = 79,503.73 MU. If Mr. Biele were to allow unlimited indebtedness, as in the extreme case, he could discount the saved rental payments after paying off the loan. The endogenous marginal interest rate is the loan interest rate of 0.1. The value is (12,000 * 1.02^10)/(0.1 - 0.02) * (1+0.1)^(-10) = 70,496.27 MU. Together this results in 70,496.27 + 79,503.73 = 150,000 MU. This is the value from the extreme case.

(For mathematicians: The marginal price was calculated using a linear program in two ways. Once by taking out an annuity loan, which was tied to the acquisition price, and once through a restriction in year 10, which prevents further debt. In the first case, the dual value of the liquidity restriction in t=1 corresponds to the present value factor of 6.63. In the second case, the endogenous marginal interest rate of 0.1 can be calculated using the dual values of the liquidity restriction.)

Table 10: Acquiring Real Estate with Growth (Annuity Loan 10 Years)

Figure 3: Discounting Acquisition - Unlimited Credit and Annuity Loan

Source: own presentation.

Acquiring Real Estate Variations

Now the examples can be extended and modified as desired. The values calculated below can no longer be determined alternatively with the above formulas, and a linear or nonlinear program (base program and valuation program) must be set up according to functional valuation theory.

First follows a variation with 10,000 MU equity from Mr. Biele. Why do the values change? In the base case of renting (base program), the withdrawal stream increases by the interest on equity (possibly corrected by a growth discount). This increased withdrawal stream must also be achieved in the case of acquisition (valuation program). In the case of acquisition (valuation program), however, it is worthwhile not to invest the money but to avoid the expensive repayment of 10%. Additionally, the loan volume can be increased. (In the end, more interest is serviced, but there is no longer a low-interest investment.) The values below just allow the same withdrawal stream as in the base program. (The decision value is the maximum acceptable acquisition price at which the advantageousness reverses.)

Table 11: Variation Acquisition with Equity

In a further variation, Mr. Biele waives 50% of his non-compulsory consumption (withdrawal) for 10 years, both when renting and acquiring.

In the case of renting (base program), he can invest the money and receive interest. He saves up to year 10 and receives compound interest.

In the case of acquisition (valuation program), two case groups are presented. Once, he and his bank allow unlimited indebtedness. Alternatively, the property is paid off within 10 years.

Table 12: Variation Acquisition and Rent with and without Growth and 50% Withdrawal Waiver for 10 Years

As you can see, there is not "the" one universally valid value for everyone. The decision value depends on the valuation subject and his situation. Nevertheless, it can be simplified. The approximation is made using the original capitalized earnings method, which is a simplification of functional valuation theory and is less complicated.