For those appraisers interested in the algebra behind the mortgage-equity valuation formula- the following is an overview of the calculations.

The mortgage-equity valuation formula can be expressed in two algebraic

equations, which set forth the mathematical relationships between known and unknown variables. The symbols used to represent these variables are listed below:

NI Net income available for debt service

V Value

M Loan-to-value ratio

i Mortgage interest rate

f Annual debt service constant

n Number of years in projection period

de Annual cash available to equity

dr Residual equity value

b Brokerage and legal cost percentage

P* Fraction of loan paid off in projection period

Fp Annual constant required to amortize the entire loan during the projection period

Rr Overall terminal capitalization rate applied to net income to calculate the total property reversion (sale price at end of the projection period)

1/Sn Current worth of $1 (discount factor) at the equity yield rate

*P = (f - i) ÷ (fp - i) where i = the interest rate of the mortgage

Using these symbols, a series of formulas can be derived to express the components making up this mortgage-equity valuation process.

Debt Service

To calculate a property’s debt service, the appraiser first determines the

amount of the mortgage, which is the total property value (V) multiplied by

the loan-to-value ratio (M). Then the amount of the mortgage is multiplied by

the annual debt service constant (f) using the following formula:

f × M × V = debt service

Net Income to Equity (Equity Dividend)

The net income to equity (de) is the property’s net income before debt service

(NI) minus the debt service. The following formula represents net income to

equity:

NI - (f × M × V) = de

Reversionary Value

The value of the hotel at the end of Year 10 is calculated by dividing the net income in Year 11 before debt service (NI11) by the terminal capitalization rate (Rr). The following formula calculates the property’s reversionary value in Year 10:

NI11/Rr = reversionary value

Broker, Legal, and Other Closing Costs

When a hotel is sold (at the end of Year 10), costs associated with the transaction normally include a broker’s commission and attorneys’ fees. For a hotel transaction, broker and legal costs typically range from 1% to 4% of the sale price. Because these expenses reduce the proceeds to the seller, they are usually deducted from the reversionary value in mortgage-equity analysis. Broker and legal costs (b) expressed as a percentage of the reversionary value (NI11/Rr) can be calculated with the following formula:

(b (NI11/Rr)) = broker and legal costs

Ending Mortgage Balance

The balance of the mortgage at the end of Year 10 must be deducted from the total reversionary value (debt and equity) to isolate the equity residual. A financial formula is used to calculate the fraction of the loan paid off, which is

expressed as a percentage of the original loan balance at a particular point in

time. The mortgage interest rate (i) is deducted from the annual debt service

constant of the loan over the entire amortization period (f), and the result is

divided by the annual constant required to amortize the entire loan over the

projection period (sub p) minus the mortgage interest rate. The formula is:

(f - i)/(fp - i) = P

If the fraction of the loan paid off expressed as a percentage of the initial

loan balance is P, then the percentage of the loan remaining can be expressed as 1 - P. Thus, the ending mortgage balance is the fraction of the loan remaining (1 - P) multiplied by the amount of the initial loan (M × V). The formula is:

(I - P) × M × V = ending mortgage balance

Equity Residual Value

The value of the equity when the property is sold at the end of the projection

period (d) is the reversionary value minus broker and legal costs and

the ending mortgage balance. The following formula represents the equity

residual value:

(NI11/Rr) - (b(NI11/Rr)) - ((1 - P) × M × V)) = dr

Annual Cash Flow to Equity

The annual cash flow to equity consists of the equity dividend for each of the

10 projection years plus the equity residual at the end of Year 10. The following formulas represent the annual cash flow to equity:

NI1- (F × M × V) = de1

NI2- (F × M × V) = de2 ....

NI10- (F × M × V) = de10

(NI11/Rr) - (b(NI11/Rr)) - ((1 - P) × M × V)) dr

Value of the Equity

If the initial amount of the mortgage is calculated by multiplying the loan-to-value ratio (M) by the value of the property (V), then the equity value will be

1 minus the loan-to-value ratio times the property value. The formula is:

(1 - M)V

Discounting the Cash Flow to Equity to Present Value

The cash flow to equity for each of the projection years is discounted to present value at the equity yield rate (1/Sn). The sum of all these cash flows is the value of the equity (1 - M)V. The following formula calculates equity as the

sum of the discounted cash flows:

(de1 × 1/S1) + (de2 × 1/S2) + … +(de10 × 1/S) + (dr × 1/S10) = (1 - M)V

Combining Equations: Annual Cash Flow to Equity and Cash Flow to Equity

Discounted to Present Value

The final step in the process is to make one overall equation that shows that

the annual cash flow to equity plus the yearly cash flows discounted to present value equal the value of the equity.

((*NI1 *- (*f *× *M *× *V*)) 1/*S*1) + … + ((*NI2 *- (*f *× *M *× *V*)) 1/*S*2) + …

+ ((*NI10 *- (*f *× *M *× *V*)) 1/*S*10) + (*NI11*/*Rr*) - (*b*(*NI11*/*Rr*)) - ((1 - *P*) × *M *× *V*) 1/*S*10) =

(1 - *M*)*V*

Since the only unknown is the property value (V), this equation is easy to solve.

For more information on the Mortgage-Equity Valuation Formula and the software that will quickly perform these calculations and show the proofs that the mortgage and equity components are obtaining their prescribed rates of return- click on the following button.