Real Estate Finance

Financial Arithmetics

Basic Concepts

Consider an initial sum of money C (e.g., euros, US dollars). Assume this sum is lent to someone who promises to pay back a sum R in each of n periods. At the end of n periods, the lender will have a sum M = n * R.

The ratio I = M / C - 1 expresses the interest paid by the borrower to the lender, representing the sum of payments exceeding the initial sum C.

Dividing I by the number of periods n gives the average interest per period: i_{avg} = I / n.

This calculation does not account for the temporal distribution of payments R, the risks associated with the debt, or variations in repayment structure and interest rates over time.

Simple Interest

If periodic payments differ (R1, …, Rn), they can be seen as the outcome of applying a sequence of interest rates to C (simple interest):

Period: 1, 2, …, n

C

R1 = C * i1, R2 = C * i2, …, Rn = C * in

Here, interest rates are "hybrid," including both principal repayments and the lender's return. Alternatively, the borrower repays C in the last period n along with Rn, in which case i1…i_n represent only "proper" interest.

The final capital M is the sum of all payments: M = R1 + … + Rn.

With simple interest, each payment is a multiplication of C by an interest rate, causing M to grow linearly with i. Thus, M = C * (i1 + … + in) = C * A.

Given a sequence of interest rates, A = (i1 + … + in) is independent of C.

If the interest rate is constant: M = C * i * n.

Compound Interest

Alternatively, consider a loan where any return is reinvested in each period. At the end of n periods, the initial capital C plus all interest is paid back in a single payment.

Period: 1, 2, …, n

C

C * (1+i_1)

C * (1+i1)(1+i2)

…

C * (1+i1)(1+i2)…(1+in) The final capital M with compound interest is: M = C (1+i1 )*…*(1+i_n ) = C * A.

Why use compound interest? From the lender's perspective, the entire sum C plus any interest can be reinvested for more returns. For example, a company reinvesting undistributed profit each year: each reinvested euro earns a return, and the next year the sum of that euro plus the return earns more returns, until profit is distributed as dividend or the company is liquidated.

Dividing (A-1) by n gives the average interest per period: i_{avg} = (A-1) / n.

Any sequence of interest rates can be transformed into an equivalent average rate by solving for i{avg} in the equation: (1+i{avg})^n = (1+i1 )*…*(1+in ).

For a constant compound interest over n periods, use the notation a{i,n}, where a{i,n} = (1+i)^n.

With a constant interest rate i during n periods, the final capital M from an initial investment C is: M = C*(1+i)^n = C*a_{i,n}.

If a constant interest rate is applied to smaller periods (e.g., a compound interest i_{1/4} applies every quarter):

M = C*(1+i_{1/4} )^4

Yearly interest rate i1 to get the same M after two years as after eight quarters applying rate i{1/4}:

(1+i{1/4} )^{4*2} = (1+i1)^2

i1 = (1+i{1/4} )^4 -1

More generally, a yearly rate i is equivalent to a rate defined on k sub-periods according to: i = (1+i_{1/k} )^k -1

These methods determine the final capital M from an initial capital C, given an interest rate sequence. The same math can be used to find the initial capital C for a final capital M, given a sequence of interest rates.

This answers questions like:

• What amount should a lender demand to sell a loan contract to another party?

This also evaluates and compares financial contracts, assets, and positions.

IRR and NPV

Introduction

The Internal Rate of Return (IRR) and Net Present Value (NPV) are common evaluation methods based on compound interest mathematics. While mathematically simple, they require care in choosing input numbers. The NPV is particularly sensitive to the choice of the discount rate.

Calculation of V

Consider an investment requiring an initial capital C and promising to repay R_k in each period k of n periods. We can compute the value at time zero of each payment:

• C (paid at time zero) enters as a negative addendum because it is a cost.

• For each payment Rk at time k, its equivalent value at time zero is given by Rk / (1+i1 )*…*(1+ik ). If the interest rate is constant, this simplifies to Rk / a{i,k}.

Thus, the value V at time zero of this sequence of payments is:

V = -C + R1 / (1+i1 ) + … + Rn / (1+i1 )*…*(1+i_n )

Internal Rate of Return (IRR)

-C + R1 / (1+i ) + … + Rn / (1+i )^n = 0

The IRR method assumes a constant interest rate, which is derived from the previous equation by solving for i. The IRR is the constant interest rate that makes the sum of actualized values at time zero equal to the invested sum C.

• If an investor is evaluating two investment opportunities requiring the same initial capital C, the one with the largest IRR is better as it is expected to have a higher return.

• If a borrower is shopping for a loan, the one with the smallest IRR is the cheapest loan (in this case, the signs in the equation are inverted).

Advantages of IRR:

• Incorporates the time value of payments.

• Requires only estimating payments in each period, making it more objective with lighter informational requirements and less sensitivity to errors (though still vulnerable if payments are estimated).

Limitations of IRR:

• Does not account for risk factors.

• Assumes a single rate of return for the entire life of the investment.

• May require solving higher-order polynomials.

Net Present Value (NPV)

NPV = -C + R1 / (1+i1 ) + … + Rn / (1+i1 )*…*(1+i_n )

The NPV method is a more flexible alternative to the IRR. NPV is the sum of all (positive and negative) payments, each actualized at time zero.

Because each payment is individually actualized, each may be discounted using a different rate, adding more flexibility compared to the IRR. The NPV results in a monetary value (in euro, USD…), while the IRR obtains a rate of return.

The rates employed to discount the flows of payments capture the opportunity cost of capital and the risk premium associated with it:

• Riskier investments entail larger discount rates.

• The cost of capital (e.g., an imputed rate of return to equity, or the cost of a loan) can also be accounted for.

Weakness of NPV:

The main weakness of the NPV is also its strength: as it relies on estimates of the discount rates, NPVs may vary wildly with such estimates. The method is therefore as precise as the estimated discount rates are.

• Most often, an appropriate discount rate can be obtained from sectoral data: the average return obtained in a given industry may provide a reasonable estimate of the opportunity cost (including the market risk premium) associated with an investment.

• For some applications, the right rate is the cost of financing. For example, if a company evaluates investing by borrowing external funds via bonds, the interest paid on the bond provides the correct rate.

A NPV can be positive or negative. Any positive NPV implies that an investment is profitable and should be undertaken. A positive NPV means that the actualized flow of payments is more valuable than the capital evaluated at time zero.

The NPV can also rank different investment opportunities by comparing the return obtained as: NPV / C

Because the NPV is sensitive to the choice of the discount rate(s) and projected payments, it is common practice to perform sensitivity analyses:

• By varying the rates or payments (e.g., all by a given % upward or downward).

• By defining best-case and worst-case scenarios.

• By also computing and reporting the IRR.

Loan

A loan is a contract where a sum of money is given to another party in exchange for repayment of the lent amount plus interest. Mortgages are loans where part of the value of the financed RE assets is given as collateral, reducing lender’s risks.

Common Loan Classifications:

• Secured Loan: Debt is guaranteed by collateral (e.g., mortgages).

• Unsecured Loan: Often for smaller amounts (e.g., bank overdraft facilities).

• Revolving Loan: Allows borrowing and repaying up to a limit (e.g., credit lines, credit cards).

• Installment Loan: Repaid over time with scheduled payments.

• Corporate Bonds: Financial assets emitted by corporations, traded over-the-counter.

Non-Revolving Secured Installment Loans:

Commonly used to finance the purchase of values with large unit value, such as:

• Mortgages for real estate.

• Loans for cars and vehicles.

• Loans for machinery and equipment.

The asset value provides the collateral: if the borrower fails to repay the loan, the lender can force a sale of the asset to repay the outstanding credit.

Mortgage Amortization Methods

Amortization is the process by which a debt is gradually extinguished through periodic payments (“installments”). Each installment repays part of the lent capital (“principal” payments) and interest.

French Amortization

Debtor pays a fixed installment.

• Schedule of increasing principal and decreasing interest.

• Simplicity is its main advantage, making it easier for the borrower to plan financial needs.

Italian (German) Amortization

Principal component stays constant while interest varies.

• Installment size is larger in earlier periods and decreases with time.

• Entails smaller interest overall because capital is reimbursed more rapidly.

American Amortization (Interest-Only Loans)

Each installment only pays interests. The entire lent amount is repaid in a single big installment at the end of the loan.

• The main advantage is the possibility to open an account where advances of the final payment may generate interest.

Example

5-year loan, lent capital of 1,000, annual interest rate of 5%:

The "Italian" method results in the cheapest loan.

In real-world applications, the base interest rate will be larger for the “Italian” method.

French Amortization Installment Formula

Installment = Capital * [ i(1 + i)^n ] / [ (1 + i)^n – 1]

For i=5%, n=5, and a lent capital of 1,000:

1,000 * [ 5\% (1 + 5\%)^5 ] / [ (1 + 5\%)^5 – 1] = 230.9747…

Computing Shares Using a Spreadsheet

1. Interest share in the first period: capital times the interest rate (e.g., 1000 * 5% = 50.00).

2. Subtract this interest from the installment to obtain the principal share (e.g., 230.97 - 50.00 = 180.97).

3. Repeat for period 2: multiply the residual capital (1000 - 180.97) by the interest rate (= 40.95), subtract it from the installment (230.97 - 40.95 = 190.02).

With each period, principal shares increase and interest shares decrease.

Alternatively, the shares can be obtained using the following formulas:

Interest share at time s: I_h = Installment * [1-(1+i)^{-(n-s+1)}]

Principal share at time s: C_h = Installment * (1+i)^{-(n-s+1)}

The residual debt at time s (sum of principal shares not yet paid): Residual debt at s = Installment * [1-(1+i)^{-(n-s)}] / i

Secured Loans

A secured loan reduces repayment risk by protecting the lender’s right with collateral.

• For loans to purchase a specific asset, the lender can enforce the sale of the asset or gain its property if a loan contract violation occurs.

• Lenders usually only finance a percentage of the purchase price to have a margin to face possible variations in the asset resell price (market risk, obsolescence).

For real estate, this share is commonly around 80%, but may vary based on the borrower’s credit reputation and market volatility.

A secured loan can also be accompanied by an insurance contract to protect the lender from specific risks and to insure against risks that are not under one’s control (unemployment, illness, disability).

These contracts follow standard insurance logic, therefore their price often varies with the age, profession, region of residence and health conditions of the insured person.

Insurance does not cover all reasons for a loan default, only selected events that are contractually defined. Therefore, usually a lender asks for both collateral and insurance.

Fixed vs Variable Interest Rate

Loans can be contracted as fixed-rate or variable-rate obligations.

• Fixed-rate loan: The borrower pays the same interest rate in each period. Alternatively, the rate changes according to a predefined schedule.

• Variable-rate loan: The interest rate is linked to some external indicator (e.g., inter-bank lending rate plus a premium).

Considerations for choosing fixed vs variable rates:

• Fixed-rate loans: More predictable, easier for the borrower to plan, reduces risks associated with incorrect expectations or market swings.

• Variable-rate loans: Protect the lender’s capital better when inflationary expectations are rising. May be cheaper for borrowers (at least in the first years).

Tax Treatment

It is important to distinguish interest and principal payments for tax and accounting purposes.

• For the borrower, interests are costs and are tax-deductible.

• For the lender, interests are taxable income.

• When a loan is for the purchase of an asset, the borrower can depreciate the asset and tax-deduct the depreciation installments.

Leasing

A leasing is a contract that provides a good under a rental agreement with an option to buy. The lessee has the right to use the good against the obligation to pay a periodic installment. At the end of the contract, the lessee can give the good back to the lessor or pay an additional sum and gain full ownership.

The lender (lessor) keeps property rights until the end of the contract.

Loan vs. Leasing

Loan

Lender -> Lent Capital -> Borrower

Borrower -> Loan Installments -> Lender

Seller -> Purchase Price -> Seller

Leasing

Lender -> Purchase Price -> Seller

Lender -> Property of the Good -> Borrower

Borrower -> Lease Rents -> Lender

Borrower -> Use of the Good -> Borrower

Leasing Contract Types

Operational Leasing (Operating Lease)

Rents a car, machine, or equipment along with connected services. The lessee may or may not buy the good at the end of the contract. Example: an “all-inclusive” car rental where a monthly payment covers rents, insurance, maintenance, and consumables.

Financial Leasing

An alternative to a secured loan used to finance the purchase of a good. The natural termination is for the lessee to buy the good at the end of the contract.

Even though it is formally a rental contract, the underlying real economic content of a financial leasing is to finance a purchase. The advantage for the lender is to retain the property of the good until the end of the contract (which provides a stronger guarantee than collateral). Consequently, the final purchase option installment is relatively small.

Tax-wise, a financial leasing is treated similarly to a loan: interests are tax-deductible costs for the lessee, while principal payments are not. The good can be depreciated by the lessee. For the lessor, interests are taxable income.

Operational leasings are used to leave the risk of obsolescence on the lender’s shoulders. A lessee can rent machinery for a short time and then change it for a newer model/version.

Tax-wise, the entire amount is considered a rent and is fully tax-deductible for the lessee. For the lessor, the entire installments are taxable income, and the good can be amortized.

Volume of Leasing Worldwide in 2017 (in billion U.S. dollars)

• North America: 445.9

• Europe: 428.3

• Asia: 354.4

• Australia/New Zealand: 31.5

• South America: 17

• Africa: 5.7

Leasing is used for different purposes.

Sale and Lease Back

A variant of the leasing contract where a subject sells an asset to another, and the latter simultaneously enters a leasing contract with the seller. These contracts allow a company to continue using an asset while transferring the ownership to a lender. The price the lender pays to buy the asset is the lent capital. The original owner of the asset then pays back installments as in a normal leasing contract.

Most often, these are employed by companies using assets of large individual value (often, internally built), for example, in the construction, shipbuilding, aviation, and transportation industries.

Sale and Lease Back Diagram

Borrower -> Purchase Price / Capital -> Lender

Lender -> Property of the Good -> Borrower

Borrower -> Lease Rents -> Lender

Sale and lease back: Keeps the use of the good.

The advantage of a sale and lease back contract for the borrower, in comparison to a more standard loan, is that a loan shows up on the company's balance sheet as debt. A sale and lease back contract, instead, does not show up as additional debt (it is registered as an asset sale), while current assets will report an increase (because of cash injected by the lender).

The main advantage for the lender is the acquisition of property over the asset, which provides a stronger guarantee than standard collateral (as per usual leasing contracts).

Mortgages

Over time, many special forms of mortgages have been developed to meet market demands and cater to specific needs. Some notable solutions are listed below.

Low-Start Loans

Allow the borrower to pay a lower interest rate than the reference rate. The difference between the lower rate and the reference rate is then capitalized.

At the end of the low-rate period, the borrower starts to repay the capitalized amount at the higher rate, thus basically having an additional loan on top of the initial mortgage.

For example, a mortgage for 100,000 Eur lasts 20 years with an interest rate of 6%. In the first 5 years, the borrower pays a reduced interest rate of 3%. The capitalized difference is (6% - 3%)100,0005 = 15,000 Eur. During the first 5 years, the borrower pays a reduced installment (the interest share is halved). Starting from the sixth year, the borrower pays the full installment, plus an additional installment (at a rate of 6%) to repay the 15,000.

Stabilized Loans

A variant of the low-start loan. Given the initial contracted interest rate, if the reference market rate rises during the course of the loan, the borrower can continue paying the contracted rate and accumulate the difference, as in the low-start loan. In this way, the borrower can smooth down unexpected rises in rates.

The capitalized difference can then either be repaid at maturity, or by extending the duration of the loan, or optionally it can be repaid before maturity.

Select-Payment Loans

In these loans, the borrower can decide to pay smaller installments in some periods, to be compensated by paying larger installments in other periods. In the end, any net difference between deficit and surplus payments has to cancel out (or, give rise to an additional payment at maturity).

Again, these types of loans serve the purpose to provide borrowers with more flexibility and reduce the risk of being illiquid.

Cap-and-Collar Loan

Offers a solution that is halfway between fixed and variable rate. The interest rate may change during the loan, based on a reference market rate. However, the borrower only pays up to a “cap” rate or benefits only up to a floor (the “collar”) rate. In other words, this type of loan is variable but only within a predefined bandwidth.

Index-Linked Loan (Indexed Loan)

Initially offered with a fixed rate (usually a low one). The rate is then increased based on either the CPI index or an index of housing prices. The idea is to keep the real interest rate constant throughout the loan.

An index-linked loan is convenient for the borrower if the linked index remains relatively low, such that the indexed rate remains lower than what could have been obtained as a fixed rate. In the case of using a housing price index, when the value of the purchased RE falls, interest payments fall as well, thus reducing both the credit risk for the borrower and the lender.

Remortgaging

Does not refer to a type of loan. Rather, it is the practice of exchanging an existing (still not fully repaid) mortgage with a new one.

Remortgaging makes sense when the underlying market and macroeconomic conditions have changed (which may easily happen, considering that mortgages usually last 20-30 years!).

• For example, a loan might have been taken with an interest linked to inflation, in a time when expected inflation rates were relatively low. Then, inflationary expectations rise. A borrower could then decide to take a new loan with a fixed rate and use it to entirely repay the outstanding debt with the first lender.

Cash-Out Refinancing

A variant where the new loan is for a larger amount than what is owed to the old lender. The excess sum can be used by the borrower to finance consumption or other (non-RE) forms of savings.

Solutions to Obtain Liquidity During the Course of a Mortgage

Home Equity Loan

An additional loan, on top of any existing mortgage over a property, is obtained, guaranteed by the same property. This loan will be subordinated, meaning that in case of default, the first mortgage has priority over any fund obtained by selling the financed RE asset. Consequently, being riskier for the lender, the interest rate is larger.

Home Equity Line of Credit

A variant where the lender offers a credit line, where the interest rate is adjusted based on a number of factors (e.g., how much of the mortgage is already repaid). The interest paid is normally lower than in home equity loans.

Reverse Mortgage

A relatively old person who owns a house receives a lump sum or a stream of payments from a lender. The borrower never repays any installment: at the time of their death, the RE asset is sold by the lender, and the proceeds from the sale are used to repay the loan, up to the obtained price (any exceeding claim from the lender is otherwise lost). Selling prices in excess of the replied loan are left to the heirs.

Financial Arithmetics

Basic Concepts

Financial arithmetic involves understanding initial capital $C$, periodic repayments $R$, total return $M$, interest $I = M/C - 1$, and average interest per period $i_{avg} = I/n$.

Simple Interest

In simple interest, each payment is derived from the initial capital. The final capital $M$ grows linearly: M = C * (i1 + \dots + in) = C * A. If the interest rate $i$ is constant, M = C * i * n.

Compound Interest

Compound interest involves reinvesting returns each period. The final capital $M$ is: M = C(1+i1) * \dots * (1+in) = C * A. With a constant rate $i$ over $n$ periods, M = C * (1+i)^n = C * a{i,n}. A yearly rate $i$ is equivalent to a rate $i{1/k}$ on $k$ sub-periods if i = (1+i_{1/k})^k - 1.

IRR and NPV

These methods evaluate investments based on compound interest. The value V at time zero for a sequence of payments (-C initial cost, Rk payments at time k) is: V = -C + R1 / (1+i1) + \dots + Rn / (1+i1) * \dots * (1+in).

Internal Rate of Return (IRR)

IRR is the constant interest rate that makes the sum of actualized values at time zero equal to the invested sum C (i.e., makes the above equation equal to 0). It's useful for comparing investment opportunities; a larger IRR is generally better for an investor, while a smaller IRR is cheaper for a borrower. Advantages include incorporating the time value of payments and reliance on payment estimates. Limitations include not accounting for risk and assuming a single rate.

Net Present Value (NPV)

NPV is the sum of all actualized payments at time zero (as shown in the V equation). It's more flexible than IRR as each payment can be discounted using a different rate, capturing opportunity cost and risk premium. A positive NPV indicates a profitable investment. NPV is sensitive to discount rate estimates. Sensitivity analyses are common, often alongside IRR.

Loan

A loan is a contract where money is exchanged for repayment plus interest. Mortgages are secured loans with real estate as collateral.

Common Loan Classifications

• Secured Loan: Backed by collateral.

• Unsecured Loan: No collateral.

• Revolving Loan: Allows borrowing and repaying up to a limit (e.g., credit cards).

• Installment Loan: Repaid over time with scheduled payments.

• Corporate Bonds: Financial assets issued by corporations.

Non-Revolving Secured Installment Loans: Common for large unit value purchases like real estate or vehicles, with the asset serving as collateral.

Mortgage Amortization Methods

• French Amortization: Fixed installment payments, leading to increasing principal and decreasing interest shares over time. Formula: Installment = Capital * [ i(1 + i)^n ] / [ (1 + i)^n – 1].

• Italian (German) Amortization: Constant principal component, resulting in larger initial installments that decrease over time.

• American Amortization (Interest-Only): Only interest is paid in installments, while the entire principal is repaid in a single lump sum at the end.

Secured Loans: Reduce lender risk via collateral, typically financing a percentage of asset value (e.g., 80% for real estate).

Fixed vs Variable Interest Rate:

• Fixed-rate: Predictable, rate may change by predefined schedule.

• Variable-rate: Linked to an external indicator, shifts risk to borrower in rising inflationary periods but may be initially cheaper.

Tax Treatment: Interest payments are tax-deductible for the borrower and taxable income for the lender. Assets purchased with loans can be depreciated by the borrower.

Leasing

A leasing contract provides the right to use a good under a rental agreement with an option to buy. The lessor (lender) retains property rights until the contract ends.

Loan vs. Leasing: Differ in ownership transfer and payment structure (loan installments vs. lease rents).

Leasing Contract Types

• Operational Leasing (Operating Lease): A rental agreement (e.g., car) often including services. Lessee may or may not buy. Risk of obsolescence is on the lessor. Rents are fully tax-deductible for the lessee.

• Financial Leasing: An alternative to a secured loan, implying eventual purchase. Lessor retains property as strong guarantee. Tax treatment is similar to a loan: interest is tax-deductible, and the good can be depreciated by the lessee.

Sale and Lease Back: A company sells an asset to a lender and simultaneously leases it back. This allows the company to retain use of the asset while injecting cash, without appearing as additional debt on the balance sheet. The lender gains stronger guarantee through asset ownership.

Mortgages

Various mortgage forms cater to specific needs:

• Low-Start Loans: Lower initial interest rate, with the difference capitalized and repaid later.

• Stabilized Loans: Allows borrower to maintain a contracted rate if market rates rise, accumulating the difference to be repaid later or by extending loan duration.

• Select-Payment Loans: Flexible installments—borrower can pay smaller amounts in some periods compensated by larger amounts in others.

• Cap-and-Collar Loan: Variable interest rate confined within predefined upper ("cap") and lower ("collar") limits.

• Index-Linked Loan: Interest rate adjusted based on CPI or housing prices to maintain a constant real interest rate.

Remortgaging: Replacing an existing mortgage with a new one, typically due to changed market conditions (e.g., seeking a better interest rate).

Cash-Out Refinancing: A type of remortgaging where the new loan is for a larger amount than the outstanding debt, allowing the borrower to access the excess cash.

Solutions to Obtain Liquidity During a Mortgage

• Home Equity Loan: A second, subordinated loan secured by the property, typically at a higher interest rate due to increased lender risk.

• Home Equity Line of Credit: A flexible credit line secured by property, with interest rates adjusting based on factors like repaid mortgage amount.

• Reverse Mortgage: Allows a qualifying homeowner to receive a lump sum or periodic payments, with the loan repaid from the sale of the home after death, without requiring regular installments from the borrower.