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• Course of Economic Evaluation of Projects, Principles of financial math 3, October 2, 2024.

Program of the course

• The course introduction and project work will be on September 18, 2024, from 8:15 - 11:15.
• September 25, 2024 from 8:15 to 11:15: The Nature of the Real Estate Market; Information on Real Estate Market; How to develop Market Analysis, Theoretical and practical example.
• 8:15-11:15 on Oct 2, 2024 covers basics of financial math and exercises.
• Oct 9, 2024, from 8:15-11:15, will cover Market Value definitions and Market Value estimation: Sales Comparison Approach (SCA) + Income Approach.
• Market Value estimation: Cost approach + cost analysis will be held on Oct 16 2024.
• Private feasibility: Criteria and decision rules will be from 8:15-11:15 on Oct 23 2024.
• Project work revision I: Cost Analysis will be held on Oct 30 2024.
• Nov 6, 2024 will cover Private feasibility: Business Plan - workshop "how to develop a business plan".
• Nov 13, 2024 workshop in class for PW Theoretical & practical example: Public feasibility: Stakeholders Analysis and Multicriteria Analysis.
• Nov 20, 2024 will cover Project work revision II: Business plan.
• Nov 27, 2024: Public feasibility; Community Impact Evaluation (CIE) method + workshop.
• Project work revision III: final review will be held on Dec 4 2024.
• Project work revision IV: final review will be held on Dec 18th 2024.

Outline

• Introduction of Financial Math
• Important definitions
• Interest
• Discounting and Postponement
• Annuity
• Exercises

Introduction

• Financial mathematics studies financial operations.
• Financial calculations include adding, subtracting, or comparing monetary values at different times.
• It is essential to solve certain economic and estimative problems. It is used to compare financial performance related to different time periods.
• Values with different time frames may be present in numerous estimation methods, such as costs and revenues of a transformation project.
• Values can change over time due to growth or decay.
• Tools are needed to operate with values positioned over time and refers to financial operations.
• The tools can allow some important estimation operations.

Definitions

• Principal (P) equals the initial money that is invested.
• Interest (I) equals the price for using monetary capital. Represents remuneration for loaning capital for a specified time.
• Accumulated amount (A) equals the Principal plus interest.
• Interest rate (r) equals interest accrued per capital unit per unit of time and can be a percentage or unit term.

Interest, Principal, Accumulated Amount

• The interest (I) is defined as the price for the use of the monetary capital. Represents the compensation due to the subject who loans capital money for a determined period of time.

Interest

• There are two types of interest: Simple and Compound.

Simple Interest

• It is directly proportional to the principal and time. Simple Interest is used when considering a period less than 1 year.
• The formula for simple interest is I = Prt, where p = principal, r = interest rate, t = time (number of days, months).
• Example: P = 2,000€, r = 5%, t = 4 months. Calculate the interest: I = (2,000)(0.05)(4/12) = 33.3€.
• Every year the interest earned is not summed to the principal.
• I = Prt is the simple interest formula.
• A = P+I, accumulated amount equals principal plus interest.
• A = P + Prt. Alternative formula for accumulated amount.
• A = P(1+rt). The other form of accumulated amount formula.
• P = A/(1+rt). Present value formula using simple discount.

Simple Interest Example

• Determine the Accumulated amount of capital of €120,000, considering a simple interest for the period of 4 months with an interest rate of 3%.
• I = Prt. The simple interest formula.
• A = P+I. Accumulated amount formula equals Principal plus interest.
• I = 120,000 * 0.03 * 4/12 equals €1,200.
• The accumulated amount is A = 120,000 + 1,200 = €121,200.

Simple Interest: Postponement and Discounting

• Postponement refers to the value of an asset at a specified date in the future that is equivalent in value to a specified sum today: A = P(1+rt).
• The discounting method calculates the present value of a future amount of money, considering the interest rate: P = A/(1+rt).
• Determining how much to pay today for an investment expected to produce future income, we have to adjust by "discounting." This reflects the true value of money.

Simple Interest Example

• Suppose an individual considers an investment that promises €15,000 after 6 months at the interest rate of 5%; what is the capital's present value?
• P = A / (1 + rt) = 15,000 / (1 + 0.05 * 6/12) = €14,634.

Compound Interest

• Used for those investments over 1 year.
• Calculate the interest yearly. In the case of compound interest, the interest that accrues within a certain period (t) is added to the initial capital and accrues new interest.
• Example: Investing €100 for two years at 5% per year.
• After the first year, earning €5, and earning 5% on €105 (€100 + €5), totaling €5.25.

Compound Interest formulas

• P = €2000, r = 5%, t = 3 years.
• I = Prt. Simple interest formula.
• I₁ = (2,000)(0.05)(1) = €100.
• P₁ = 2,000 + 100 = €2,100.
• I₂ = (2,100)(0.05)(1) = €105.
• P₂ = 2,100 + 105 = €2,205.
• I₃ = (2,205)(0.05)(1) = €110.25.
• P₃ = 2,205 + 110.25 = €2,315.25.
• P₁ = Po+Por
• P₁ = Po (1+r)
• P2 = P1 + P1r
• P2 = P1(1+r)
• P2 = Po(1+r) (1+r) = Po(1+r)²
• A = Po(1+r)^t. Using compound interest.
• Po = A/(1+r)^t

Compound Interest: Postponement and Discounting

• Postponement is the value of an asset at a specified date in the future that is equivalent in value to a specified sum today: A = P(1+r)^t.
• Discounting: Calculating present value considering the interest rate, which is P = A/(1+r)^t.

Coefficient of Discounting / Postponement

• Simple Interest: Postponement = (1 + rt), Discounting = 1 / (1 + rt).
• Compound Interest: Postponement = (1 + r)^t, Discounting = 1 / (1 + r)^t.

Annuity

• A sequence of equal payments made at equal periods.
• The time between payments is called the Payment period.
• The term of the annuity is the time from the beginning of the first payment to the end of the last period.
• Annuities are used to accumulate funds (regular deposits) and pay out funds (pension after retirement).

Annuities

• A not limited annuity: an infinite number of annuity payments.
• A limited annuity: payments with a limited number.
• Ordinary annuity: payments made at the end of each period, where the frequency of payment is the same as the frequency of compounding interest.
• Annuity due: payments are made at the beginning of each period.

Recall Annuities

• Annuity: series of equal payments at equal intervals.
• Saving for retirement.
• Getting paid from a pension plan.
• Paying the rent for a house.
• Ordinary annuity: payment is made at the end of the period.
• Due annuity: payment made at the beginning of the period.

Ordinary limited annuity example

• This formula is of fundamental importance in real estate evaluation, employed for determining the market value of buildings on the basis of the income it produces over its economic life: A= a/r

Conclusion

• The following principles must be taken into account when solving any estimative question that requires the application of financial mathematics:
• Each value cannot be moved over time without considering the relative interest or discount.
• Addition/subtraction and comparisons between values referring to different periods cannot be performed; to do this, the values must first be made homogeneous, that is, they are reported simultaneously.