Topic 1. Accumulation and Discounting

Time Factor in Quantitative Analysis of Financial Operations Key elements for financial modeling are time and money. In essence, financial models in some way reflect quantitative relations between sums of money that refer to various time points One and the same sum of money has various values at various time points. Contrariwise, with reference to certain conditions different sums of money at different periods of time may be equivalent in the financial and economic context. Necessity in considering the time factor is expressed in a form of the principle of money disparities that refer to various time points.

Interests and Interest Rates Let us use the following symbols: – point of lending money (present time point); or – life of the loan; – sum provided as a loan in the time point – sum of the debt being discharged at the point ; – interest rate (of accumulation); – discount rate; – interests or interest money.

- interests (interest money) - interest rate - discount rate -normalized interest(1) (discount) rate Interests and Interest Rates

(2) (3) Two principles of interest calculation: 1) Interests and Interest Rates

2) Interests and Interest Rates

Accumulation with Simple Interest (4) accumulation factor with a simple rate

Variable rates (5) Reinvestment Accumulation with Simple Interest

Compound Interest Capitalization process occurs in the following model: (6) accumulation factor with compound rates

The relation of accumulation factors with simple and compound rates: Variable rates:

Nominal and Effective Interest Rates The nominal rate is an annual interest rate at interest calculations once a year. (7) The effective rate is an annual rate of interests calculated once a year that gives the same financial result as, which is a single annual interest calculation with the nominal rate. (8) Where is the effective rate

(9) (10) Nominal and Effective Interest Rates

Determining Loan Duration and Interest Rates Simple rate 1) are known, find duration: (11) 2) are known, find rate: (12)

Example 1.9. The annual rate of simple interest equals 12.5%. How many years will it take the initial sum to double? Solution. The following inequation must be solved: ( n ) > 2, i.e n > 1. We obtain n > 1/ Answer: 8 years. Determining Loan Duration and Interest Rates

Compound rate 1) are known, find duration: (13) (14) Determining Loan Duration and Interest Rates

2) are known, find rate: (15) (16) Determining Loan Duration and Interest Rates

The Notion of Discounting Discounting with simple rates present value(17) discount factor with a simple rate

Discounting Compound rate (18) discount factor with a compound rate (19)

Discounting with a Compound Interest Rate discount ofsum Let us show that the accumulated sum for a specific intermediary time point equals the present value of the payment at the same time point

Inflation Accounting with Interest Accumulation Fisher equation (20) (21)

Continuous Accumulation and Discounting (Continuous Interest) Constant growth rate (22) growh rate is the nominal rate with (23) (24) (25)

(23) (24) Discounting based on continuous interest (25) Continuous Accumulation and Discounting (Continuous Interest)

Growth rate variable,are given (26) (27) (28) Continuous Accumulation and Discounting (continuous interest)

Simple and Compound Interest Rate Equivalency (29) (30) (31) (32)

Change of Contract Terms,, consolidated payment options,are given Variant 1. Term is given. Find size

(33) (34) Change of Contract Terms

Variant 2. The amount of payment is known. Its term must be found (35) (36)

A general case of contract terms change paymentswith terms of the new commitment substitutable payments with terms (old commitment) (37) discount factor Change of Contract Terms

Discounting and Accumulating at Discount Rate

Simple Discount Rate

Example A promissory note to be paid off by January 1, 2008, was discounted 10 months before its discharge for a sum of 180 currency units. What is the value of the annual discount rate if the monthly discount is 2 currency units? Solution. Since interest is deducted for every month, so 1 month may be taken as the unit for measuring time. - monthly discount rate - discounts for all the period д.е. are given

Compound Discount Rate or

Compound Discount Rate. Continued

Example A government bond was discounted five years before its discharge. What is the sum paid for the bond if the discounts for the last and last but one year before discharge were 2000 and 1600 currency units correspondingly? Solution Let us use the obtained relations for compound discounts. If the unit for measuring time is 1 year, then years. currency units.

Discounting with a Nominal Discount Rate Definition. The annual discount rate g is called nominal if a compound discount rate g/m is used for discounting during the 1/m part of the year. where m≥1. If m=1, then g=d.

Continuous Discounting with a Compound Discount Rate Continuous discounting is discounting at infinitely small time intervals, i.e. at 1/m→0 (or m→∞ ).

Variable Discount Rate Let us consider discrete variable interest rates. Let n be the life of the loan divided into k parts – the duration of j interval that uses the discount rate here k – number of periods.

Accumulation with Discount Rate If the discount rate is variable, then If a problem contrary to bank discounting is solved, then the discount rate is used to find the sum of the dischargeable debt.

Comparison of Accumulation Methods With the simple interest rate i Accumulation MethodFormula Accumulation Factor With the compound interest rate i With the nominal interest rate j With the constant growth rate δ With then nominal discount rate δ With the compound discount rate d With the simple discount rate d

Definition. The number indicating the number of times that the accumulated sum of the debt exceeds the initial one is called the accumulation factor (or cumulation factor). The economic substance of the accumulation factor is the following. If the life of the loan is n units of time, then the accumulation factor indicates the future cost of 1 currency unit to be accumulated by the n moment, that will be invested at the t = 0 moment for the n term. It is obvious that the accumulation factor is larger than 1. Comparison of Accumulation Methods

The intensity of accumulation process is determined by the accumulation factor Comparison of Accumulation Methods. Continued

Comparison of Discounting Methods Discounting MethodFormulaDiscounting Factor With the simple discount rate d With the compound discount rate d With the nominal discount rate g With the constant growth rate δ With the nominal interest rate j With the compound interest rate i With the simple interest rate i

Definition. The number indicating what portion of the dischargeable debt sum is comprised by its present value is called the discount factor. The economic substance of the discount factor is the following. If the life of the loan is n units of time, then the discount factor is the present cost of 1 currency unit that must be paid after the n time period. It is obvious that the discount factor is smaller than 1. Comparison of Discounting Methods. Continued

The intensity of the discounting process is determined by the discount factor. Comparison of Discounting Methods