Math 177 UCLA Winter 2023 Lecture notes Interest rate measurement 1 Introduction What is interest Time value of money The compensation a lender receives from a borrower for not having access to the lent money during the time it was borrowed Typically we talk about interest rates usually as an annual percentage The annual interest rate is multiplied by the amount invested to calculate the amount of interest accrued in a one year period Interest accumulation Generally as interest is accrued it is considered to be reinvested This means the principal amount of the investment increases as interest is earned 2 This is known as compound interest Consider an initial investment of C with an interest rate of i per annum year How does the investment grow in value Example 1 Alice borrows 100 dollars with an interest rate of 10 per annum with interest credited annually How much money will she owe at the end of 4 years 3 4 The rate of interest may change from one year to the next Example 2 Suppose Adam makes an investment that in year 1 has interest rate i1 0 1 in year 2 has interest rate i2 0 5 and in year 3 has interest rate i3 1 2 How much would a 1 dollar investment earn 5 De nition 3 Consider an investment earning compound interest If the interest rate in year 1 is i1 the interest rate in year 2 is i2 and the interest rate in year n is in then the average annual return over those n year is the interest rate i such that 1 in 1 in 1 1 i1 1 i n E ective rates of interest As interest is accrued we say the lender is credited with the interest and the borrower is charged with the interest 6 There are many examples likes this where terminology changes depending on whether we are discussing the lender s perspective or the borrower s per spective Previously we were considering examples where the interest is credited or charged at the end of every year long period usually by convention we say it happens on December 31 However in practice interest may be credited or charged more frequently than once per year Notational convention the textbook usually uses i to denote an annual in terest rate and j denotes an interest rate for a period of time other than a year Even if interest is accruing say monthly we often want to consider what annual rate would give us the same outcome 7 This helps us directly compare two investments that may have di erent com pounding periods De nition 4 The annual e ective rate of interest earned by an invest ment during a one year period is the percentage change in the value of the investment from the beginning to the end of the year value at end of year value at beginning of year value at beginning of year Example 5 If we deposit 100 into a bank account which earns 5 interest per month what is the annual e ective rate of interest 8 De nition 6 Two rates of interest are said to be equivalent if a given amount of principal invested for the same length of time at each of the rates produces the same accumulated values Example 7 Suppose again our bank account earns 5 interest per month What is the equivalent interest rate for an investment that compounds semiannually every six months 9 Compound interests with deposits and withdrawals Banks accounts earn interest but of course deposits and withdrawals can be made The next example shows how to account for these transactions Example 8 Andrea deposits 1000 into an account on January 1 2011 The account credits interest at an annual e ective interest rate of 10 every December 31 Andrea withdraws 100 on January 1 2013 She deposits 200 on January 1 2015 She deposits 50 on January 1 2016 What is the balance in the account just after interest is credited on December 31 2017 10 11 De nition 9 For a time t we denote by a t the accumulated value at time t of an investment of 1 made at time 0 We call a t the accumu lation factor from time 0 to time t If the initial investment is A 0 then the accumulated value at time t of the investment is A t A 0 a t the initial investment times the accumulation factor A t is called the accumulation amount function Example 10 What is the accumulation factor a t for compound in terest at rate i per period Let A t be an accumulated amount function with t measured in years 12 The amount that the investment grows from t1 to t2 is A t2 A t1 We can also use A t to nd the annual e ective interest rate for a one year period from u to u 1 iu 1 A u 1 A u A u Periods of months and days years In most of our examples so far we have measured period of time in terms of 13 However many types of investments use periods of months or days When time is measured in months it is common in practice to formulate m months as a faction of a year This fraction is used even though not all months are exactly 1 12 of a year When time is measured in days it is common in practice to formulate d many days as a faction of a year t m 12 t d 365 Some investments use the denominator of 360 t d 360 Simple interest In compound interest the interest earned is considered to be reinvested This way the principal of the investment grows as interest is earned 14 However in simple interest the principal is considered to stay the same Example 11 Consider an initial investment of X earning simple inter est at an annual rate of i per year What is the accumulated value of this investment after 1 year 2 years 3 years De nition 12 The accumulation from time 0 to time t at annual simple interest rate i where t is measured in years is 15 Thus the accumulated amount function is a t 1 it A t A 0 1 it Example 13 On Jan 31 Smith borrows 5000 from Brown and gives Brown a promissory note The notes states that the loan will be repaid on April 30 of the same year with interest at 12 per annum On March 1 Brown sells the promissory notes to Jones who pays Brown a sum of money in return for the right to collect the payment from Smith on April 30 Jones pays Brown an amount such that Jones yield interest rate earned from March 1 to the maturity date can be stated as an annual rate of interest of 15 a Determine the amound Smith was to have paid Brown on April 30 b Determine the aound that Jones paid to Bornw and the yield rate interest rate Brown earned quoted on an annual basis Assume all …