Math 177 UCLA Winter 2023 1 Lecture notes Valuation of annuities Introduction annuity A nancial transaction that involves a series of period payments is called an Some examples of annuities periodic dividend payments to a stock owner Monthly payments on a loan Annual interest payments on a coupon bond 2 An important formula for computing the value of an annuity is the partial sum formula for a geometric series Proposition 1 For any real number x 1 x x2 xk 1 xk 1 1 x xk 1 1 x 1 Note that we can also get expressions for sums like x4 x5 x100 3 Example 2 Starting in 2011 Andy s grandmother gives him 100 every year on his birthday which is January 1 Every year Andy immediately deposits the 100 into a savings account earning interest at the nominal annual rate of 24 compounded monthly What is the accumulated value on January 1 2015 immediately after he receives his birthday gift for that year 4 Level payment annuities De nition 3 The symbol sn i denotes the accumulated value at the time of and including the nal payment of a series of n payments of 1 each made at equally spaced intervals of time where the rate of interest per payment period is i sn i 1 1 i 1 i n 1 1 i t 1 i n 1 i n 1 cid 88 t 0 The number n of payments is called the term of the annuity and the time between payments is called the payment period or frequency Note that sn i is accurate when we are calculating the value at the time of and including the nal payment The term annuity immediate is used to indicate that the accumulated value we are calculating is at the exact moment of the nal payment payment is Note that if the payments have value X each with everything else being the same then the accumulated value of the annuity at the time of the nal X sn i 5 Example 4 What level amount must be deposited on May 1 and November 1 each year from 2010 to 2016 inclusive to accumulate to 1000 on November 1 2020 if the nominal annual rate of interest com pounded semiannually is 8 6 Accumulated value some time after the nal payment Example 5 On each January 1 Smith deposits 50 into a savings ac count from 2010 to 2020 inclusive After the payment on January 1 2020 Smith does not deposit any more money The account earns an e ective annual interest rate of 9 What is the accumulated value of the savings when Smith retires in 2045 Consider an annuity with n payments of 1 each earning interest rate i per period The accumulated value k period after the nal payment is 7 sn i 1 i k There is another useful way to express this quantity Accumulated value with non level interest rates Suppose the interest rate changes in the middle of the annuity How can we calculate the accumulated value at the end of the annuity 8 Accumulated value with a changing payment Suppose now the interest rate stays level but the amount of the payment changes 9 Present value of an annuity Consider an annuity of n payments of 1 each made at equally spaced intervals of time where the rate of interest per payment period is i 10 What is the present value of the annuity 1 period before the rst payment 11 De nition 6 The symbol an i denotes the present value of a series of equally spaced payment of amount 1 each when the valuation point is one payment period before the payment begins an i v v2 vn vt n cid 88 t 1 1 vn i When we are valuing the annuity 1 period before the rst payment we refer to it as a annuity immediate 12 Example 7 Smith has bought a new car and requires a loan of 12 000 to pay for it The car dealer o ers Smith a loan with monthly payments for 3 years starting one month after the purchase with an annual interest rate of 12 compounded monthly Find the total amount that Smith will pay for the car 13 Present value some time before payments begin Example 8 A bank account earning 10 nominal annual interest will get deposits of 100 on the rst of every month for the years 2025 and 2026 What is the present value on January 1 2023 of the amount of money in the account on December 1 2026 after the deposit for that month has been made 14 We can generalize the previous example De nition 9 Suppose an n payment annuity of 1 per period is to be valued k 1 payment periods before the rst payment is made Then the present value is vk v v2 vn vk an i This is called a deferred annuity Speci cally it is a k period deferred n payment annuity immediate of 1 per period Its present value is denoted by the symbol k an i k an i vk an i Alternative expressions for accumulated and present value When the interest i is understood and there is no risk of confusion then we 15 can use the notation sn an k an We also have the equalities sn k sn 1 i k sk an k vk an ak Moreover we can relate accumulated value and present value using 16 sn 1 i n an Valuation of perpetuities 17 Suppose a series of payments of 1 will be received inde nitely What is the present value 1 payment period before the start of the payments We denote the present value by a i and it is de ned as a limit a i lim n an i We can compute this limit This in nite period annuity is called a perpetuity Since we are valuing it one period before the rst payment it is referred to as a perpetuity immediate 18 Example 10 A perpetuity immediate pays X per year Brian receives the rst n payments Colleen receives the remaining payments Suppose that Brian s share of the present value of the perpetuity is 40 What is Colleen s share of the present value 19 Annuity due We have symbols sn i an i for the accumulated and present value annuity immediate valuations For the accumulated value sn i this means we are valuing at the exact time of the nal payment st payment is made For the present value an i this means we are valuing one period before the However the accumulated and present valuations are sometime made at dif ferent times For the present value case we saw this with deferred annuities De nition 11 For n payment annuities with equally spaced payments of 1 each the annuity due present value has valuation at the time of the rst payment and is denoted an i 1 v vn 1 1 vn 1 v The annuity due accumulated value is one period after the nal payment sn i 1 i 1 i 2 1 …