The winner of a lottery chooses to receive annual payments of $160,000 at the end of each year for 25 years. If the current interest rate is 5.9%, find the present value (in dollars) of the payments. (Round your answer to the nearest cent. See Example 5 in this section.)<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n Let’s solve this step-by-step.<\/p>\n\n\n\n A lottery winner receives $160,000<\/strong> at the end of each year for 25 years<\/strong>. The present value is the current worth of a stream of future payments, discounted back to the present using the interest rate. It tells us how much the series of future payments is worth right now.<\/p>\n\n\n\n Since the payments are the same every year, this is an ordinary annuity<\/strong> problem.<\/p>\n\n\n\n PV=P\u00d71\u2212(1+r)\u2212nrPV = P \\times \\frac{1 – (1 + r)^{-n}}{r}<\/p>\n\n\n\n Where:<\/p>\n\n\n\n 1\u2212(1+r)\u2212nr=1\u2212(1+0.059)\u2212250.059\\frac{1 – (1 + r)^{-n}}{r} = \\frac{1 – (1 + 0.059)^{-25}}{0.059}<\/p>\n\n\n\n First, calculate (1+0.059)\u221225(1 + 0.059)^{-25}: 1+0.059=1.0591 + 0.059 = 1.059 1.059\u221225=11.059251.059^{-25} = \\frac{1}{1.059^{25}}<\/p>\n\n\n\n Calculate 1.059251.059^{25}:<\/p>\n\n\n\n Using a calculator: 1.05925\u22484.2561.059^{25} \\approx 4.256<\/p>\n\n\n\n So, 1.059\u221225=14.256\u22480.2351.059^{-25} = \\frac{1}{4.256} \\approx 0.235<\/p>\n\n\n\n Now substitute back: 1\u22120.2350.059=0.7650.059\u224812.97\\frac{1 – 0.235}{0.059} = \\frac{0.765}{0.059} \\approx 12.97<\/p>\n\n\n\n PV=160,000\u00d712.97=2,075,200PV = 160,000 \\times 12.97 = 2,075,200<\/p>\n\n\n\n $2,075,200.00\\boxed{\\$2,075,200.00}<\/p>\n\n\n\n The present value calculation discounts each of the $160,000 payments back to today’s value using the 5.9% interest rate. Because the payments continue for 25 years, we use the annuity formula to sum the discounted value of all payments. This shows that receiving $160,000 yearly for 25 years is equivalent to receiving about $2,075,200 as a lump sum today, assuming the money could be invested at a 5.9% annual rate.<\/p>\n\n\n\n This present value is useful for comparing options: for instance, if the lottery winner could take the entire amount now instead of annual payments, they’d expect to get around $2.075 million to be financially equivalent. If they could invest at a rate higher than 5.9%, the lump sum might be lower, and vice versa.<\/p>\n","protected":false},"excerpt":{"rendered":" The winner of a lottery chooses to receive annual payments of $160,000 at the end of each year for 25 years. If the current interest rate is 5.9%, find the present value (in dollars) of the payments. (Round your answer to the nearest cent. See Example 5 in this section.) The correct answer and explanation […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-21729","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/21729","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=21729"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/21729\/revisions"}],"predecessor-version":[{"id":21730,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/21729\/revisions\/21730"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=21729"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=21729"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=21729"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nProblem Summary:<\/h3>\n\n\n\n
The current interest rate (discount rate) is 5.9% per year<\/strong>.
We want to find the present value (PV)<\/strong> of these payments.<\/p>\n\n\n\n
\n\n\n\nWhat is Present Value?<\/h3>\n\n\n\n
\n\n\n\nPresent Value Formula for an Ordinary Annuity:<\/h3>\n\n\n\n
\n
\n\n\n\nStep 1: Calculate the present value factor<\/h3>\n\n\n\n
\n\n\n\nStep 2: Calculate the present value<\/h3>\n\n\n\n
\n\n\n\nFinal answer:<\/h3>\n\n\n\n
\n\n\n\nExplanation:<\/h3>\n\n\n\n