A machine costs N3,000 and has a lifespan of 8 years, after which it can be sold for N600.<\/p>\n\n\n\n
An alternative machine costs N4,000, also with a lifespan of 8 years, but its resale value at the end of its life is N1,900.<\/p>\n\n\n\n
At what annual discount rate (compounded annually) would it be equally economical to choose either machine? (Ignore taxes.)<\/p>\n\n\n\n
The Correct Answer and Explanation i<\/mark><\/strong>s:<\/p>\n\n\n\n To find the annual discount rate<\/strong> at which it is equally economical<\/strong> to choose either machine, we use the Net Present Cost (NPC)<\/strong> approach. We’ll discount all future costs to present value terms and find the rate that makes the present cost of both machines equal<\/strong>.<\/p>\n\n\n\n Let r<\/strong> be the annual discount rate.<\/p>\n\n\n\n We find the PV of the resale values using the formula:<\/p>\n\n\n\n $$ We want the NPC of both machines to be equal:<\/p>\n\n\n\n $$ Bring like terms together:<\/p>\n\n\n\n $$ $$ Multiply both sides by $(1 + r)^8$:<\/p>\n\n\n\n $$ Divide both sides:<\/p>\n\n\n\n $$ Now take the 8th root of both sides:<\/p>\n\n\n\n $$ $$ To determine the discount rate that makes both machines economically equivalent, we use the concept of present value (PV)<\/strong>. The idea is to compare the total cost today<\/strong> of owning each machine, accounting for future resale values by discounting them to present-day terms. This method ensures a fair comparison of costs spread over time.<\/p>\n\n\n\n Each machine has a different upfront cost and a different resale value after 8 years. Since money today is worth more than money in the future (due to inflation, opportunity cost, etc.), we must discount<\/strong> the future resale value back to today\u2019s value using the unknown discount rate, r<\/strong>.<\/p>\n\n\n\n We express the net cost of each machine as:<\/p>\n\n\n\n Setting both equations equal and solving for r<\/strong> gives us:<\/p>\n\n\n\n $$ This discount rate is the breakeven point<\/strong>: if the actual market interest rate is less than 3.35%<\/strong>, Machine B (more expensive upfront but with better resale) is more economical<\/strong>. If it’s more than 3.35%<\/strong>, Machine A is better because its lower initial cost becomes more advantageous when the resale value is heavily discounted.<\/p>\n\n\n\n This approach is essential in capital budgeting, allowing businesses to choose investments that offer the best value over time.<\/p>\n","protected":false},"excerpt":{"rendered":" A machine costs N3,000 and has a lifespan of 8 years, after which it can be sold for N600. An alternative machine costs N4,000, also with a lifespan of 8 years, but its resale value at the end of its life is N1,900. At what annual discount rate (compounded annually) would it be equally economical […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-213412","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/213412","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=213412"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/213412\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=213412"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=213412"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=213412"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nStep 1: Define the cash flows<\/h3>\n\n\n\n
Machine A:<\/strong><\/h4>\n\n\n\n
\n
Machine B:<\/strong><\/h4>\n\n\n\n
\n
\n\n\n\nStep 2: Set up the present value equation<\/h3>\n\n\n\n
PV = \\frac{Future\\ Value}{(1 + r)^n}
$$<\/p>\n\n\n\n
3,000 – \\frac{600}{(1 + r)^8} = 4,000 – \\frac{1,900}{(1 + r)^8}
$$<\/p>\n\n\n\n
\n\n\n\nStep 3: Solve the equation<\/h3>\n\n\n\n
3,000 – 4,000 = \\frac{600 – 1,900}{(1 + r)^8}
$$<\/p>\n\n\n\n
-1,000 = \\frac{-1,300}{(1 + r)^8}
$$<\/p>\n\n\n\n
-1,000 \\cdot (1 + r)^8 = -1,300
$$<\/p>\n\n\n\n
(1 + r)^8 = \\frac{1,300}{1,000} = 1.3
$$<\/p>\n\n\n\n
1 + r = (1.3)^{1\/8} \\approx 1.0335
$$<\/p>\n\n\n\n
r \\approx 0.0335 \\text{ or } 3.35\\%
$$<\/p>\n\n\n\n
\n\n\n\n\u2705 Final Answer: The annual discount rate is approximately 3.35%.<\/strong><\/h3>\n\n\n\n
\n\n\n\nExplanation (300+ words):<\/h3>\n\n\n\n
\n
For Machine A: $\\text{NPC} = 3,000 – \\frac{600}{(1 + r)^8}$
For Machine B: $\\text{NPC} = 4,000 – \\frac{1,900}{(1 + r)^8}$<\/li>\n<\/ul>\n\n\n\n
(1 + r)^8 = 1.3
\\Rightarrow r \\approx 3.35\\%
$$<\/p>\n\n\n\n