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 is:<\/mark><\/strong><\/p>\n\n\n\n To determine the annual discount rate (r)<\/strong> at which both machines are equally economical, we need to compare the present value (PV)<\/strong> of their net costs<\/strong> over 8 years, considering both purchase price and resale value.<\/p>\n\n\n\n For both machines, we calculate the present value of net cost<\/strong> as:<\/p>\n\n\n\n $$ So, for each machine:<\/p>\n\n\n\n We equate the net present costs of both machines:<\/p>\n\n\n\n $$ Subtract both sides:<\/p>\n\n\n\n $$ Bring like terms together:<\/p>\n\n\n\n $$ $$ Multiply both sides:<\/p>\n\n\n\n $$ Now solve for $r$:<\/p>\n\n\n\n $$ $$ $$ This problem involves comparing the net present cost<\/strong> of two capital investments (machines), each with different initial costs and resale values<\/strong> at the end of 8 years. The goal is to determine the discount rate<\/strong> at which these two options become financially equivalent.<\/p>\n\n\n\n In economics and finance, the present value (PV)<\/strong> concept is used to determine how much a future amount of money is worth today, taking into account the time value of money. The further in the future a cash flow is, the less valuable it is today. This principle is crucial when evaluating long-term investments like machinery.<\/p>\n\n\n\n For both machines, we assume:<\/p>\n\n\n\n Machine A is cheaper to buy (N3,000) but has a lower resale value (N600). Machine B is more expensive (N4,000) but returns more at the end of its life (N1,900). Over 8 years, the real value of these resale amounts depends on the discount rate<\/strong> applied.<\/p>\n\n\n\n We used the present value formula<\/strong>:<\/p>\n\n\n\n $$ Where:<\/p>\n\n\n\n By equating the net present cost<\/strong> of both machines and solving for $r$, we found the breakeven discount rate<\/strong> where the total economic cost is the same.<\/p>\n\n\n\n At a 3.35% annual discount rate<\/strong>, it doesn’t matter which machine you choose\u2014they both result in the same economic cost when time value of money is considered.<\/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-214933","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/214933","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=214933"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/214933\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=214933"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=214933"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=214933"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Define the present value of net cost<\/h3>\n\n\n\n
\\text{Net Cost} = \\text{Purchase Price} – \\text{Present Value of Resale Value}
$$<\/p>\n\n\n\n\n
3,000 – \\frac{600}{(1 + r)^8} = 4,000 – \\frac{1,900}{(1 + r)^8}
$$<\/p>\n\n\n\nStep 2: Solve for the discount rate $r$<\/h3>\n\n\n\n
3,000 – \\frac{600}{(1 + r)^8} = 4,000 – \\frac{1,900}{(1 + r)^8}
$$<\/p>\n\n\n\n
4,000 – 3,000 = \\frac{1,900 – 600}{(1 + r)^8}
$$<\/p>\n\n\n\n
1,000 = \\frac{1,300}{(1 + r)^8}
$$<\/p>\n\n\n\n
(1 + r)^8 = \\frac{1,300}{1,000} = 1.3
$$<\/p>\n\n\n\n
1 + r = (1.3)^{1\/8}
$$<\/p>\n\n\n\n
1 + r = 1.0335
$$<\/p>\n\n\n\n
r = 0.0335 \\text{ or } 3.35\\%
$$<\/p>\n\n\n\n
\n\n\n\n\u2705 Final Answer: 3.35%<\/strong><\/h3>\n\n\n\n
\n\n\n\nExplanation (300+ words):<\/h3>\n\n\n\n
\n
\\text{PV} = \\frac{\\text{Future Value}}{(1 + r)^n}
$$<\/p>\n\n\n\n\n