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 find the annual discount rate<\/strong> at which both machines are equally economical<\/strong>, we compare their present values (PVs)<\/strong> over their 8-year lifespan.<\/p>\n\n\n\n Let:<\/p>\n\n\n\n We use the Net Present Cost (NPC)<\/strong> formula:<\/p>\n\n\n\n $$ So:<\/p>\n\n\n\n To find the rate at which both machines are equally economical<\/strong>, we equate their NPCs:<\/p>\n\n\n\n $$ Solve:<\/p>\n\n\n\n $$ $$ $$ Now solve for $r$:<\/p>\n\n\n\n $$ $$ $$ In evaluating capital investments like machines, financial analysts often use discounted cash flow (DCF)<\/strong> techniques to consider the time value of money. Here, two machines have different costs and salvage values but the same useful life of 8 years. To determine which machine is more economical under varying interest (discount) rates, we calculate each machine\u2019s Net Present Cost (NPC)<\/strong>.<\/p>\n\n\n\n The NPC accounts for the initial purchase cost minus<\/strong> the present value of the resale value, which is discounted back to the present using the formula:<\/p>\n\n\n\n $$ A lower NPC is preferable because it indicates less total cost over the machine\u2019s lifetime.<\/p>\n\n\n\n In this problem, we equate the NPCs of both machines to find the discount rate at which both options are equally economical. This gives us:<\/p>\n\n\n\n $$ Solving this equation leads us to:<\/p>\n\n\n\n $$ This means at a 3.33% annual discount rate<\/strong>, the present value of both machines\u2019 net costs is the same. If the actual market discount rate is higher than 3.33%<\/strong>, the first machine becomes more economical due to its lower upfront cost. If the rate is lower<\/strong>, the second machine is preferable because of its higher salvage value..<\/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-214931","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/214931","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=214931"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/214931\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=214931"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=214931"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=214931"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\\text{NPC} = \\text{Initial Cost} – \\frac{\\text{Salvage Value}}{(1 + r)^n}
$$<\/p>\n\n\n\n\n
3000 – \\frac{600}{(1 + r)^8} = 4000 – \\frac{1900}{(1 + r)^8}
$$<\/p>\n\n\n\n
3000 – 4000 = \\frac{600 – 1900}{(1 + r)^8}
$$<\/p>\n\n\n\n
-1000 = \\frac{-1300}{(1 + r)^8}
$$<\/p>\n\n\n\n
(1 + r)^8 = \\frac{1300}{1000} = 1.3
$$<\/p>\n\n\n\n
1 + r = (1.3)^{1\/8}
$$<\/p>\n\n\n\n
1 + r \u2248 1.0333
$$<\/p>\n\n\n\n
r \u2248 0.0333 = 3.33\\%
$$<\/p>\n\n\n\n
\n\n\n\n\u2705 Final Answer: 3.33% annual discount rate<\/strong><\/h3>\n\n\n\n
\n\n\n\n\u270d\ufe0f Explanation (300+ words):<\/h3>\n\n\n\n
\\text{Present Value of Salvage} = \\frac{\\text{Salvage}}{(1 + r)^n}
$$<\/p>\n\n\n\n
3000 – \\frac{600}{(1 + r)^8} = 4000 – \\frac{1900}{(1 + r)^8}
$$<\/p>\n\n\n\n
(1 + r)^8 = 1.3 \\Rightarrow r = (1.3)^{1\/8} – 1 \u2248 0.0333 \\text{ or } 3.33\\%
$$<\/p>\n\n\n\n