{"id":19019,"date":"2025-06-13T12:01:50","date_gmt":"2025-06-13T12:01:50","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=19019"},"modified":"2025-06-13T12:01:56","modified_gmt":"2025-06-13T12:01:56","slug":"the-rent-for-the-next-five-years-of-an-eighty-five-year-property-contract-is-set-at-4000-permonth","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-rent-for-the-next-five-years-of-an-eighty-five-year-property-contract-is-set-at-4000-permonth\/","title":{"rendered":"The rent for the next five years of an eighty Five-year property contract is set at \u00a34,000 permonth"},"content":{"rendered":"\n<p>The rent for the next five years of an eighty Five-year property contract is set at \u00a34,000 permonth. Thereafter the rent will increase by 20% compound every five years. Whatprice would be paid for the contract in order to achieve a yield of 8% per annumeffective?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s carefully analyze the problem step-by-step:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Problem Summary:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Property contract length:<\/strong> 85 years<\/li>\n\n\n\n<li><strong>Initial rent:<\/strong> \u00a34,000 per month for the first 5 years (so \u00a348,000 per year)<\/li>\n\n\n\n<li><strong>After first 5 years:<\/strong> rent increases by 20% compounded every 5 years<\/li>\n\n\n\n<li><strong>Required yield:<\/strong> 8% effective annual yield<\/li>\n\n\n\n<li><strong>Goal:<\/strong> Find the <strong>price<\/strong> to pay now for this contract to achieve an 8% yield.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Understand the cash flows<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For years 1\u20135: annual rent = \u00a34,000 \u00d7 12 = \u00a348,000<\/li>\n\n\n\n<li>From year 6\u201310: rent increases by 20% \u2192 \u00a348,000 \u00d7 1.2 = \u00a357,600<\/li>\n\n\n\n<li>From year 11\u201315: rent increases another 20% on previous increase \u2192 \u00a357,600 \u00d7 1.2 = \u00a369,120<\/li>\n\n\n\n<li>And so on, rent increases every 5 years by 20% compound.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Define rent at each 5-year period<\/h3>\n\n\n\n<p>Let R0=48,000R_0 = 48,000 be the annual rent for the first 5 years.<\/p>\n\n\n\n<p>The rent for each 5-year period nn (where n=0,1,2,&#8230;n = 0, 1, 2, &#8230;) is: Rn=R0\u00d7(1.2)nR_n = R_0 \\times (1.2)^n<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Present Value of each 5-year rent block<\/h3>\n\n\n\n<p>We want to calculate the present value (PV) of the rent payments.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Discount rate i=8%i = 8\\% per year (effective).<\/li>\n\n\n\n<li>Rent is paid annually.<\/li>\n\n\n\n<li>Rent amount RnR_n is paid for 5 years, starting at year 5n+15n + 1 through 5n+55n + 5.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Calculate PV of each 5-year block<\/h3>\n\n\n\n<p>The PV of the nthn^{th} block of 5 years of rent, starting at year 5n+15n + 1, is: PVn=Rn\u00d7\u2211t=151(1+i)5n+tPV_n = R_n \\times \\sum_{t=1}^{5} \\frac{1}{(1 + i)^{5n + t}}<\/p>\n\n\n\n<p>This is a 5-year annuity starting at time 5n+15n + 1.<\/p>\n\n\n\n<p>Rewrite sum inside as: \u2211t=15(1+i)\u2212(5n+t)=(1+i)\u22125n\u00d7\u2211t=15(1+i)\u2212t\\sum_{t=1}^{5} (1+i)^{-(5n+t)} = (1+i)^{-5n} \\times \\sum_{t=1}^{5} (1+i)^{-t}<\/p>\n\n\n\n<p>The term \u2211t=15(1+i)\u2212t\\sum_{t=1}^5 (1+i)^{-t} is a standard 5-year annuity factor: a5\u203e\u2223i=1\u2212(1+i)\u22125ia_{\\overline{5}|i} = \\frac{1 &#8211; (1+i)^{-5}}{i}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 5: Total PV of all 17 blocks (since 85 years \/ 5 = 17 blocks)<\/h3>\n\n\n\n<p>PV=\u2211n=016Rn\u00d7(1+i)\u22125n\u00d7a5\u203e\u2223iPV = \\sum_{n=0}^{16} R_n \\times (1+i)^{-5n} \\times a_{\\overline{5}|i}<\/p>\n\n\n\n<p>Substitute Rn=R0(1.2)nR_n = R_0 (1.2)^n: PV=a5\u203e\u2223i\u00d7R0\u00d7\u2211n=016(1.2)n(1+i)\u22125nPV = a_{\\overline{5}|i} \\times R_0 \\times \\sum_{n=0}^{16} (1.2)^n (1+i)^{-5n}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 6: Calculate annuity factor a5\u203e\u2223ia_{\\overline{5}|i}<\/h3>\n\n\n\n<p>At i=8%=0.08i = 8\\% = 0.08: a5\u203e\u22230.08=1\u2212(1.08)\u221250.08a_{\\overline{5}|0.08} = \\frac{1 &#8211; (1.08)^{-5}}{0.08}<\/p>\n\n\n\n<p>Calculate (1.08)\u22125(1.08)^{-5}: 1.085=1.4693\u2005\u200a\u27f9\u2005\u200a(1.08)\u22125=11.4693=0.68061.08^{5} = 1.4693 \\implies (1.08)^{-5} = \\frac{1}{1.4693} = 0.6806<\/p>\n\n\n\n<p>Therefore: a5\u203e\u22230.08=1\u22120.68060.08=0.31940.08=3.9925a_{\\overline{5}|0.08} = \\frac{1 &#8211; 0.6806}{0.08} = \\frac{0.3194}{0.08} = 3.9925<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 7: Calculate the sum<\/h3>\n\n\n\n<p>Define: S=\u2211n=016(1.2)n(1.08)\u22125n=\u2211n=016(1.21.085)nS = \\sum_{n=0}^{16} (1.2)^n (1.08)^{-5n} = \\sum_{n=0}^{16} \\left( \\frac{1.2}{1.08^5} \\right)^n<\/p>\n\n\n\n<p>Calculate 1.0851.08^5 again: 1.085=1.46931.08^5 = 1.4693<\/p>\n\n\n\n<p>Therefore: 1.21.4693=0.8166\\frac{1.2}{1.4693} = 0.8166<\/p>\n\n\n\n<p>So, S=\u2211n=016(0.8166)nS = \\sum_{n=0}^{16} (0.8166)^n<\/p>\n\n\n\n<p>This is a geometric series with ratio r=0.8166&lt;1r = 0.8166 &lt; 1, so: S=1\u2212r171\u2212rS = \\frac{1 &#8211; r^{17}}{1 &#8211; r}<\/p>\n\n\n\n<p>Calculate r17=0.816617r^{17} = 0.8166^{17}:<\/p>\n\n\n\n<p>Using a calculator, log\u206110(0.816617)=17\u00d7log\u206110(0.8166)=17\u00d7(\u22120.0889)=\u22121.511\\log_{10}(0.8166^{17}) = 17 \\times \\log_{10}(0.8166) = 17 \\times (-0.0889) = -1.511<\/p>\n\n\n\n<p>So, r17=10\u22121.511=0.0308r^{17} = 10^{-1.511} = 0.0308<\/p>\n\n\n\n<p>Hence, S=1\u22120.03081\u22120.8166=0.96920.1834=5.285S = \\frac{1 &#8211; 0.0308}{1 &#8211; 0.8166} = \\frac{0.9692}{0.1834} = 5.285<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 8: Calculate total present value<\/h3>\n\n\n\n<p>PV=a5\u203e\u2223i\u00d7R0\u00d7S=3.9925\u00d748,000\u00d75.285PV = a_{\\overline{5}|i} \\times R_0 \\times S = 3.9925 \\times 48,000 \\times 5.285<\/p>\n\n\n\n<p>Calculate stepwise: 3.9925\u00d748,000=191,6403.9925 \\times 48,000 = 191,640<\/p>\n\n\n\n<p>Then: 191,640\u00d75.285=1,012,130191,640 \\times 5.285 = 1,012,130<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Answer:<\/strong><\/h3>\n\n\n\n<p>The price to pay for the contract to achieve an 8% yield is <strong>approximately \u00a31,012,130<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h3>\n\n\n\n<p>This problem involves valuing a long-term lease contract with rent that increases in a stepwise manner every 5 years by 20%. To find the contract\u2019s value at an 8% annual effective yield, we discount all future cash flows back to the present.<\/p>\n\n\n\n<p>Because rent increases every 5 years by 20%, the total rent stream can be split into 5-year blocks where the rent is constant, but each subsequent block grows by 20% over the previous one. For each 5-year block, we calculate the present value of those 5 years&#8217; rents (an annuity), then discount that entire block back to today.<\/p>\n\n\n\n<p>Summing the present values of all 17 blocks (since 85 years \/ 5 years per block = 17 blocks) gives the total present value. Since the rent increase is geometric and the discount factor is geometric, the sum of the present values reduces to a geometric series that we can calculate explicitly.<\/p>\n\n\n\n<p>This approach yields the price to pay now to achieve an 8% yield on the property lease income stream with the specified rent escalation pattern.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The rent for the next five years of an eighty Five-year property contract is set at \u00a34,000 permonth. Thereafter the rent will increase by 20% compound every five years. Whatprice would be paid for the contract in order to achieve a yield of 8% per annumeffective? The correct answer and explanation is: Let&#8217;s carefully analyze [&hellip;]<\/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-19019","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/19019","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=19019"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/19019\/revisions"}],"predecessor-version":[{"id":19020,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/19019\/revisions\/19020"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=19019"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=19019"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=19019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}