{"id":584,"date":"2021-11-15T18:53:25","date_gmt":"2021-11-15T10:53:25","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=584"},"modified":"2022-07-15T17:28:24","modified_gmt":"2022-07-15T09:28:24","slug":"modigliani-miller-mm-theorem","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=584","title":{"rendered":"Modigliani-Miller (M&#038;M) Theorem"},"content":{"rendered":"\n<p><strong>M&amp;M theorem<\/strong> (Modigliani and Miller, 1958) is used to value a firm. It states that a firm&#8217;s value is based on its ability to earn revenue plus its risk of underlying assets. The way a firm finances its operations should not affect its value.<\/p>\n\n\n\n<p>At its most basic level, the theorem argues that, with certain assumptions in place, it is irrelevant whether a company finances its growth by borrowing, by issuing stock shares, or by reinvesting its profits.<\/p>\n\n\n\n<p><strong>Assumptions <\/strong>are 1. the markets are completely efficient; 2. there are no costs of bankruptcy or agency dynamics and no taxes.<\/p>\n\n\n\n<p>However, there are of course taxes and costs in the reality, and the<strong> assumptions do not hold<\/strong>. Therefore, the M&amp;M theorem implies that firms are more valuable if financed by debts than financed by equities. The reason is the tax shield effects of debts. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Mathematic Example<\/h4>\n\n\n\n<p>Consider two companies, same risks, same expected cash flow before interest, \\( Y\\).<\/p>\n\n\n\n<ol><li><strong>Co1<\/strong>, has debt with market value of \\(D_1\\). Total market value \\(V_1=E_1+D_1\\).<\/li><li><strong>Co2<\/strong>, has no debt. \\(V_2=E_2\\), market value of equity.<\/li><\/ol>\n\n\n\n<p>We can invest,<\/p>\n\n\n\n<ul><li><strong>Investment A<\/strong>: We own a fraction \\(a\\) (e.g. 6%) of shares in Co1z. They worth \\(aE_1\\) (e.g. $6,000). Expected cash flow from investment A, \\(y_A\\), is: <\/li><\/ul>\n\n\n\n<p>$$(y_A=\\underbrace{a}_{SharesOwned} \\times \\underbrace{(Y-R_D D_1)}_{Co1&#8217;s EarningAfterInterest}$$<\/p>\n\n\n\n<p>Co1 needs to pay an interest rate of its debt, \\(R_D D_1\\). Purchasing Co1 means only purchasing the equity of Co1, which is EV-Debts.<\/p>\n\n\n\n<ul><li><strong>Investment B<\/strong>: We sell the shares of Co1. We receive amount \\(aE_1\\) ($6,000). Then, we use this to buy shares in Co2, which is ungeared.<\/li><\/ul>\n\n\n\n<p>To Produce the same gearing and risk as investment A, we borrow a further amount worth \\(aD_1\\) (&#8216;home-made gearing&#8217; or &#8216;artificially gearing&#8217;), at interest rate \\(R_D\\), and buy more shares in Co2.<\/p>\n\n\n\n<p>E.G. if gearing of Co1 is \\( \\frac{D_1}{D_1+E_1}=0.25\\), then to get same gear for our investment B, we borrow \\(<span class=\"katex math inline\">6,000\\times \\frac{0.25}{0.75}=<\/span>2,000\\).<\/p>\n\n\n\n<p>Expected cash flow from investment B, \\( y_B\\), is:<\/p>\n\n\n\n<p>$$ y_B=\\frac{aE_1+aD_1}{E_2}Y-R_D\\times aD_1 = a\\frac{V_1}{V_2}Y-R_D\\times aD_1$$<\/p>\n\n\n\n<p>In equilibrium, \\( y_A=y_B\\). Otherwise, arbitrage opportunity emerges. Therefore, we get \\(V_1=V_2\\).<\/p>\n\n\n\n<p>In conclusion, gearing does not affect value (Equity plus Debt).<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Implication<\/h4>\n\n\n\n<p>Since expected net cash flow (\\(Y\\)) and company value are the same for each company, the cost of equity for ungeared Co2 and WACC for geared Co1 must be equal. <strong>So WACC must be constant with respect to gearing.<\/strong><\/p>\n\n\n\n<p>Let the cost of equity for a company with no debt be \\(R_{ungeared}\\).<\/p>\n\n\n\n<p>$$ R_{ungeared}=R_D\\frac{D}{V}+R_E\\frac{E}{V}=WACC$$<\/p>\n\n\n\n<p>$$ R_{ungeared}=R_D\\frac{D}{D+E}+R_E\\frac{E}{D+E}=WACC$$<\/p>\n\n\n\n<p>WACC is constant w.r.t. \\(\\frac{D}{V}\\).<\/p>\n\n\n\n<p>$$ R_{ungeared}(D+E)=R_DD+R_EE$$<\/p>\n\n\n\n<p>$$ R_EE=R_{ungeared}(D+E)-R_DD $$<\/p>\n\n\n\n<p>$$ R_E=R_{ungeared}+(R_{ungeared}-R_D)\\frac{D}{E} $$<\/p>\n\n\n\n<p>So we get a linear relation between cost of equity and D\/E, assuming a constant cost of debt, assuming a constant cost of debt (and assuming \\(R_{ungeared}&gt;R_D\\)). Implicly, changes in gearing structure (or how to finance the business) do not affect the WACC for a company.<\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/11\/IMG_5225-1024x528.jpeg\" alt=\"\" class=\"wp-image-1108\" width=\"690\" height=\"355\" srcset=\"http:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/11\/IMG_5225-1024x528.jpeg 1024w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/11\/IMG_5225-300x155.jpeg 300w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/11\/IMG_5225-768x396.jpeg 768w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/11\/IMG_5225-1536x792.jpeg 1536w, http:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/11\/IMG_5225.jpeg 1589w\" sizes=\"(max-width: 690px) 100vw, 690px\" \/><figcaption>The relation is shown in this figure<\/figcaption><\/figure><\/div>\n\n\n<p>Violation of the constant debt assumption of course would make WACC unconstant.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">MM Theory &amp; Beta<\/h4>\n\n\n\n<p>In the CAPM, expected returns on assets differ because their betas differ.<\/p>\n\n\n\n<p>So we can write,<\/p>\n\n\n\n<p>$$ \\beta_{ungeared}=\\beta_{debt}\\frac{D}{D+E}+\\beta_{geared}\\frac{E}{D+E} $$<\/p>\n\n\n\n<p>$$\\beta_{geared}=\\beta_{ungeared}+(\\beta_{ungeared}-\\beta_{debt})\\frac{D}{E}$$<\/p>\n\n\n\n<p>, where \\(\\beta_{ungeared}\\) denotes asset beta, and \\( \\beta_{geared}\\) denotes actual beta of shares of a geared company (estimated from the market data).<\/p>\n\n\n\n<p>Calculations are similar to those as above.<\/p>\n\n\n\n<p>If assume \\(\\beta_{debt}=0\\), then<\/p>\n\n\n\n<p>$$ \\beta_{ungeared}=\\beta_{geared} \\frac{E}{V}$$<\/p>\n\n\n\n<p>$$ \\beta_{geared}=\\beta_{ungeared}(1+\\frac{D}{E}) $$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">A Modigliani-Miller Theorem for Open-Market Operations<\/h4>\n\n\n\n<p>Monetary policy determines the composition of the government&#8217;s portfolio. Fiscal policy (the size of the deficit on the current account) determines the path of net government indebtedness. Wallace showed that alternative paths of the government&#8217;s portfolio consistent with a single path of fiscal policy can be irrelevant. <strong>The irrelevance means that both the equilibrium consumption allocation and the path of the price level are independent of the path of the government&#8217;s portfolio.<\/strong><\/p>\n\n\n\n<div data-wp-interactive=\"\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!selectors.core.file.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/11\/Wallance1981.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Embed of Wallance1981..\"><\/object><a id=\"wp-block-file--media-3710b5d3-c8cc-4fde-88f4-d8f559d9f873\" href=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/11\/Wallance1981.pdf\">Wallance1981<\/a><a href=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/11\/Wallance1981.pdf\" class=\"wp-block-file__button\" download aria-describedby=\"wp-block-file--media-3710b5d3-c8cc-4fde-88f4-d8f559d9f873\">Download<\/a><\/div>\n\n\n\n<p>Typos are there. See the original paper issued by Fed in 1979.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Reference<\/h4>\n\n\n\n<p>Modigliani, F. and Miller, M.H., 1958. The cost of capital, corporation finance and the theory of investment.&nbsp;<em>The American economic review<\/em>,&nbsp;<em>48<\/em>(3), pp.261-297.<\/p>\n\n\n\n<p>Wallace, N., 1981. A Modigliani-Miller theorem for open-market operations.&nbsp;<em>The American Economic Review<\/em>,&nbsp;<em>71<\/em>(3), pp.267-274.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>M&amp;M theorem (Modigliani and Miller, 1958) is used to value a firm. It states that a firm&#8217;s value is based on its ability to earn revenue plus its risk of underlying assets. The way a firm finances its operations should not affect its value. At its most basic level, the theorem argues that, with certain &hellip; <a href=\"https:\/\/fanyuzhao.com\/?p=584\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Modigliani-Miller (M&#038;M) Theorem<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,11,6],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/584"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=584"}],"version-history":[{"count":97,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/584\/revisions"}],"predecessor-version":[{"id":3847,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/584\/revisions\/3847"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=584"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=584"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=584"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}