{"id":187526,"date":"2023-04-04T10:05:25","date_gmt":"2023-04-04T14:05:25","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=187526"},"modified":"2023-04-04T10:05:43","modified_gmt":"2023-04-04T14:05:43","slug":"the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/","title":{"rendered":"The Mathematics of Bonds: Simulating the Returns of Constant Maturity Government Bond ETFs"},"content":{"rendered":"\n<p><em>The article &#8220;The Mathematics of Bonds: Simulating the Returns of Constant Maturity Government Bond ETFs&#8221; first appeared on <a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/\">Portfolio Optimizer<\/a> blog.<\/em><\/p>\n\n\n\n<p><em>Excerpt<\/em><\/p>\n\n\n\n<p>With more than&nbsp;$1.2 trillion under management in the U.S. as of mid-July 2022<sup>1<\/sup>, investors are more and more using bond ETFs as building blocks in their asset allocation.<\/p>\n\n\n\n<p>One issue with such instruments, though, is that their price history dates back to at best 2002<sup>1<\/sup>, which is problematic in some applications like trading strategy backtesting or portfolio historical stress-testing.<\/p>\n\n\n\n<p>In this post, which builds on the paper&nbsp;<em>Treasury Bond Return Data Starting in 1962<\/em>&nbsp;from Laurens Swinkels<sup>2<\/sup>, I will show that the returns of specific bond ETFs &#8211; those seeking a constant maturity exposure to government-issued bonds &#8211; can be simulated using&nbsp;<em>standard textbook formulas<\/em><sup>2<\/sup>&nbsp;together with appropriate yields to maturity.<\/p>\n\n\n\n<p>This allows in particular to extend the price history of these ETFs by several tens of years thanks to publicly available yield to maturity series published by governments, government-affiliated agencies, researchers\u2026<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong><em>Notes:<\/em><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A Google sheet corresponding to this post is available&nbsp;<a href=\"https:\/\/docs.google.com\/spreadsheets\/d\/12RYYOAqGfMlEJOX_ztMe7eIbVdP6BOd1aYeQ0J7ym0Q\/edit?usp=sharing\">here<\/a><\/li>\n<\/ul>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"mathematical-preliminaries\">Mathematical preliminaries<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"bond-yield-formula\">Bond yield formula<\/h3>\n\n\n\n<p>In what comes next, I will make heavy use of the formula expressing the price of a bond as a function of its yield to maturity.<\/p>\n\n\n\n<p>This formula can be found in the appendix&nbsp;<em>A3.1 Yield to maturity for settlement dates other than coupon payment dates<\/em>&nbsp;of Tuckman and Serrat<sup>3<\/sup>, and is reproduced below for convenience.<\/p>\n\n\n\n<p>Let be a bond<sup>4<\/sup>&nbsp;at a date&nbsp;<em>t<\/em>, with a remaining maturity equal to&nbsp;<em>T<\/em>, a yield to maturity equal to&nbsp;<em>y<sub>t<\/sub><\/em>\u200b&nbsp;and a coupon rate equal to&nbsp;<em>c<sub>t<\/sub><\/em>\u200b.<\/p>\n\n\n\n<p>Then, its price&nbsp;<em>P<sub>t<\/sub><\/em>\u200b(<em>c<sub>t<\/sub><\/em>\u200b,<em>y<sub>t<\/sub><\/em>\u200b,<em>T<\/em>)&nbsp;per 100 face amount is equal to<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"892\" height=\"150\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer.png\" alt=\"\" class=\"wp-image-187533 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer.png 892w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-700x118.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-300x50.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-768x129.png 768w\" data-sizes=\"(max-width: 892px) 100vw, 892px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 892px; aspect-ratio: 892\/150;\" \/><\/figure>\n\n\n\n<p>, where&nbsp;<em>\u03c4<sub>t<\/sub><\/em>\u200b&nbsp;is the fraction of a semiannual period until the next coupon payment.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"par-bond-total-return-formula\">Par bond total return formula<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#par-bond-total-return-formula\"><\/a><\/h2>\n\n\n\n<p>Using the bond yield formula, it is possible to approximate the total return&nbsp;<em>TR<\/em>&nbsp;of a par bond over a specific period using only its remaining maturity at the beginning of the period, its yield to maturity at the beginning of the period and its yield to maturity at the end of the period.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"par-bond-total-return-formula-for-a-monthly-period\">Par bond total return formula for a monthly period<\/h3>\n\n\n\n<p>In the case of a monthly period, let be a bond such that:<\/p>\n\n\n\n<p>\u2014Its remaining maturity at the end of the month&nbsp;<em>t\u22121<\/em>&nbsp;is equal to&nbsp;<em>T<\/em>. <br><br>\u2014Its yield to maturity at the end of the month&nbsp;<em>t<\/em>\u22121, for a remaining maturity equal to&nbsp;<em>T<\/em>, is&nbsp;<em>y<sub>t\u22121<\/sub><\/em>\u200b<br><br>\u2014Its yield to maturity at the end of the month&nbsp;<em>t<\/em>, for a remaining maturity equal to&nbsp;<em>T<\/em>, is&nbsp;<em>y<sub>t<\/sub><\/em><\/p>\n\n\n\n<p><br>Then, assuming that<\/p>\n\n\n\n<p>\u2014The bond trades at par at the end of the month&nbsp;<em>t<\/em>\u22121<br> \u2014The bond yield curve at the end of the month <em>t<\/em> is flat for remaining maturities between <img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-3.png\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\"> and <em>T<\/em>.<\/p>\n\n\n\n<p>, the total return&nbsp;<em>TR<sub>t<\/sub><\/em>\u200b&nbsp;of this bond from the end of the month&nbsp;<em>t<\/em>\u22121&nbsp;to the end of the month&nbsp;<em>t<\/em>&nbsp;can be approximated by<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"917\" height=\"190\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-2.png\" alt=\"\" class=\"wp-image-187535 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-2.png 917w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-2-700x145.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-2-300x62.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-2-768x159.png 768w\" data-sizes=\"(max-width: 917px) 100vw, 917px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 917px; aspect-ratio: 917\/190;\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Demonstration of the par bond total return formula for a monthly period<\/h3>\n\n\n\n<p>A&nbsp;possible demonstration for the previous formula goes as follows.<\/p>\n\n\n\n<p>At the end of the month&nbsp;<em>t<\/em>\u22121, the bond has the following characteristics:<\/p>\n\n\n\n<p>\u2014 Its remaining maturity is equal to&nbsp;<em>T<\/em> <br>\u2014 Its coupon rate&nbsp;<em>c<sub>t\u22121<\/sub><\/em>\u200b&nbsp;is equal to its yield to maturity&nbsp;<em>y<sub>t<\/sub><\/em>\u22121\u200b, because of the assumption that the bond trades at par at the end of the month&nbsp;<em>t<\/em>\u22121<\/p>\n\n\n\n<p>Its price&nbsp;<em>P<sub>t<\/sub><\/em>\u22121\u200b(<em>c<sub>t<\/sub><\/em>\u22121\u200b,<em>y<sub>t<\/sub><\/em>\u22121\u200b,<em>T<\/em>)&nbsp;is then equal, through the bond yield formula\u200b<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"616\" height=\"90\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/04\/portfolio-optimizer-formula.png\" alt=\"\" class=\"wp-image-187971 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/04\/portfolio-optimizer-formula.png 616w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/04\/portfolio-optimizer-formula-300x44.png 300w\" data-sizes=\"(max-width: 616px) 100vw, 616px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 616px; aspect-ratio: 616\/90;\" \/><\/figure>\n\n\n\n<p>, with&nbsp;<em>\u03c4<sub>t\u22121\u200b<\/sub><\/em>&nbsp;the fraction of a semiannual period until the next coupon payment at the end of month&nbsp;<em>t<\/em>\u22121.<\/p>\n\n\n\n<p>At the end of the month&nbsp;<em>t<\/em>, the bond has the following characteristics:<\/p>\n\n\n\n<p>\u2014 Its remaining maturity is equal to&nbsp;<img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-3.png\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\">\u200b, one month short of its initial remaining maturity&nbsp;<em>T<\/em><br>\u2014 Its coupon rate&nbsp;<em>c<sub>t<\/sub><\/em>\u200b&nbsp;is equal to&nbsp;<em>c<sub>t\u22121\u200b<\/sub><\/em>, that is, its initial yield to maturity&nbsp;<em>y<sub>t\u22121<\/sub><\/em>\u200b<br>\u2014 Its yield to maturity is equal to&nbsp;<em>y<sub>t<\/sub><\/em>\u200b, because of the assumption on the bond yield curve at the end of the month&nbsp;<em>t<\/em><\/p>\n\n\n\n<p>Its price&nbsp;<img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/04\/portfolio-optimizer-bond-formula.png\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\">&nbsp;is then equal, through the bond yield formula, to<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"954\" height=\"165\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-4.png\" alt=\"\" class=\"wp-image-187539 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-4.png 954w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-4-700x121.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-4-300x52.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-4-768x133.png 768w\" data-sizes=\"(max-width: 954px) 100vw, 954px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 954px; aspect-ratio: 954\/165;\" \/><\/figure>\n\n\n\n<p><a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#par-bond-total-return-formula\"><\/a><\/p>\n\n\n\n<p>, with&nbsp;<em>\u03c4<sub>t<\/sub><\/em>\u200b&nbsp;the fraction of a semiannual period until the next coupon payment at the end of month&nbsp;<em>t<\/em>.<\/p>\n\n\n\n<p>The total return&nbsp;<em>TR<sub>t<\/sub><\/em>\u200b&nbsp;of this bond from the end of the month&nbsp;<em>t<\/em>\u22121&nbsp;to the end of the month <em>t<\/em>&nbsp;is then by definition equal to<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"922\" height=\"125\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-5.png\" alt=\"\" class=\"wp-image-187541 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-5.png 922w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-5-700x95.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-5-300x41.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-5-768x104.png 768w\" data-sizes=\"(max-width: 922px) 100vw, 922px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 922px; aspect-ratio: 922\/125;\" \/><\/figure>\n\n\n\n<p>, that is<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"972\" height=\"183\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-6.png\" alt=\"\" class=\"wp-image-187542 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-6.png 972w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-6-700x132.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-6-300x56.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-6-768x145.png 768w\" data-sizes=\"(max-width: 972px) 100vw, 972px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 972px; aspect-ratio: 972\/183;\" \/><\/figure>\n\n\n\n<p>The first term of this expression corresponds to the re-investment of the accrued interest.<\/p>\n\n\n\n<p>Under the practical assumptions that<\/p>\n\n\n\n<p>\u2014 There is only a single rate for accrued interest, chosen equal to&nbsp;<em>y<sub>t\u22121<\/sub><\/em><sup>5<\/sup><\/p>\n\n\n\n<p>\u2014 The accrued interest is not re-invested<sup>6<\/sup><\/p>\n\n\n\n<p>and noticing that <img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/04\/portfolio-optimizer-bond-formula-2.png\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\"> \u200b\u200b<sup>7<\/sup>, this expression becomes<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"911\" height=\"254\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-7.png\" alt=\"\" class=\"wp-image-187544 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-7.png 911w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-7-700x195.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-7-300x84.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-7-768x214.png 768w\" data-sizes=\"(max-width: 911px) 100vw, 911px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 911px; aspect-ratio: 911\/254;\" \/><\/figure>\n\n\n\n<p>Finally, by linearizing the accrued interest through the first-order Taylor approximation&nbsp;<img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/04\/portfolio-optimizer-bond-formula-3.png\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\">\u200b\u200b, this expression becomes<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"903\" height=\"245\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-8.png\" alt=\"\" class=\"wp-image-187546 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-8.png 903w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-8-700x190.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-8-300x81.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/math-bond-formula-portfolio-optimizer-8-768x208.png 768w\" data-sizes=\"(max-width: 903px) 100vw, 903px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 903px; aspect-ratio: 903\/245;\" \/><\/figure>\n\n\n\n<p><strong><em>Remark:<\/em><\/strong><\/p>\n\n\n\n<p><em>The formula above is based on a suggestion by Dr Winfried Hallerbach to improve the accuracy of the initial formula used in Swinkels<sup>2<\/sup>&nbsp;which is based on a second-order Taylor approximation of the bond yield formula, c.f. Swinkels<sup>8<\/sup>.<\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol class=\"wp-block-list\">\n<li>See&nbsp;<a href=\"https:\/\/etfdb.com\/news\/2022\/07\/25\/happy-20th-anniversary-of-bond-etfs\/\">Happy 20th Anniversary of Bond ETFs<\/a>.&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:1\">\u21a9<\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:1:1\">\u21a9<sup>2<\/sup><\/a><\/li>\n\n\n\n<li>See&nbsp;<a href=\"https:\/\/doi.org\/10.3390\/data4030091\">Swinkels, L., 2019, Treasury Bond Return Data Starting in 1962, Data 4(3), 91<\/a>.&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2\">\u21a9<\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:1\">\u21a9<sup>2<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:2\">\u21a9<sup>3<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:3\">\u21a9<sup>4<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:4\">\u21a9<sup>5<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:5\">\u21a9<sup>6<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:6\">\u21a9<sup>7<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:7\">\u21a9<sup>8<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:8\">\u21a9<sup>9<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:9\">\u21a9<sup>10<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:10\">\u21a9<sup>11<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:11\">\u21a9<sup>12<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:12\">\u21a9<sup>13<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:13\">\u21a9<sup>14<\/sup><\/a>&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:2:14\">\u21a9<sup>15<\/sup><\/a><\/li>\n\n\n\n<li>See&nbsp;<a href=\"https:\/\/www.wiley.com\/en-sg\/Fixed+Income+Securities:+Tools+for+Today's+Markets,+4th+Edition-p-9781119835554\">Tuckman, B., and Serrat A., 2022, Fixed Income Securities Tools for Today\u2019s Markets, 4th edition, John Wiley And Sons Ltd<\/a>.&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:3\">\u21a9<\/a><\/li>\n\n\n\n<li>In this post, I use the same conventions as in Tuckman and Serrat<sup><a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fn:3\">3<\/a><\/sup>: bonds are assumed to be paying semiannual coupons, their coupon rate is assumed to be annual, their yield to maturity is assumed to be provided as semiannually compounded and their maturity is assumed to be expressed in years.&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:5\">\u21a9<\/a><\/li>\n\n\n\n<li>Another sensible choice would be to use a rate equal to&nbsp;\u22121+22<em>yt<\/em>\u22121\u200b+<em>yt<\/em>\u200b\u200b.&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:6\">\u21a9<\/a><\/li>\n\n\n\n<li>Since bonds with semi-annual coupons are paying coupons every six months, these coupons are anyway hardly collected and re-invested every month in practice, so that this is a sensible simplifying assumption.&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:7\">\u21a9<\/a><\/li>\n\n\n\n<li>C.f. Tuckman and Serrat<sup><a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fn:3\">3<\/a><\/sup>&nbsp;for explanations about the term&nbsp;1661\u200b.&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:8\">\u21a9<\/a><\/li>\n\n\n\n<li>See&nbsp;<a href=\"https:\/\/doi.org\/10.25397\/eur.8152748\">Swinkels, L., 2023, Historical Data: International monthly government bond returns, Erasmus University Rotterdam (EUR)<\/a>.&nbsp;<a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/#fnref:9\">\u21a9<\/a><\/li>\n<\/ol>\n\n\n\n<p><em>Visit <a href=\"https:\/\/portfoliooptimizer.io\/blog\/the-mathematics-of-bonds-simulating-the-returns-of-constant-maturity-government-bond-etfs\/\">Portfolio Optimizer<\/a> blog to read the full article.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I will show that the returns of specific bond ETFs &#8211; those seeking a constant maturity exposure to government-issued bonds &#8211; can be simulated using\u00a0standard textbook formulas\u00a0together with appropriate yields to maturity.<\/p>\n","protected":false},"author":186,"featured_media":181436,"comment_status":"closed","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,338,341,352,344],"tags":[4922,4941,8213],"contributors-categories":[15016],"class_list":{"0":"post-187526","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-ibkr-quant-news","9":"category-quant-development","10":"category-quant-north-america","11":"category-quant-regions","12":"tag-econometrics","13":"tag-financial-mathematics","14":"tag-portfolio-optimisation","15":"contributors-categories-portfolio-optimizer"},"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v27.3) - 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