interest per annum as a percentage (add the symbol afterwards)
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", "group": "Ungrouped variables", "name": "growthpp", "definition": "growth/period[1]"}, "par": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "par", "definition": "random(list(10000..90000#10000)+list(100000..500000#100000)+list(1000..9000#1000))\n"}, "growthppdec": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "growthppdec", "definition": "growthpp/100"}, "coupondec": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "coupondec", "definition": "couponinterest/100"}, "period": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "period", "definition": "random([random('half-yearly','semiannually'),2,'half'],['quarterly', 4,'a quarter'],['monthly',12,'a twelfth'])"}, "cfc": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "cfc", "definition": "gcf(couponinterest*par,period[1]*100)"}, "couponppdec": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "couponppdec", "definition": "couponpp/100"}, "Prounded": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "Prounded", "definition": "precround(P,2)"}, "n": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "n", "definition": "years*period[1]"}, "couponamount": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "couponamount", "definition": "couponppdec*par"}, "numC": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "numC", "definition": "couponinterest*par/cfc"}, "denc": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "denc", "definition": "period[1]*100/cfc"}, "couponpp": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "couponpp", "definition": "couponinterest/period[1]"}, "compound": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "compound", "definition": "par/(1+growthppdec)^n"}, "P": {"templateType": "anything", "description": "present value
", "group": "Ungrouped variables", "name": "P", "definition": "par/(1+growthppdec)^n+couponamount/growthppdec*(1-1/(1+growthppdec)^n)"}, "coupons": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "coupons", "definition": "couponamount/growthppdec*(1-1/(1+growthppdec)^n)"}}, "tags": [], "parts": [{"useCustomName": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "gaps": [{"mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "maxValue": "P", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "precision": "2", "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "useCustomName": false, "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the nearest cent.", "minValue": "P", "precisionType": "dp", "mustBeReducedPC": 0, "strictPrecision": false, "extendBaseMarkingAlgorithm": true, "customName": "", "scripts": {}, "correctAnswerStyle": "plain", "allowFractions": false, "unitTests": [], "correctAnswerFraction": false, "marks": 1, "customMarkingAlgorithm": "", "type": "numberentry"}], "customName": "", "scripts": {}, "prompt": "An A $\\var{couponinterest}\\%$ $\\$\\var{par}$ bond has $\\var{years}$ years until maturity and pays interest {period[0]}. Calculate the current price required to yield an investor $\\var{growth}\\%$ per annum compounded {period[0]}.
\n\n$\\$\\,$[[0]] (to the nearest cent)
", "variableReplacements": [], "unitTests": [], "sortAnswers": false, "marks": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "type": "gapfill"}], "rulesets": {}, "metadata": {"description": "Financial maths. Determine the present value for a bond to obtain a certain yield. This involves annuities and compound interest.", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "ungrouped_variables": ["P", "Prounded", "years", "period", "n", "growth", "growthdec", "growthpp", "growthppdec", "par", "couponinterest", "coupondec", "couponpp", "couponppdec", "couponamount", "cfc", "numC", "denc", "coupons", "compound"], "preamble": {"js": "", "css": ""}, "statement": "If you are unsure of how to do a question, click on Reveal answers to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again. Do each question repeatedly to ensure you have mastered it.
", "variable_groups": [], "extensions": [], "functions": {}, "advice": "We are asked to find the present value of a bond. Bonds have a face value (or par value), which is the amount the bondholder will get back at maturity, and a coupon amount, which is the interest paid to the bondholder each time period. To work out the present value of a bond we think of the regular interest payments as an ordinary annuity and use the present value of an ordinary annuity formula:
\n$\\displaystyle P=\\frac{C}{i}\\left(1-\\frac{1}{(1+i)^n}\\right)$
\nbut we also need to discount the face value payment which will occur at maturity, for this, we need the compound interest present value formula:
\n$\\displaystyle P=\\frac{S}{(1+i)^n}$
\nWe need to add the two resulting values so our formula is really
\n$\\displaystyle P=\\frac{C}{i}\\left(1-\\frac{1}{(1+i)^n}\\right)+\\frac{S}{(1+i)^n}$
\nwhere $P$ is the present value, $C$ is the coupon amount, $i$ is the investor's yield (or the required rate of return) per time period, $n$ is the number of time periods and $S$ is the face value of the bond.
\n\nNote that \"An \"A $\\var{couponinterest}\\%$ $\\$\\var{par}$ bond\" has a face value of $\\$\\var{par}$ and a coupon interest rate of $\\var{couponinterest}\\%$ of $\\$\\var{par}$ per annum. Since the coupons/interest are paid {period[0]} the coupon amount is actually {period[2]} of this amount, that is:
\n$\\begin{align} C&=\\frac{\\var{couponinterest}\\% \\text{ of } \\var{par}}{\\var{period[1]}}\\\\&=\\frac{\\var{coupondec}\\times\\var{par}}{\\var{period[1]}}\\\\&=\\simplify[unitDenominator]{{numC}/{denC}}\\end{align}$
\nwe also have
\n$\\displaystyle i=\\frac{\\var{growth}\\%}{\\var{period[1]}}=\\simplify[unitDenominator]{{growthdec}/{period[1]}}$,
\n$n=\\var{years}\\times\\var{period[1]}=\\var{n}$,
\n$S=\\var{par}$
\nand therefore
\n$\\begin{align}P&=\\frac{\\left(\\simplify[unitDenominator]{{numC}/{denC}}\\right)}{\\left(\\frac{\\var{growthdec}}{\\var{period[1]}}\\right)}\\left(1-\\frac{1}{\\left(1+\\frac{\\var{growthdec}}{\\var{period[1]}}\\right)^\\var{n}}\\right)+\\frac{\\var{par}}{\\left(1+\\frac{\\var{growthdec}}{\\var{period[1]}}\\right)^\\var{n}}\\\\&=\\$ \\var{Prounded} \\text{ (to the nearest cent)}\\end{align}$
\n\nNotice that the present value is greater than the face value, this is because the coupon interest rate is greater than the investor's yield (or the required rate of return).
\nNotice that the present value is less than the face value, this is because the coupon interest rate is less than the investor's yield (or the required rate of return).
\n", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}