present value
", "templateType": "anything", "definition": "par/(1+growthppdec)^n+couponamount/growthppdec*(1-1/(1+growthppdec)^n)", "name": "P", "group": "Ungrouped variables"}, "coupons": {"description": "", "templateType": "anything", "definition": "couponamount/growthppdec*(1-1/(1+growthppdec)^n)", "name": "coupons", "group": "Ungrouped variables"}, "growthpp": {"description": "interest per period, only use for debugging, use fractions in display and calculations.
", "templateType": "anything", "definition": "growth/period[1]", "name": "growthpp", "group": "Ungrouped variables"}, "period": {"description": "", "templateType": "anything", "definition": "random([random('half-yearly','semiannually'),2,'half'],['quarterly', 4,'a quarter'],['monthly',12,'a twelfth'])", "name": "period", "group": "Ungrouped variables"}, "Prounded": {"description": "", "templateType": "anything", "definition": "precround(P,2)", "name": "Prounded", "group": "Ungrouped variables"}, "growthppdec": {"description": "", "templateType": "anything", "definition": "growthpp/100", "name": "growthppdec", "group": "Ungrouped variables"}, "couponpp": {"description": "", "templateType": "anything", "definition": "couponinterest/period[1]", "name": "couponpp", "group": "Ungrouped variables"}, "par": {"description": "", "templateType": "anything", "definition": "random(list(10000..90000#10000)+list(100000..500000#100000)+list(1000..9000#1000))\n", "name": "par", "group": "Ungrouped variables"}, "n": {"description": "", "templateType": "anything", "definition": "years*period[1]", "name": "n", "group": "Ungrouped variables"}, "denc": {"description": "", "templateType": "anything", "definition": "period[1]*100/cfc", "name": "denc", "group": "Ungrouped variables"}, "numC": {"description": "", "templateType": "anything", "definition": "couponinterest*par/cfc", "name": "numC", "group": "Ungrouped variables"}, "couponinterest": {"description": "", "templateType": "anything", "definition": "random(5..10 except growth)", "name": "couponinterest", "group": "Ungrouped variables"}, "growthdec": {"description": "", "templateType": "anything", "definition": "growth/100", "name": "growthdec", "group": "Ungrouped variables"}, "growth": {"description": "interest per annum as a percentage (add the symbol afterwards)
", "templateType": "anything", "definition": "random(5..10)", "name": "growth", "group": "Ungrouped variables"}, "years": {"description": "", "templateType": "anything", "definition": "random(3..10)", "name": "years", "group": "Ungrouped variables"}, "couponppdec": {"description": "", "templateType": "anything", "definition": "couponpp/100", "name": "couponppdec", "group": "Ungrouped variables"}}, "variable_groups": [], "tags": [], "advice": "We are asked to find the present value of a debenture. Debentures have a face value (or par value), which is the amount the holder will get back at maturity, and a coupon amount, which is the interest paid to the holder each time period. To work out the present value of a debenture 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, or the market interest rate) per time period, $n$ is the number of time periods and $S$ is the face value of the debenture.
\n\nNote that the coupon interest rate, is a percentage of the face value 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, or the market interest rate).
\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, or the market interest rate).
\n", "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.
", "preamble": {"js": "", "css": ""}, "name": "present value - bonds -2", "rulesets": {}, "extensions": [], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["P", "Prounded", "years", "period", "n", "growth", "growthdec", "growthpp", "growthppdec", "par", "couponinterest", "coupondec", "couponpp", "couponppdec", "couponamount", "cfc", "numC", "denc", "coupons", "compound"], "parts": [{"showFeedbackIcon": true, "gaps": [{"showPrecisionHint": false, "precisionMessage": "You have not given your answer to the nearest cent.", "mustBeReducedPC": 0, "allowFractions": false, "marks": 1, "minValue": "P", "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "unitTests": [], "customName": "", "variableReplacements": [], "precision": "2", "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "precisionType": "dp", "showFeedbackIcon": true, "type": "numberentry", "customMarkingAlgorithm": "", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "maxValue": "P", "scripts": {}, "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": false, "showCorrectAnswer": true}], "type": "gapfill", "prompt": "Determine the current price of a debenture with a $\\$\\var{par}$ face value, a coupon rate of $\\var{couponinterest}\\%$ paid {period[0]}, with $\\var{years}$ years remaining to maturity and market interest rates at $\\var{growth}\\%$.
\n\n$\\$\\,$[[0]] (to the nearest cent)
", "customMarkingAlgorithm": "", "marks": 0, "useCustomName": false, "scripts": {}, "unitTests": [], "customName": "", "variableReplacements": [], "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}], "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/"}]}