// Numbas version: exam_results_page_options
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"Ungrouped variables", "definition": "{prob1}*(1+{in1}/100)^5+{prob2}*(1+{in2}/100)^5+{prob3}*(1+{in3}/100)^5", "templateType": "anything", "description": ""}, "ami": {"name": "ami", "group": "Ungrouped variables", "definition": "30000/(1+{ei3})^5", "templateType": "anything", "description": ""}, "int4": {"name": "int4", "group": "Ungrouped variables", "definition": "random(3..3#1)", "templateType": "randrange", "description": ""}, "int3": {"name": "int3", "group": "Ungrouped variables", "definition": "{int1}+3", "templateType": "anything", "description": ""}, "int1": {"name": "int1", "group": "Ungrouped variables", "definition": "random(5..7#1)", "templateType": "randrange", "description": ""}, "var": {"name": "var", "group": "Ungrouped variables", "definition": "{value}^2*({ei22}-{ei2}^2)", "templateType": "anything", "description": ""}, "premium": {"name": "premium", "group": "Ungrouped variables", "definition": "10000/(1+{Ei})^10", "templateType": "anything", "description": ""}, 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"ei22", "group": "Ungrouped variables", "definition": "((1/3)((1+{int4}/100)^2+(1+{int5}/100)^2+(1+{int6}/100)^2))(0.6(1+{int7}/100)^2+0.4(1+{int8}/100)^2)", "templateType": "anything", "description": ""}, "ei2": {"name": "ei2", "group": "Ungrouped variables", "definition": "((1/3)(3+({int4}+{int5}+{int6})/100))(0.6(1+{int7}/100)+0.4(1+{int8}/100))", "templateType": "anything", "description": ""}, "Ei": {"name": "Ei", "group": "Ungrouped variables", "definition": "({int1}/100)(0.3)+({int2}/100)(0.5)+({int3}/100)(0.2)", "templateType": "anything", "description": ""}, "int8": {"name": "int8", "group": "Ungrouped variables", "definition": "{int7}-1", "templateType": "anything", "description": ""}, "ei3": {"name": "ei3", "group": "Ungrouped variables", "definition": "{prob1}*({in1}/100)+{prob2}*({in2}/100)+{prob3}*({in3}/100)", "templateType": "anything", "description": ""}, "int6": {"name": "int6", "group": "Ungrouped variables", "definition": "{int5}+2", "templateType": "anything", "description": ""}, "prob3": {"name": "prob3", "group": "Ungrouped variables", "definition": "0.35", "templateType": "anything", "description": ""}, "in2": {"name": "in2", "group": "Ungrouped variables", "definition": "{in1}+2", "templateType": "anything", "description": ""}, "profit": {"name": "profit", "group": "Ungrouped variables", "definition": "({premium}*(0.3(1+({int1}/100))^10+0.5(1+({int2}/100))^10+0.2(1+({int3}/100))^10))-10000", "templateType": "anything", "description": ""}, "profit1": {"name": "profit1", "group": "Ungrouped variables", "definition": "{ami}*{ei33}-30000", "templateType": "anything", "description": ""}}, "advice": "<p>$S_n=(1+i_1)(1+i_2)...(1+i_n)$</p>\n<p>$E(S_n)=(1+j)^n$</p>\n<p>$var(S_n)=((1+j)^2+s^2)^n-(1+j)^{2n}$</p>\n<p>$E[(S_n)^k]=(E[(1+i_t)^k])^n$</p>", "statement": "", "parts": [{"showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "scripts": {}, "prompt": "<p>An insurance company calculates the single premium for a contract paying&nbsp;&pound;10,000 in ten years time as the present value of the benefit payable at the expected rate of interest it will earn on its fund. The annual effective rate of interest over the whole of the next ten years will be {int1}%, {int2}%, or {int3}% with probabilities 0.3, 0.5, and 0.2 respectively.</p>\n<p>a) Calculate the single premium.&nbsp;&pound;[[0]]</p>\n<p>b) Calculate the expected profit at the end of the term of the contract.&nbsp;&pound;[[1]]</p>", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "gaps": [{"showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "{premium}-1", "mustBeReducedPC": 0, "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{premium}+1", "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": ""}, {"showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "{profit}-1", "mustBeReducedPC": 0, "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{profit}+1", "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": ""}], "marks": 0}, {"showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "scripts": {}, "prompt": "<p>&pound;{value} is invested in a bank account which pays the interest at the end of each year. The rate of interest is fixed randomly at the beginning of each year and remains unchanged until the beginning of the next year. The rate of interest applicable in any one year is independent of the rate applicable in any other year. During the first year, the rate of interest per annum effective will be {int4}%, {int5}% or {int6}% with equal probability. During the second year, the rate of interest per annum effective will be either {int7}% with probability with 0.6 or {int8}% with probability 0.4</p>\n<p>a) Assuming that interest is always reinvested in the account, Calculate the expected accumulated amount in the bank account at the end of two years.&nbsp;&pound;[[0]]</p>\n<p>b) Calculate the variance of the accumulated amount in the bank account at the end of two years.&nbsp;&pound;[[1]]</p>", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "gaps": [{"showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "{eacc}-1", "mustBeReducedPC": 0, "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{eacc}+1", "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": ""}, {"showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "{var}-1", "mustBeReducedPC": 0, "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{var}+1", "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": ""}], "marks": 0}, {"showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "scripts": {}, "prompt": "<p>The rate of interest earned on a particular investment in a different years are independently and identically distributed. Each year the distribution of (1+ i<sub>t</sub> ) is log-normal, where it is the return in year t. The annual rates have mean of 5% and standard deviation of 10%.</p>\n<p>a) Calculate the expected value of an investment of&nbsp;&pound;2,000 after 10 years.&nbsp;&pound;[[0]]</p>\n<p>b) Calculate the probability that the accumulated of the investment after 10 years will be less than 90% of the expected value from a). [[1]]%</p>", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "gaps": [{"showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "3257", "mustBeReducedPC": 0, "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "3258", "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": ""}, {"showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "41", "mustBeReducedPC": 0, "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "43", "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": ""}], "marks": 0}, {"showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "scripts": {}, "prompt": "<p>A fund has a liability to pay&nbsp;&pound;30,000 at the end of five years. To meets its liability, the fund manager invests in five-year zero coupon bonds which provides any one of the annual effective return of {in1}% or {in2}% or {in3}% with probability {prob1}, {prob2}, and {prob3} respectively over the next five years. The fund manager invests the present value of the liability at the expected rate of interest it would earn on the zero-coupon bond.</p>\n<p>Calculate the amount invested and the expected profit of the fund due to this investment at the end of five years.</p>\n<p>Amount invested:&pound;[[0]]</p>\n<p>Expected profit:&pound;[[1]]</p>", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "gaps": [{"showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "{ami}-1", "mustBeReducedPC": 0, "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{ami}+1", "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": ""}, {"showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "{profit1}-1", "mustBeReducedPC": 0, "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{profit1}+1", "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": ""}], "marks": 0}], "tags": [], "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "preamble": {"js": "", "css": ""}, "extensions": [], "rulesets": {}, "type": "question", "contributors": [{"name": "Swa Lup", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2902/"}]}]}], "contributors": [{"name": "Swa Lup", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2902/"}]}