Chapter 3 Financial Mathematics

SECTION A MULTIPLE CHOICE QUESTIONS Assuming that inflation is not negative, which of the following statements is true?(a) The real interest rate will always be greater than the nominal interest rate.(b) The nominal interest rate will always be greater than the real interest rate.(c) The real interest rate cannot be less than the nominal interest rate.(d) The nominal interest rate cannot be less than the real interest rate.Which of the following does the RBA focus on when it implements fiscal policy?(a) Shortterm interest rates(b) Longterm interest rates(c) Both (a) and (b)(d) None of the aboveIf interest rates are based purely on expectations, what would we expect to see?(a) Shortterm rates will be the geometric average of expected future shortterm rates.(b) There will be an upward bias in the yield curve.(c) Longterm rates will be accurate predictors of future shortterm rates.(d) None of the aboveIf investing long term is seen as riskier than investing short term, what do we expect to see?(a) There will be an upward bias in the yield curve.(b) It is not possible for the yield curve to be downwardsloping.(c) Both (a) and (b)(d) None of the aboveLa Trobe University 4SECTION B SHORT ANSWER QUESTIONSWhat is meant by the term structure of interest rates?What is meant by a normal yield curve? Why is it referred to as normal?Briefly explain how long term rates become the average of expected future short term rates under the PureExpectations Theory.Briefly explain how the yield curve has an upward bias under the Liquidity Premium Theory.Briefly describe how interest rates are determined in Australia.La Trobe University 5SECTION C ESSAY QUESTIONSDescribe the assumptions and predictions of the expectations theory of the term structure of interest rates?La Trobe University 6Describe the assumptions and predictions of the liquidity premium theory of the term structure of interestrates?La Trobe University 7Why is an upward sloping yield curve observed more often than any other shape?La Trobe University 8SECTION D CALCULATION QUESTIONSWhat is the present value of an annuity where the payment is $600 per year for 4 years and the interest rateis 11% p.a.? *What is the present value of an annuity consisting of 4 annual payments of $600, with the first payment to bereceived immediately, and an interest rate of 11% p.a.? *What is the present value of an annuity consisting of 4 annual payments of $600, with the first payment to bereceived 5 years from now, and an interest rate of 11% p.a.? *You intend to retire at age 65. You will place your retirement savings into a bank account paying 12% p.a.,compounded monthly. How much do you need to save for your retirement in order to make monthlywithdrawals of $2000 from the account for the rest of your life if you expect to live to 100 years old exactly?The first withdrawal would be one month after your 65th birthday and the last withdrawal would be on your100th birthday. *La Trobe University 9You intend to retire at age 65. You will place your retirement savings into a bank account paying 12% p.a.,compounded monthly. How much do you need to save for your retirement in order to make monthlywithdrawals of $2000 from the account for the rest of your life if you expect to live to 100 years old exactly?The first withdrawal would be on your 65th birthday and the last withdrawal would be on your 100th birthday.*You intend to retire at age 65. You will place your retirement savings into a bank account paying 12% p.a.,compounded monthly. How much do you need to save for your retirement in order to make monthlywithdrawals of $2000 from the account for the rest of your life if you expect to live to 100 years old exactly?The first withdrawal would be on your 65th birthday and the last withdrawal would be one month before your100th birthday (since you dont actually need any cash on that day). *What is the future value of an annuity where the payment is $600 per year for 4 years and the interest rate is11% p.a.?What is the future value of an annuity consisting of 4 annual payments of $600, with the first payment to bereceived immediately, and an interest rate of 11% p.a.? *La Trobe University 10What is the future value of an annuity consisting of 4 annual payments of $600, with the first payment to bereceived 5 years from now, and an interest rate of 11% p.a.? *What is the future value of an annuity consisting of payments of $3000 every quarter for 7 years, if the interestrate is 5% p.a., compounded quarterly?What is the future value of an annuity consisting of payments of $150 every week for 4 years, if the interestrate is 8% p.a., compounded weekly?If a sum of money grows from $100 to $500 in 10 years, what is the rate of return or growth rate?La Trobe University 11You retire on your 65th birthday with a lump sum superannuation payout of $300,000. You expect to live toIf you place this money into an account paying 12% p.a., compounding monthly, how much can you affordto withdraw each month, if the first withdrawal is one month after your 65th birthday and the last withdrawalis on your 90th birthday? *You decide that you need to save $300,000 for your retirement. How much do you need to save each year inorder to save this amount by your 65th birthday, if the first annual payment is on you 23rd birthday and thelast payment is on your 65th birthday, and the interest rate is 7.75% p.a., compounded annually? *What is the Equivalent Annual Rate if the Annual Percentage Rate is 14%, compounded fortnightly?You borrow $20,000 for 8 years at an interest rate of 6.6% compounded quarterly. What is the EffectiveAnnual Rate?You are shopping around for term loans. ANZ offers you an interest rate of 8% p.a., compounded annually;Westpac offers 7.8% p.a., compounded semiannually; NAB offers 7.7% p.a., compounded quarterly; andCommonwealth Bank offers 7.6% compounded monthly. Which banks offer is best?La Trobe University 12Refer to the following information in answering Questions 28 to 30.You take out a $320,000 mortgage loan, repayable over 20 years at an initial interest rate of 9% p.a., compoundedmonthly. After 2 years of repayments, the bank advises you that the interest rate will increase to 10% p.a.,compounded monthly.What is the monthly repayment when you first take out the loan? *How much do you owe the bank just after making the repayment due 2 years after taking out the loan? *What will your monthly repayment be after the change in interest rate, for the remaining 18 years ofrepayments? *La Trobe University 13If the nominal interest rate is 11% and the expected inflation rate is 2.5%, what is the approximate real interestrate?If the nominal interest rate is 11% and the expected inflation rate is 2.5%, what is the exact real interest rate?La Trobe University 14Hint, tips, advice and guidanceSECTION D CALCULATION QUESTIONSWhat is the present value of an annuity where the payment is $600 per year for 4 years and the interest rateis 11% p.a.?This requires the present value of an annuity.What is the present value of an annuity consisting of 4 annual payments of $600, with the first payment to bereceived immediately, and an interest rate of 11% p.a.?All payments after the first payment constitute an ordinary annuity, so you should calculate the present valueof that annuity (note carefully how many payments there are in that annuity) and then add that to the presentvalue of the first payment.What is the present value of an annuity consisting of 4 annual payments of $600, with the first payment to bereceived 5 years from now, and an interest rate of 11% p.a.?This is what is called a deferred annuity. You should use the formula for the present value of an annuity, usingthe information given in the question, but bear in mind that the answer you get applies to the beginning of theannuity period, which is one period before the first payment. If the annuity is not deferred, the beginning ofthe annuity is today, and the first occurs one period in the future. Look at the question, work out how manyperiods the annuity is deferred, and then treat the present value of the annuity as a single sum occurring thatmany periods in the future. Discount it to a present value at the interest rate given in the question.Any time there is anything slightly unusual happening with the timing of cash flows, a time line is stronglyrecommended.You intend to retire at age 65. You will place your retirement savings into a bank account paying 12% p.a.,compounded monthly. How much do you need to save for your retirement in order to make monthlywithdrawals of $2000 from the account for the rest of your life if you expect to live to 100 years old exactly?The first withdrawal would be one month after your 65th birthday and the last withdrawal would be on your100th birthday.This requires the present value of an annuity, where the payment is $2000 per month and the interest rate is12% p.a. (which you need to adjust for monthly compounding). Be careful in calculating the number ofpayments in the annuity.You intend to retire at age 65. You will place your retirement savings into a bank account paying 12% p.a.,compounded monthly. How much do you need to save for your retirement in order to make monthlywithdrawals of $2000 from the account for the rest of your life if you expect to live to 100 years old exactly?The first withdrawal would be on your 65th birthday and the last withdrawal would be on your 100th birthday.This is similar to the previous question, but the number of payments is different. In addition to the paymentsin the previous question, there is an additional payment on your 65th birthday, which needs to be added to thepresent value of the annuity of payments received after your 65th birthday to get the total value (as at age 65)of all payments.La Trobe University 15You intend to retire at age 65. You will place your retirement savings into a bank account paying 12% p.a.,compounded monthly. How much do you need to save for your retirement in order to make monthlywithdrawals of $2000 from the account for the rest of your life if you expect to live to 100 years old exactly?The first withdrawal would be on your 65th birthday and the last withdrawal would be one month before your100th birthday (since you dont actually need any cash on that day).Again, this is similar to the previous question, but the number of payments is different. You need to work outhow many payments there are after your 65th birthday (because these payments constitute an annuity whopresent value is as at your 65th birthday), calculate the present value of that annuity, and then add the extrapayment that occurs on your 65th birthday to determine the total value of all payments.What is the future value of an annuity consisting of 4 payments of $600, with the first payment to be receivedimmediately, and an interest rate of 11% p.a.?If you are asked for the future value of an annuity, without any other information as to the date of thevaluation, this should be interpreted as the future value as at the day of the last payment. Unlike a presentvalue calculation, it doesnt matter when the payments begin, because they will just grow over the life of theannuity (in this case, 4 years).What is the future value of an annuity consisting of 4 payments of $600, with the first payment to be received5 years from now, and an interest rate of 11% p.a.?If you are asked for the future value of an annuity, without any other information as to the date of thevaluation, this should be interpreted as the future value as at the day of the last payment. Unlike a presentvalue calculation, it doesnt matter when the payments begin, because they will just grow over the life of theannuity (in this case, 4 years).You retire on your 65th birthday with a lump sum superannuation payout of $300,000. You expect to live toIf you place this money into an account paying 12% p.a., compounding monthly, how much can you affordto withdraw each month, if the first withdrawal is one month after your 65th birthday and the last withdrawalis on your 90th birthday?This is based on the formula for the present value of an annuity, but we know the present value ($300,000).You need to rearrange the formula to solve for the payment amount.You decide that you need to save $300,000 for your retirement. How much do you need to save each year inorder to save this amount by your 65th birthday, if the first annual payment is on you 23rd birthday and thelast payment is on your 65th birthday, and the interest rate is 7.75% p.a., compounded annually?This is based on the formula for the future value of an annuity, but we know the future value ($300,000). Youneed to rearrange the formula to solve for the payment amount.Be careful when determining how many payments there are this is tricky. You may find a timeline helpful.La Trobe University 16Refer to the following information in answering Questions 28 to 30.You take out a $320,000 mortgage loan, repayable over 20 years at an initial interest rate of 9% p.a., compoundedmonthly. After 2 years of repayments, the bank advises you that the interest rate will increase to 10% p.a.,compounded monthly.What is the monthly repayment when you first take out the loan?This is based on the formula for the present value of an annuity, but we know the present value the amountborrowed. You need to rearrange the formula to solve for the payment.How much do you owe the bank just after making the repayment due 2 years after taking out the loan?The amount owed at any time is the present value of the remaining repayments at the original interest rate.You know the original interest rate and the payment amount, and you can work out how many paymentsremain.What will your monthly repayment be after the change in interest rate, for the remaining 18 years ofrepayments?This is based on the formula for the present value of an annuity, but we know the present value (it is the answerto Question 87). You need to rearrange the formula to solve for the payment amount based on the new interestrate.La Trobe University 17SolutionsSECTION A MULTIPLE CHOICE QUESTIONSAssuming that inflation is not negative, which of the following statements is true?(a) The real interest rate will always be greater than the nominal interest rate.(b) The nominal interest rate will always be greater than the real interest rate.(c) The real interest rate cannot be less than the nominal interest rate.(d) The nominal interest rate cannot be less than the real interest rate.Which of the following does the RBA focus on when it implements fiscal policy?(a) Shortterm interest rates(b) Longterm interest rates(c) Both (a) and (b)(d) None of the aboveIf interest rates are based purely on expectations, what would we expect to see?(a) Shortterm rates will be the geometric average of expected future shortterm rates.(b) There will be an upward bias in the yield curve.(c) Longterm rates will be accurate predictors of future shortterm rates.(d) None of the aboveIf investing long term is seen as riskier than investing short term, what do we expect to see?(a) There will be an upward bias in the yield curve.(b) It is not possible for the yield curve to be downwardsloping.(c) Both (a) and (b)(d) None of the aboveLa Trobe University 18SECTION B SHORT ANSWER QUESTIONSWhat is meant by the term structure of interest rates?The pattern of interest rates that is currently available for investments with different terms to maturity. Howinterest rates vary from shortterm to mediumterm to longterm. Can be illustrated using a yield curve. Onlyuseful if the securities or investments are identical in every respect except for term to maturity.What is meant by a normal yield curve? Why is it referred to as normal?Upwardsloping. It is referred to as normal because it is the one most commonly observed.The reason is probably because longterm securities are seen as riskier than shortterm securities, so investorshave a preference for shortterm and longterm borrowers need to offer higher rates for longterminvestments, resulting in an upward bias of the yield curve.Briefly explain how long term rates become the average of expected future short term rates under the PureExpectations Theory.If there is difference between longterm rates and expected average shortterm rates, borrowers and investorswill tend to invest or borrow either short term or long term wherever they perceive an advantage (i.e. lowerborrowing costs or higher returns). Market forces will tend to move longterm rates until they are geometricaverage of expected short term rates, eliminating any perceived advantage.Briefly explain how the yield curve has an upward bias under the Liquidity Premium Theory.Investors see long term investments as riskier than short term, and will therefore tend to prefer short term.This could be because long term investments are more sensitive to interest rate changes (because the cashflows are further into the future and are discounted more to get a present value, and hence a change in thediscount rate has a greater effect on the present value). Borrowers who need to borrow long term will needto offer slightly higher rates higher than what would be indicated by pure expectations in order to enticeinvestors to borrow long term.Briefly describe how interest rates are determined in Australia.The RBA Board meets monthly and monitors the economy. It targets moderate economic growth, lowunemployment, an inflation rate between 2 and 3%, and a value for the dollar low enough to make exportscompetitive. It attempts to influence these aspects of the economy through monetary policy adjusting themoney supply to influence short term interest rates, which then generally flow on to longer term rates. Themoney supply is managed by buying or selling government securities in the secondary market.La Trobe University 19SECTION D CALCULATION QUESTIONSWhat is the present value of an annuity where the payment is $600 per year for 4 years and the interest rateis 11% p.a.? 41 1 1 600 1 1 1 $1861.471 0.11 1.11 n PV Cr r What is the present value of an annuity consisting of 4 payments of $600, with the first payment to be receivedimmediately, and an interest rate of 11% p.a.? 31 1 1 600 1 1 1 600 $2066.231 0.11 1.11 n PV C Cr r What is the present value of an annuity consisting of 4 payments of $600, with the first payment to be received5 years from now, and an interest rate of 11% p.a.? 4 441 1 1 1 600 1 1 1 1.11 $1226.211 0.11 1.11 n PV C rr r Note: The cash flow in the previous question is an ordinary annuity deferred by 4 years (because the firstpayment is in 5 years instead of 1 year).You intend to retire at age 65. You will place your retirement savings into a bank account paying 12% p.a.,compounded monthly. How much do you need to save for your retirement in order to make monthlywithdrawals of $2000 from the account for the rest of your life if you expect to live to 100 years old exactly?The first withdrawal would be one month after your 65th birthday and the last withdrawal would be on your100th birthday. 4201 1 1 2000 1 1 1 $196,937.661 0.01 1.01 n PV Cr r You intend to retire at age 65. You will place your retirement savings into a bank account paying 12% p.a.,compounded monthly. How much do you need to save for your retirement in order to make monthlywithdrawals of $2000 from the account for the rest of your life if you expect to live to 100 years old exactly?The first withdrawal would be on your 65th birthday and the last withdrawal would be on your 100th birthday. 1 4201 1 1 2000 1 1 1 2000 $198,937.661 0.01 1.01 n PV C Cr r1 3 4 5 8 96002 6 7600 600 600La Trobe University 20You intend to retire at age 65. You will place your retirement savings into a bank account paying 12% p.a.,compounded monthly. How much do you need to save for your retirement in order to make monthlywithdrawals of $2000 from the account for the rest of your life if you expect to live to 100 years old exactly?The first withdrawal would be on your 65th birthday and the last withdrawal would be one month before your100th birthday (since you dont actually need any cash on that day). 4191 1 1 2000 1 1 1 2000 $198,907.041 0.01 1.01 n PV C Cr r What is the future value of an annuity where the payment is $600 per year for 4 years and the interest rate is11% p.a.? 1 1 1 600 1 1.11 4 1 $2825.840.11n FV C rrWhat is the future value of an annuity consisting of 4 annual payments of $600, with the first payment to bereceived immediately, and an interest rate of 11% p.a.? 1 1 1 600 1 1.11 4 1 $2825.840.11n FV C rrNote: If you are asked for the future value of an annuity, without any other information as to the date of thevaluation, this should be interpreted as the future value as at the day of the last payment. Unlike a presentvalue calculation, it doesnt matter when the payments begin, because they will just grow over the life of theannuity (in this case, 4 years).What is the future value of an annuity consisting of 4 annual payments of $600, with the first payment to bereceived 5 years from now, and an interest rate of 11% p.a.? 1 1 1 600 1 1.11 4 1 $2825.840.11n FV C rrNote: If you are asked for the future value of an annuity, without any other information as to the date of thevaluation, this should be interpreted as the future value as at the day of the last payment. Unlike a presentvalue calculation, it doesnt matter when the payments begin, because they will just grow over the life of theannuity (in this case, 4 years).What is the future value of an annuity consisting of payments of $3000 every quarter for 7 years, if the interestrate is 5% p.a., compounded quarterly?11 1 3000 1 1.012528 1 $99,838.150.0125n FV C rr What is the future value of an annuity consisting of payments of $150 every week for 4 years, if the interestrate is 8% p.a., compounded weekly? 4 52 1 1 1 150 1 1 0.08 1 $36,736.940.08 / 52 52n FV C rr If a sum of money grows from $100 to $500 in 10 years, what is the rate of return or growth rate? 1/ 1/10 1 1 500 1 17.46%100nFV PV r n r FVPV La Trobe University 21You retire on your 65th birthday with a lump sum superannuation payout of $300,000. You expect to live toIf you place this money into an account paying 12% p.a., compounding monthly, how much can you affordto withdraw each month, if the first withdrawal is one month after your 65th birthday and the last withdrawalis on your 90th birthday? 3001 1 1 300,000 $3159.671 1 1 1 1 1 1 0.01 1.01 1nnPV C C PVr rr r You decide that you need to save $300,000 for your retirement. How much do you need to save each year inorder to save this amount by your 65th birthday, if the first annual payment is on you 23rd birthday and thelast payment is on your 65th birthday, and the interest rate is 7.75% p.a., compounded annually? 43 1 1 1 300,000 $978.08 1 1 1 1 1.0775 10.0775nnFV C r C FVr rr What is the Equivalent Annual Rate if the Annual Percentage Rate is 14%, compounded fortnightly?26 1 1 1 0.14 1 14.98%26m EAR APRm You borrow $20,000 for 8 years at an interest rate of 6.6% compounded quarterly. What is the EffectiveAnnual Rate? 4 1 1 1 0.066 1 6.77%4m EAR APRmYou are shopping around for term loans. ANZ offers you an interest rate of 8% p.a., compounded annually;Westpac offers 7.8% p.a., compounded semiannually; NAB offers 7.7% p.a., compounded quarterly; andCommonwealth Bank offers 7.6% compounded monthly. Which banks offer is best?8.00% A EAR 2 1 0.078 1 7.95%W 2 EAR 4 1 0.077 1 7.93%N 4 EAR 12 1 0.076 1 7.87%C 12 EAR CBA offers the lowest effective annual interest rate.La Trobe University 22Refer to the following information in answering Questions 28 to 32.You take out a $320,000 mortgage loan, repayable over 20 years at an initial interest rate of 9% p.a., compoundedmonthly. After 2 years of repayments, the bank advises you that the interest rate will increase to 10% p.a.,compounded monthly.What is the monthly repayment when you first take out the loan? 2401 1 11320,000 $2879.121 1 1 1 1 1 0.0075 1.0075 1nnPV Cr rC PVr r How much do you owe the bank just after making the repayment due 2 years after taking out the loan? 2161 1 1 2879.12 1 1 1 $307,452.171 0.0075 1.0075 n PV Cr r What will your monthly repayment be after the change in interest rate, for the remaining 18 years ofrepayments? 2161 1 11307,452.17 $3074.041 1 1 1 1 1 0.008333 1.008333 1nnPV Cr rC PVr r If the nominal interest rate is 11% and the expected inflation rate is 2.5%, what is the approximate real interestrate?0.11 0.025 8.5% n r r n r r r r If the nominal interest rate is 11% and the expected inflation rate is 2.5%, what is the exact real interest rate?1 1 1 1 1 1.11 1 8.29%1 1.025nn r rr r r r

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