Chapter 3 – Financial Mathematics
[ad_1]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) Short‐term interest rates
(b) Long‐term interest rates
(c) Both (a) and (b)
(d) None of the above - If interest rates are based purely on expectations, what would we expect to see?
(a) Short‐term rates will be the geometric average of expected future short‐term rates.
(b) There will be an “upward bias” in the yield curve.
(c) Long‐term rates will be accurate predictors of future short‐term rates.
(d) None of the above - If 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 downward‐sloping.
(c) Both (a) and (b)
(d) None of the above
La Trobe University 4
SECTION B – SHORT ANSWER QUESTIONS - What 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 Pure
Expectations 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 5
SECTION C – ESSAY QUESTIONS - Describe the assumptions and predictions of the expectations theory of the term structure of interest rates?
La Trobe University 6 - Describe the assumptions and predictions of the liquidity premium theory of the term structure of interest
rates?
La Trobe University 7 - Why is an upward sloping yield curve observed more often than any other shape?
La Trobe University 8
SECTION D – CALCULATION QUESTIONS - What is the present value of an annuity where the payment is $600 per year for 4 years and the interest rate
is 11% p.a.? * - What is the present value of an annuity consisting of 4 annual payments of $600, with the first payment to be
received 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 be
received 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 monthly
withdrawals 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 your
100th birthday. *
La Trobe University 9 - 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 monthly
withdrawals 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 monthly
withdrawals 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 your
100th birthday (since you don’t 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 is
11% p.a.? - What is the future value of an annuity consisting of 4 annual payments of $600, with the first payment to be
received immediately, and an interest rate of 11% p.a.? *
La Trobe University 10 - What is the future value of an annuity consisting of 4 annual payments of $600, with the first payment to be
received 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 interest
rate 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 interest
rate 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 11 - You retire on your 65th birthday with a lump sum superannuation payout of $300,000. You expect to live to
- If you place this money into an account paying 12% p.a., compounding monthly, how much can you afford
to withdraw each month, if the first withdrawal is one month after your 65th birthday and the last withdrawal
is 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 in
order to save this amount by your 65th birthday, if the first annual payment is on you 23rd birthday and the
last 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 Effective
Annual 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 semi‐annually; NAB offers 7.7% p.a., compounded quarterly; and
Commonwealth Bank offers 7.6% compounded monthly. Which bank’s offer is best?
La Trobe University 12
Refer 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., compounded
monthly. 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 of
repayments? *
La Trobe University 13 - If the nominal interest rate is 11% and the expected inflation rate is 2.5%, what is the approximate real interest
rate? - 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 14
Hint, tips, advice and guidance
SECTION D – CALCULATION QUESTIONS - What is the present value of an annuity where the payment is $600 per year for 4 years and the interest rate
is 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 be
received 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 value
of that annuity (note carefully how many payments there are in that annuity) and then add that to the present
value of the first payment. - What is the present value of an annuity consisting of 4 annual payments of $600, with the first payment to be
received 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, using
the information given in the question, but bear in mind that the answer you get applies to the beginning of the
annuity period, which is one period before the first payment. If the annuity is not deferred, the beginning of
the annuity is today, and the first occurs one period in the future. Look at the question, work out how many
periods the annuity is deferred, and then treat the present value of the annuity as a single sum occurring that
many 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 strongly
recommended. - 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 monthly
withdrawals 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 your
100th birthday.
This requires the present value of an annuity, where the payment is $2000 per month and the interest rate is
12% p.a. (which you need to adjust for monthly compounding). Be careful in calculating the number of
payments 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 monthly
withdrawals 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 payments
in the previous question, there is an additional payment on your 65th birthday, which needs to be added to the
present 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 15 - 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 monthly
withdrawals 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 your
100th birthday (since you don’t 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 out
how many payments there are after your 65th birthday (because these payments constitute an annuity who
present value is as at your 65th birthday), calculate the present value of that annuity, and then add the extra
payment 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 received
immediately, 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 the
valuation, this should be interpreted as the future value as at the day of the last payment. Unlike a present
value calculation, it doesn’t matter when the payments begin, because they will just grow over the life of the
annuity (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 received
5 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 the
valuation, this should be interpreted as the future value as at the day of the last payment. Unlike a present
value calculation, it doesn’t matter when the payments begin, because they will just grow over the life of the
annuity (in this case, 4 years). - You retire on your 65th birthday with a lump sum superannuation payout of $300,000. You expect to live to
- If you place this money into an account paying 12% p.a., compounding monthly, how much can you afford
to withdraw each month, if the first withdrawal is one month after your 65th birthday and the last withdrawal
is 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 in
order to save this amount by your 65th birthday, if the first annual payment is on you 23rd birthday and the
last 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). You
need 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 16
Refer 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., compounded
monthly. 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 amount
borrowed. 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 payments
remain. - What will your monthly repayment be after the change in interest rate, for the remaining 18 years of
repayments?
This is based on the formula for the present value of an annuity, but we know the present value (it is the answer
to Question 87). You need to rearrange the formula to solve for the payment amount based on the new interest
rate.
La Trobe University 17
Solutions
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) Short‐term interest rates
(b) Long‐term interest rates
(c) Both (a) and (b)
(d) None of the above - If interest rates are based purely on expectations, what would we expect to see?
(a) Short‐term rates will be the geometric average of expected future short‐term rates.
(b) There will be an “upward bias” in the yield curve.
(c) Long‐term rates will be accurate predictors of future short‐term rates.
(d) None of the above - If 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 downward‐sloping.
(c) Both (a) and (b)
(d) None of the above
La Trobe University 18
SECTION B – SHORT ANSWER QUESTIONS - What 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. How
interest rates vary from short‐term to medium‐term to long‐term. Can be illustrated using a yield curve. Only
useful 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”?
Upward‐sloping. It is referred to as normal because it is the one most commonly observed.
The reason is probably because long‐term securities are seen as riskier than short‐term securities, so investors
have a preference for short‐term and long‐term borrowers need to offer higher rates for long‐term
investments, 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 Pure
Expectations Theory.
If there is difference between long‐term rates and expected average short‐term rates, borrowers and investors
will tend to invest or borrow either short term or long term – wherever they perceive an advantage (i.e. lower
borrowing costs or higher returns). Market forces will tend to move long‐term rates until they are geometric
average 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 cash
flows are further into the future and are discounted more to get a present value, and hence a change in the
discount rate has a greater effect on the present value). Borrowers who need to borrow long term will need
to offer slightly higher rates – higher than what would be indicated by pure expectations – in order to entice
investors 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, low
unemployment, an inflation rate between 2 and 3%, and a value for the dollar low enough to make exports
competitive. It attempts to influence these aspects of the economy through monetary policy – adjusting the
money supply to influence short term interest rates, which then generally flow on to longer term rates. The
money supply is managed by buying or selling government securities in the secondary market.
La Trobe University 19
SECTION D – CALCULATION QUESTIONS - What is the present value of an annuity where the payment is $600 per year for 4 years and the interest rate
is 11% p.a.?
4
1 1 1 600 1 1 1 $1861.47
1 0.11 1.11 n PV C
r r
- What is the present value of an annuity consisting of 4 payments of $600, with the first payment to be received
immediately, and an interest rate of 11% p.a.?
3
1 1 1 600 1 1 1 600 $2066.23
1 0.11 1.11 n PV C C
r r
- What is the present value of an annuity consisting of 4 payments of $600, with the first payment to be received
5 years from now, and an interest rate of 11% p.a.?
4 4
4
1 1 1 1 600 1 1 1 1.11 $1226.21
1 0.11 1.11 n PV C r
r r
Note: The cash flow in the previous question is an ordinary annuity deferred by 4 years (because the first
payment 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 monthly
withdrawals 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 your
100th birthday.
420
1 1 1 2000 1 1 1 $196,937.66
1 0.01 1.01 n PV C
r 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 monthly
withdrawals 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.
1 420
1 1 1 2000 1 1 1 2000 $198,937.66
1 0.01 1.01 n PV C C
r r
1 3 4 5 8 9
600
2 6 7
600 600 600
La Trobe University 20 - 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 monthly
withdrawals 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 your
100th birthday (since you don’t actually need any cash on that day).
419
1 1 1 2000 1 1 1 2000 $198,907.04
1 0.01 1.01 n PV C C
r r
- What is the future value of an annuity where the payment is $600 per year for 4 years and the interest rate is
11% p.a.?
1 1 1 600 1 1.11 4 1 $2825.84
0.11
n FV C r
r - What is the future value of an annuity consisting of 4 annual payments of $600, with the first payment to be
received immediately, and an interest rate of 11% p.a.?
1 1 1 600 1 1.11 4 1 $2825.84
0.11
n FV C r
r
Note: If you are asked for the “future value of an annuity”, without any other information as to the date of the
valuation, this should be interpreted as the future value as at the day of the last payment. Unlike a present
value calculation, it doesn’t matter when the payments begin, because they will just grow over the life of the
annuity (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 be
received 5 years from now, and an interest rate of 11% p.a.?
1 1 1 600 1 1.11 4 1 $2825.84
0.11
n FV C r
r
Note: If you are asked for the “future value of an annuity”, without any other information as to the date of the
valuation, this should be interpreted as the future value as at the day of the last payment. Unlike a present
value calculation, it doesn’t matter when the payments begin, because they will just grow over the life of the
annuity (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 interest
rate is 5% p.a., compounded quarterly?
11 1 3000 1 1.012528 1 $99,838.15
0.0125
n FV C r
r
- What is the future value of an annuity consisting of payments of $150 every week for 4 years, if the interest
rate is 8% p.a., compounded weekly?
4 52 1 1 1 150 1 1 0.08 1 $36,736.94
0.08 / 52 52
n FV C r
r
- 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%
100
n
FV PV r n r FV
PV
La Trobe University 21 - You retire on your 65th birthday with a lump sum superannuation payout of $300,000. You expect to live to
- If you place this money into an account paying 12% p.a., compounding monthly, how much can you afford
to withdraw each month, if the first withdrawal is one month after your 65th birthday and the last withdrawal
is on your 90th birthday?
300
1 1 1 300,000 $3159.67
1 1 1 1 1 1 1 0.01 1.01 1
n
n
PV C C PV
r r
r r
- You decide that you need to save $300,000 for your retirement. How much do you need to save each year in
order to save this amount by your 65th birthday, if the first annual payment is on you 23rd birthday and the
last 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 1
0.0775
n
n
FV C r C FV
r r
r
- What is the Equivalent Annual Rate if the Annual Percentage Rate is 14%, compounded fortnightly?
26 1 1 1 0.14 1 14.98%
26
m EAR APR
m
- You borrow $20,000 for 8 years at an interest rate of 6.6% compounded quarterly. What is the Effective
Annual Rate?
4 1 1 1 0.066 1 6.77%
4
m EAR APR
m - 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 semi‐annually; NAB offers 7.7% p.a., compounded quarterly; and
Commonwealth Bank offers 7.6% compounded monthly. Which bank’s 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 22
Refer 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., compounded
monthly. 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?
240
1 1 1
1
320,000 $2879.12
1 1 1 1 1 1 0.0075 1.0075 1
n
n
PV C
r r
C PV
r r
- How much do you owe the bank just after making the repayment due 2 years after taking out the loan?
216
1 1 1 2879.12 1 1 1 $307,452.17
1 0.0075 1.0075 n PV C
r r
- What will your monthly repayment be after the change in interest rate, for the remaining 18 years of
repayments?
216
1 1 1
1
307,452.17 $3074.04
1 1 1 1 1 1 0.008333 1.008333 1
n
n
PV C
r r
C PV
r r
- If the nominal interest rate is 11% and the expected inflation rate is 2.5%, what is the approximate real interest
rate?
0.11 0.025 8.5% n r r n r r πr r π - 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 1.025
n
n r r
r r π r r
π
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