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Recursion and Finance, GEN2 2024 VCAA 6

Emi invested profits of $10 000 into a savings account that earns interest compounding fortnightly, for one year.

The effective interest rate, rounded to two decimal places, is 5.07%.

Assume that there are exactly 26 fortnights in a year.

  1. What is the nominal percentage rate of interest for the account?
  2. Round your answer to two decimal places.   (1 mark)

  3. Explain why the nominal interest rate appears lower than the effective interest rate.  (1 mark)

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a.    \(4.95\%\ \text{(2 d.p.)}\)

b.    \(\text{Nominal and effective interest rates are equal if there is only one}\)

\(\text{compounding period per year. If there are more compounding}\)

\(\text{periods (as in this example) the effective rate will be higher than}\)

\(\text{the nominal rate.}\)

Show Worked Solution

a.    \(\text{Using CAS: FINANCE}\ \rightarrow \ \text{Interest Conversion}\ \rightarrow \ \text{Nominal Interest Rate}\)

\(\text{nom}(5.07, 26) = 4.95037\%\approx 4.95\%\ \text{(2 d.p.)}\)
 

\(\text{Using formula:}\)

Mean mark (a) 51%.
\(r_e\) \(=\left[\left(1+\dfrac{r}{100n}\right)^n-1\right]\times 100\%\)
\(\dfrac{5.07}{100}\) \(=\left[\left(1+\dfrac{r}{2600}\right)^{26}-1\right]\)
\(\left(1+\dfrac{r}{2600}\right)^{26}\) \(=1.0507\)
\(\dfrac{r}{2600}\) \(=(1.0507)^\frac{1}{26}-1\)
\(r\) \(=\left((1.0507)^\frac{1}{26}-1\right)\times 2600\)
  \(=4.95036\dots\approx 4.95\%\ \text{(2 d.p.)}\)

 
b.    \(\text{Nominal and effective interest rates are equal if there is only one}\)

\(\text{compounding period per year. If there are more compounding}\)

\(\text{periods (as in this example) the effective rate will be higher than}\)

\(\text{the nominal rate.}\)

♦♦♦ Mean mark (b) 26%.

Recursion and Finance, GEN1 2022 VCAA 21 MC

Consider the following four statements regarding nominal and effective interest rates as they apply to compound interest investments and loans:

  • An effective interest rate is the same as a nominal interest rate if interest compounds annually.
  • Effective interest rates increase as the number of compounding periods per year increases.
  • A nominal rate of 12% per annum is equivalent to a nominal rate of 1% per month.
  • An effective interest rate can be lower than a nominal interest rate.

How many of these four statements are true?

  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
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\(D\)

Show Worked Solution

\(\text{Statement 1}\)

\(\text{Let}\ \ n = 1\ \ \text{and}\ \ r = 12\% :\)

\(R_{eff}=\Big{(}1 + \dfrac{12}{100 \times 1}\Big{)}^1-1=12\%\ \ \checkmark\)

 
\(\text{Statement 2}\)

\(\text{Let}\ \ n = 2\ \ \text{and}\ \ r = 12\% :\)

\(R_{eff}=\Big{(}1 + \dfrac{12}{100 \times 2}\Big{)}^2-1=12.36\%\ \ \checkmark \)

 
\(\text{Statement 3}\)

\(\dfrac{12\text{%}}{12\ \text{months}} = 1\text{%} \ \text{per month}\ \ \checkmark\)

  
\(\text{Statement 4}\)

\(\text{Statements 1 and 2 make the 4th statement incorrect.}\)

\(\Rightarrow D\)


♦♦ Mean mark 33%.

CORE, FUR1 2021 VCAA 21 MC

Enrico invests $3000 in an account that pays interest compounding monthly.

After four years, the balance of the account is $3728.92

The effective annual interest rate for this investment, rounded to two decimal places, is

  1. 5.45%
  2. 5.52%
  3. 5.56%
  4. 5.59%
  5. 5.60%
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`D`

Show Worked Solution

`text{By TVM Solver:}`

♦ Mean mark 40%.
`N` `= 48`
`I(%)` `= ?`
`PV` `= -3000`
`PMT` `= 0`
`FV` `= 3728.92`
`text(P/Y)` `= text(C/Y) = 12`

 
`=> I(%) = 5.45text(%)`

`r_text{effective}` `= [(1 + {5.45}/{100 xx 12})^12 -1] xx 100text(%)`
  `= 5.588 …%`

`=> D`

CORE, FUR1 2020 VCAA 28 MC

The nominal interest rate for a loan is 8% per annum.

When rounded to two decimal places, the effective interest rate for this loan is not

  1. 8.33% per annum when interest is charged daily.
  2. 8.32% per annum when interest is charged weekly.
  3. 8.31% per annum when interest is charged fortnightly.
  4. 8.30% per annum when interest is charged monthly.
  5. 8.24% per annum when interest is charged quarterly.
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`C`

Show Worked Solution

`text(Consider option C:)`

`r_text(effective)` `= [(1 + 8/(100 xx 26))^26 – 1] xx 100`
  `~~ 8.3154`
  `~~ 8.32%`

 
 `=>  C`

CORE, FUR1 2019 VCAA 24 MC

Millie invested $20 000 in an account at her bank with interest compounding monthly.

After one year, the balance of Millie’s account was $20 732.

The difference between the rate of interest per annum used by her bank and the effective annual rate of interest for Millie’s investment is closest to

  1. 0.04%
  2. 0.06%
  3. 0.08%
  4. 0.10%
  5. 0.12%
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`B`

Show Worked Solution

`text(Let)\ \ r = text(bank annual rate)`

`20\ 000 (1 + r/12)^12` `= 20\ 732`
`(1 + r/12)^12` `= (20\ 732)/(20\ 000)`
`1 + r/12` `= 1.003`
`r` `= 0.03600`
  `= 3.6text(%)`

 

`r_text(effective)` `= [(1 + r/(100 xx 12))^12 – 1] xx 100text(%)`
  `= 3.66text(%)`

 
`:.\ text(Difference) = 0.06text(%)`

`=>  B`