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Date	May 2022	Marks available	3	Reference code	22M.1.SL.TZ2.13
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Determine	Question number	13	Adapted from	N/A

Question

Juliana plans to invest money for $10$ years in an account paying $3.5 %$ interest, compounded annually. She expects the annual inflation rate to be $2 %$ per year throughout the $10$ -year period.

Juliana would like her investment to be worth a real value of $$ 4000$ , compared to current values, at the end of the $10$ -year period. She is considering two options.

Option 1: Make a one-time investment at the start of the $10$ -year period.

Option 2: Invest $$ 1000$ at the start of the $10$ -year period and then invest $$ x$ into the account
at the end of each year (including the first and last years).

For option 1, determine the minimum amount Juliana would need to invest. Give your answer to the nearest dollar.

[3]

For option 2, find the minimum value of $x$ that Juliana would need to invest each year. Give your answer to the nearest dollar.

[3]

Markscheme

METHOD 1 – (with $F V = 4000$ )

EITHER

$N = 10$
$I = 1.5$
$FV = 4000$
$P/Y = 1$
$C/Y = 1$ (A1)(M1)

Note: Award A1 for $(3.5 - 2 =) 1.5$ seen and M1 for all other entries correct.

$4000 = A {(1 + 0.015)}^{10}$ (A1)(M1)

Note: Award A1 for $1.5$ or $0.015$ seen, M1 for attempt to substitute into compound interest formula and equating to $4000$ .

THEN

$(PV =) $ 3447$ A1

Note: Award A0 if not rounded to a whole number or a negative sign given.

METHOD 2 – (With $F V$ including inflation)

calculate $FV$ with inflation

$4000 \times 1 . 02^{10}$ (A1)

$(= 4875.977 \dots)$

EITHER

$4000 \times 1 . 02^{10} = PV \times 1 . 035^{10}$ (M1)

$N = 10$
$I = 3.5$
$FV = 4875.977 \dots$
$P/Y = 1$
$C/Y = 1$ (M1)

Note: Award M1 for their $FV$ and all other entries correct.

THEN

$(PV =) $ 3457$ A1

Note: Award A0 if not rounded to a whole number or a negative sign given.

METHOD 3 – (Using formula to calculate real rate of return)

(real rate of return =) $1.47058 \dots (%)$ (A1)

EITHER

$4000 = PV \times 1.0147058 \dots^{10}$ (A1)

$N = 10$
$I = 1.47058 \dots$
$FV = 4000$
$P/Y = 1$
$C/Y = 1$ (M1)

Note: Award M1 for all entries correct.

THEN

$(PV =) $ 3457$ A1

[3 marks]

METHOD 1 – (Finding the future value of the investment using $P V$ from part (a))

$N = 10$
$I = 3.5$
$PV = 3446.66 \dots$ (from Method 1) OR $3456.67 \dots$ (from Methods 2, 3)
$P/Y = 1$
$C/Y = 1$ (M1)

Note: Award M1 for interest rate $3.5$ and answer to part (a) as $PV$ .

$(FV =) $ 4861.87$ OR $$ 4875.97$ (A1)

so payment required (from TVM) will be $$ 294$ OR $$ 295$ A1

Note: Award A0 if a negative sign given, unless already penalized in part (a).

METHOD 2 – (Using $F V$ )

$N = 10$
$I = 3.5$
$PV = - 1000$
$FV = 4875.977 \dots$
$P/Y = 1$
$C/Y = 1$ (A1)(M1)

Note: Award A1 for $I = 3.5$ and $FV = \pm 4875.977 \dots$ , M1 for all other entries correct and opposite $PV$ and $FV$ signs.

$(PMT =) $ 295 (295.393)$ A1

Note: Correct 3sf answer is $295$ , however accept an answer of $296$ given that the context supports rounding up. Award A0 if a negative sign given, unless already penalized in part (a).

[3 marks]

Examiners report

Very few candidates appeared to be familiar with real rate of return on an investment and attempted the question by ignoring the inflation rate. There was a mix of candidates who attempted to use the financial app on their graphic display calculator and those who attempted to use the compound interest formula, both of which are accepted methods. The preferred method of the IB for calculating the real rate of return is by simply subtracting the inflation rate from the nominal interest rate, however the exact formula is of course accepted too.

Was very poorly done, both in recognition of an appropriate future value with inflation and realizing that PV and FV must have opposite signs when using the financial app on their graphic display calculator. In both parts of the question, there seemed to be little consideration as to the appropriateness of the answers, and often unreasonable answers were presented.

Syllabus sections

Topic 1—Number and algebra » SL 1.4—Financial apps – compound interest, annual depreciation

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