// Numbas version: finer_feedback_settings {"name": "Net Present Value", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["value1", "value2", "value3", "value4", "n1", "n2", "n3", "npvone", "npv20", "irr"], "name": "Net Present Value", "tags": [], "advice": "

The NPV is the net present value. This is the sum of all the discounted cash flows at time $t=0$.

Suppose you gain value 1 at time $t=1$, value 2 at time $t=2$,... and value n at time $t=n$, then the NPV is given by;

$NPV=x_1v+x_2v^2+...+x_nv^n$

where:

$x_k$= value gained at time $t=k$
$v={1 \\over 1+i}$ (see Numbas- Discounting)
$i$= annual effective rate

In the above question, the NPV is given by:

$NPV=-{x_1}-x_2v^{t_2}+x_3v^{t_3}+x_4v^{t_4}$

where:

$x_1$=£{value1} paid
$x_2$=£{value2} paid
$x_3$=£{value3} received
$x_4$=£{value4} received
$t_2$={n1}
$t_3$={n2}
$t_4$={n3}
the interest rate $i$ would be either 1% or 20% depending on the question.

Substituting these values into the NPV expression gives the net present value for 1% or 20%.

Internal Rate of Return

The internal rate of return (IRR) is the interest rate at which the NPV is zero (i.e. $NPV(i)=0$). Another name for this is yield.

In order to be able to use linear interpolation, one must have an interest rate that gives a negative NPV and an interest rate that gives a positive NPV. To approximate the internal rate of return, use the following:

$i_*=i_1+{{i_2-i_1} \\over {y_2-y_1}}(y_*-y_1)$

where:

$i_1$= the interest rate that gives a positive NPV
$i_2$= the interest rate that gives a negative NPV
$y_1$= the positive NPV
$y_2$= the negative NPV
$i_*$= the interest rate that gives a NPV of zero
$y_*$= 0

Substituting $i_1=0.01, i_2=0.2$, the corresponding NPV's and $y_*=0$ into the above equation gives the internal rate of return.

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What is the NPV (to two decimal places) on the basis of an interest rate of 1%?

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What is the NPV (to two decimal places) on the basis of an interest rate of 20%?

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Approximate the internal rate of return (% to two deicmal places) of this transaction using linear interpolation with the values above.

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An investor enters into an agreement to pay £{value1} immediately and £{value2} at the end of {n1} years in exchange for the receipt of £{value3} at the end of year {n2} and £{value4} at the end of year {n3}.

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