Math  /  Algebra

Question مبادئ التّمويل(1) نُظري - طولكرم
Question 29. Not yet answered Marked out of 1.00
Flag question If the interest rate is positive, then the future value of an annuity due will be greater than the future value of an ordinary annuity.
Select one: True False

Studdy Solution

STEP 1

1. An annuity due is a series of equal payments made at the beginning of each period.
2. An ordinary annuity is a series of equal payments made at the end of each period.
3. Future value (FV) calculations for both types of annuities consider the interest rate.
4. The interest rate is positive.

STEP 2

1. Define the formula for the future value of an ordinary annuity.
2. Define the formula for the future value of an annuity due.
3. Compare the future values of both annuities.
4. Conclude whether the statement is true or false.

STEP 3

Define the formula for the future value of an ordinary annuity.
The future value of an ordinary annuity (FVOA) can be calculated using the formula: FVOA=P(1+r)n1r \text{FVOA} = P \frac{(1 + r)^n - 1}{r}
where: - P P is the payment per period - r r is the interest rate per period - n n is the number of periods

STEP 4

Define the formula for the future value of an annuity due.
The future value of an annuity due (FVAD) can be calculated using the formula: FVAD=P(1+r)n1r(1+r) \text{FVAD} = P \frac{(1 + r)^n - 1}{r} (1 + r)

STEP 5

Compare the future values of both annuities.
For an ordinary annuity: FVOA=P(1+r)n1r \text{FVOA} = P \frac{(1 + r)^n - 1}{r}
For an annuity due: FVAD=P(1+r)n1r(1+r) \text{FVAD} = P \frac{(1 + r)^n - 1}{r} (1 + r)
Notice that: FVAD=FVOA×(1+r) \text{FVAD} = \text{FVOA} \times (1 + r)
Since r r is positive, (1+r)>1 (1 + r) > 1 .
Therefore: FVAD>FVOA \text{FVAD} > \text{FVOA}

STEP 6

Based on the comparison, the statement "If the interest rate is positive, then the future value of an annuity due will be greater than the future value of an ordinary annuity" is:
True \text{True}

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