答案保留两位小数

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a. What's the present value of a 4-year ordinary annuity of $2,200 per year plus an additional$2,500 at the end of Year 4 if the interest rate is 5%? (8 marks)
b. What's the present value of $1,600 discounted back 5 years if the appropriate interest rate is 6%, compounded quarterly? (12 marks)
c. Master Card and other credit card issuers must by law print the Annual Percentage Rate (APR) on their monthly statements. If the APR is stated to be 24%, with interest paid monthly, what is the card's EFF%? (5 marks)
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Answer from Sia

Posted 2 months ago

Solution

a

Present Value of Annuity: To find the present value of a 4-year ordinary annuity of $2,200 per year plus an additional$2,500 at the end of Year 4 with an interest rate of 5%, we use the formula for the present value of an annuity and the present value of a lump sum.
1. Calculate the present value of the annuity:
$PV_{\text{annuity}} = PMT \times \left(1 - (1 + r)^{-n}\right) / r$
where $PMT = 2200$, $r = 0.05$, and $n = 4$.
$PV_{\text{annuity}} = 2200 \times \left(1 - (1 + 0.05)^{-4}\right) / 0.05$
$PV_{\text{annuity}} = 2200 \times \left(1 - 0.8227\right) / 0.05$
$PV_{\text{annuity}} = 2200 \times 0.1773 / 0.05$
$PV_{\text{annuity}} = 2200 \times 3.546$
$PV_{\text{annuity}} = 7801.2$
2. Calculate the present value of the additional $2,500 at the end of Year 4:
\[ PV_{\text{lump sum}} = FV / (1 + r)^n \]
\[ PV_{\text{lump sum}} = 2500 / (1 + 0.05)^4 \]
\[ PV_{\text{lump sum}} = 2500 / 1.2155 \]
\[ PV_{\text{lump sum}} = 2056.5 \]
3. Add the present values:
\[ PV_{\text{total}} = PV_{\text{annuity}} + PV_{\text{lump sum}} \]
\[ PV_{\text{total}} = 7801.2 + 2056.5 \]
\[ PV_{\text{total}} = 9857.7 \]

b

Present Value of a Lump Sum: To find the present value of $1,600 discounted back 5 years at an interest rate of 6%, compounded quarterly, we use the formula for the present value of a lump sum with quarterly compounding.
1. Convert the annual interest rate to a quarterly rate:
\[ r_{\text{quarterly}} = 0.06 / 4 = 0.015 \]
2. Calculate the number of quarters:
\[ n_{\text{quarters}} = 5 \times 4 = 20 \]
3. Use the present value formula:
\[ PV = FV / (1 + r_{\text{quarterly}})^{n_{\text{quarters}}} \]
\[ PV = 1600 / (1 + 0.015)^{20} \]
\[ PV = 1600 / (1.015)^{20} \]
\[ PV = 1600 / 1.3469 \]
\[ PV = 1187.6 \]

c

Effective Annual Rate (EFF%): To find the effective annual rate (EFF%) when the APR is 24% with monthly compounding, we use the formula for the effective annual rate.
1. Convert the APR to a monthly rate:
$r_{\text{monthly}} = 0.24 / 12 = 0.02$
2. Use the EFF% formula:
$EFF\% = (1 + r_{\text{monthly}})^{12} - 1$
$EFF\% = (1 + 0.02)^{12} - 1$
$EFF\% = 1.2682 - 1$
$EFF\% = 0.2682$
$EFF\% = 26.82\%$

Answer

a. $9,857.7

b. $1,187.6

c. 26.82%

Key Concept

Present Value Calculation

Explanation

The present value of future cash flows is calculated by discounting them back to the present using the appropriate interest rate. For annuities, the present value formula accounts for multiple periods of equal payments, while for lump sums, it considers a single future amount. The effective annual rate (EFF%) adjusts the nominal rate to reflect compounding within the year.

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