Simple Interest
1.2 Future and Present Values
A. Computing Future Value
When you borrow money, you are to repay, sometime in the future, both the original amount borrowed (the principal or present value) and the amount of interest charged for the loan period. The sum of the original amount and the interest charged is the future value [asciimath]S[/asciimath] (maturity value or accumulated value).
[asciimath]S=P+I[/asciimath] Formula 1.2
Substituting [asciimath]I=Prt[/asciimath] (Formula 1.1) into Formula 1.2 yields
[asciimath]S=P+P*r*t[/asciimath]
Factoring [asciimath]P[/asciimath] gives
[asciimath]S=P(1+r*t)[/asciimath] Formula 1.3a
You invest $3000 in a short-term deposit at 4.5% p.a. for 165 days. a) What will be the future value of this investment? b) What will be the amount of interest earned?
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Given information:
- The investment principal: [asciimath]P=$3000[/asciimath]
- Nominal Interest rate per year: [asciimath]r = 4.5%[/asciimath]
- Time period of the investment: [asciimath]t = 165[/asciimath] days [asciimath]= 165/365[/asciimath] years
a) Substituting the values into equation (iii), we get
[asciimath]S=P(1+r*t)[/asciimath]
[asciimath]S=3000(1+(4.5%)(165/365))[/asciimath]
[asciimath]=3061.027397...[/asciimath]
[asciimath]~~ $3061.03[/asciimath] (Rounded to two decimal places)
b) The amount of interest can be obtained using either Formula 1.1 or 1.2. Given the future value is known now, we use Formula 1.2 as it involves fewer operations:
[asciimath]S=P+I[/asciimath]
[asciimath]->[/asciimath] [asciimath]I=S-P[/asciimath]
[asciimath]I=3061.03-3000[/asciimath]
[asciimath]=$61.03[/asciimath]
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B. Computing Present Value
Sometimes you may know the future value and want to determine the principal amount required to reach that future value at a specific interest rate over a certain period. In that case, Formula 1.3a can be rearranged for [asciimath]P[/asciimath].
[asciimath]P=S/(1+r*t)[/asciimath] Formula 1.3b
which using the negative-exponent notation is equivalent to
[asciimath]P=S(1+r*t)^-1[/asciimath] Formula 1.3c
To help determine whether a problem is asking you to find the present value (principal) or future value, look for specific keywords or phrases. Below are examples of how questions might be phrased when they are seeking the present value:
- What principal will have a future value of $8,000?
- What amount of money will accumulate to $14,000?
- Compute the present value of $4000.
- Calculate the amount of money that will grow to $4500.
- Find the amount paid for an investment that will mature to $20,000.
- How much money was borrowed if the maturity value of the loan is $12,000?
What amount of money will mature to $4195.25 in 110 days at 4.5% p.a. interest?
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Given information:
- The maturity value: [asciimath]S=$4195.25[/asciimath]
- Nominal Interest rate per year: [asciimath]r = 4.5%[/asciimath]
- Time period: [asciimath]t = 110[/asciimath] days [asciimath]= 110/365[/asciimath] years
Substituting the values into Formula 1.3b, we obtain
[asciimath]P=S/(1+r*t)[/asciimath]
[asciimath]P=4195.25/(1+(4.5%)(110/365))[/asciimath]
[asciimath]=4139.116772...[/asciimath]
[asciimath]~~ $4139.12[/asciimath] (Rounded to the nearest cents)
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Section 1.2. Exercises
- What is the maturity value of $37,150 invested in an account earning 2.38% p.a. for 3 years and 11 months?
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S = $40,613.00
- What amount of money will grow to $76,750in 2 years and 11 months at the interest rate of 3.95% p.a.?
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P = $68,821.22
- Suppose you invest $58,650 in a short-term deposit at 6.5% for 42 days. a) Calculate the accumulated value of this amount at the end of the term. b) Calculate the amount of interest earned.
Show/Hide Answer
a) S = $59,088.67; b) I = $438.67
