Question

You deposit $9000 in an account that pays 5% interest compounded quarterly.

A. Find the value of one year

B.Use the future value formula for simple interest to determine the effective annual yield
(Formula: A=P(1 +rt)

Answer 1

A) $9458.51

B) 5.09% (2 d.p.)

Explanation:

Part A

Compound Interest Formula

`\large \text{$ \sf A=P\left(1+(r)/(n)\right)^(nt) $}`

where:

• A = final amount
• P = principal amount
• r = interest rate (in decimal form)
• n = number of times interest applied per time period
• t = number of time periods elapsed

Given:

• P = $9000
• r = 5% = 0.05
• n = 4 (quarterly)
• t = 1 year

Substitute the given values into the formula and solve for A:

`\implies \sf A=9000\left(1+(0.05)/(4)\right)^(4 * 1)`

`\implies \sf A=9000(1.0125)^4`

`\implies \sf A=9458.508032...`

Therefore, the value of the account after 1 year is $9458.51.

Part B

Simple Interest Formula

A = P(1 + rt)

where:

• A = final amount
• P = principal
• r = interest rate (in decimal form)
• t = time (in years)

Given:

• A = $9458.51
• P = $9000
• t = 1 year

Substitute the given values into the formula and solve for r:

`\implies \sf 9458.51=9000(1+r(1))`

`\implies \sf 9458.51=9000(1+r)`

`\implies \sf (9458.51)/(9000)=1+r`

`\implies \sf r=(9458.51)/(9000)-1`

`\implies \sf r=0.050945...`

`\implies \sf r=5.0945... \%`

Therefore, the effective annual yield is 5.09% (2 d.p.).

Answer 2

Explanation:

Given:

• Deposit P = $9000
• Interest rate r = 5% = 0.05
• Time t = 1 year
• Number of compounds n = 4

A. Find the value of one year

• 
  `A = P(1 + r/n)^(nt) = 9000(1+0.05/4)^4=9458.51`

B. Use the future value formula for simple interest to determine the effective annual yield

• 
  `A=P(1+rt)\\`
• 
  `r=(A/P-1)/t`
• 
  `r=(9458.51/9000-1)/1=0.0509`
• r = 5.1%