**Step 1 of 3**

According to question, John Rider wants to accumulate $135,000 to be used for his daughter’s college education and he would like to have the amount available on December 31, 2023. Assume that the funds will accumulate in a certificate of deposit paying 8% interest compounded annually.

How much would he have to invest on December 31, 2018, If John were to deposit a single amount.

what is the required amount of each deposit, If John were to make five equal deposits on each December 31, beginning a year later, on December 31, 2019 and what is the required amount of each deposit, if John were to make five equal deposits on each December 31, beginning now, on December 31, 2018.

in order to answer the question, first understand the concept of time value of money.

Time value of Money :

A financial concept known as the time value of money acknowledges that the value of money fluctuates over time. The idea is predicated on the idea that a logical investor would rather have a specific sum of money now than the same amount later.

the Key components of the time value of money include

1)Present Value

2)Future Value

3)Interest Rates

4) Compounding

5)Discounting

**Explanation:**

in this question, above the time value of money is explained because the asked question will solve using the concept and formula of time value of money. Based on the given situation, question will be answered.

**Step 2 of 3**

1)

John wants to accumulate $135,000, the interest rate is 8% (0.08), and the number of years is 5 (from December 31, 2018, to December 31, 2023).

The future value (FV) formula is give

FV=PV×(1+r)n

Where:

- FV = the future value,
- PV = the present value (the initial investment),
- r = the interest rate per period,
*n = the number of periods.*

*135,000=PV×(1+0.08)5*

PV= 135,000(1+0.08)5

PV= 135,0001.4693

PV=91,878.73

**Explanation:**

John wants to deposit a single amount of $135000 on 8% interest rate for 5 year then the present value( December 31, 2018 ) is calculated by using the basic future value formula.

**Step 3 of 3**

2)

John makes five equal deposits, the interest rate is 8%, and the number of years is 4 (from December 31, 2019, to December 31, 2023).

will be used , is given :

FV=PMT×(1+r)n−1r

Where:

- PMT =the equal payment made each period,
- r = the interest rate per period (0.08)
- n
*=*the number of periods 4 year(December 31, 2019- December 31, 2023)

FVAD=PMT×(1+r)^n -1r)

135,000=PMT×((1+0.08)4−10.08)

*PMT*=135,000(1+0.08)4−10.08

PMT = 135,0004.5061

PMT =29,959.3086

3) The same formula will be used but the number of year will be taken 4 because the first deposit is made on December 31, 2018).

The future value of an ordinary annuity (FVAD) formula will be used , is given :

FVAD=PMT×([(1+r)^n-1] r)

- PMT =the equal payment made each period,
- r = the interest rate per period (0.08)
- n
*=*the number of periods (5 year) December 31, 2018 TO December 31, 2023

FVAD=PMT×([(1+r)^n-1]r)

135,000=PMT×([(1+0.08)^5-1]0.08)

PMT=135,0005.8666

PMT =23,011.6213

**Explanation:**

John wants to deposit a single amount of $135000 on 8% interest rate for 5 year and 4 year then the present value( December 31, 2018 ) is calculated by using the future value of an ordinary annuity (FVAD) formula .

**Final solution**

- If John were to deposit a single amount on December 31, 2018, he would need to invest approximately 91,878.73
- If John were to make five equal deposits starting on December 31, 2019, the required amount of each deposit would be approximately 29,959.3086
- If John were to make five equal deposits starting on December 31, 2018, the required amount of each deposit would be approximately 23,011.6213