Introductory to Finance - Math Problem Example

The paper  “Introductory to Finance”  is an engrossing example of a finance & accounting math problem. The rate of returns during the 30 years that am accumulating funds is 8% and at retirement to earn a 10% rate of return.
Determining the annual end-of-year savings needed to fund retirement,
Estimated annual expenditure in retirement is computed as;           
=0.80 * $84,000 = $67,200.

Additional annual retirement income needed is calculated as follows;
 =$67200-$5300=$14200.

Thus,$14200 is the amount without inflation adjustment.
Therefore the lump sum needed for 30 years to fund the additional annual retirement income would be;

=$14200/0.10*{1-[1/ (1+0.1) ^27])} =$142000*0.923722=$131,168.524.

The annual-end-year savings required to fund lump sum;

=$131,168.524/ {[(1+0.08) ^30- 1]/0.008}=131,168.524/113.28325.

=$1157.881011

Therefore, to fund my retirement objective for the period of 27 years, I need to save approximately $1158 at the end of each of the next 30 years. This implies that if I earn lower returns, then I have to save more for each year.

QUESTION TWO
Calculating the bond price for Greece in December 2009 and how the price distorted six months afterward.
The annual compounding rate formula is given as:

= Annual rate=, where  is the rate and  is the  compounding frequency when the rate is 4%

= (1+0.04/2) ^2 -1 = (1+0.02) ^2 -1

=1.0404-1=0.04045%.

The rate at 7% when the bond rises from 4% to 7% is also computed as follows;

= (1+0.07/2) ^2-1= (1+0.035) ^2-1

=1.071225-1=0.071225%.

The difference between the annual compounded rate in December 2009 at 4% and 7% gives the margin increase in the price of the bond.

=0.071225%-0.04045%
=0.030775% price change in 2009.  

QUESTION THREE
Given that the dividend increases by 20%, and afterward grow at the rate of 5%.
The dividend was $2 per share and the market required rate of return on this stock at 14%.
Then the value of this stock would be as follows;

Calculating the dividends:

D1 = $2.

D2 = $2 ´ 1.20 = $2.40.

D3 = $2.40 ´ 1.20 = $2.88.

D4 = $2.88 ´ 1.05 = $3.024.

Calculating the price of the stock at Year 3, when it becomes a constant growth stock:

Po = D4/ (k - g)

     = $3.024/ (0.14 - 0.05)

     = $504.

Calculating the price of the stock today:

Po = ($2/1.14) + $2.40/(1.14)2 + ($2.88 + $504)/(1.14)3

     = $1.754386 + $1.22449 + $184.723032

     = $187.7001908.

QUESTION FOUR
INVESTMENT A: Invest a lump sum of $2,750 today in an account that pays 6% annual interest and leave the funds on deposit for exactly 15 yearsThe formula for the future value of a lump sum investment is given as;

FV=PV*(1+r) ^n, Where
FV=Future value of an investment.
PV=Present value of an investment.
R=Interest rate per year.
 N=the number of years the lump sum is invested.

So, the future value of investing $2,750 with an interest of 6% per year for 15 years will be;

FV=$2,750*(1+0.06) ^15
=$2,750* 2.396558
=$6590.5345

INVESTMENT B: The investments are made for each year and the value is computed independently for each year of investment.

Beginning of Year

Amount

1

$900

2

1,000

3

1,200

4

1,500

5

1,800

Year one he invests $900 which will earn;

            =$900 *(1+0.09) =$981

Year two he invests $100 which will earn;

            =$1000*(1+0.09) =$1090

Year three he invests $1200 which will earn;

            =$1200*(1+0.09) =$1308

Year four he invests $1500 which will earn;

            =$1500*(1+0.09) =$1635

And finally, in year five he invests $1800 which will earn;

            =$1800*(1+0.09) =$1962

Thus, the accumulated value will be;

            =$981+=$1090+$1308+$1635+$1962

            =$6,976.

Investment C: This is an ordinary annuity where payment is made at the end of the year. The formula for ordinary annuity thus is given as;

                              =$1200*15.93742

                               =$19124.904

Investment D: This is an annuity- due where payment is made at the beginning of each year. The formula for annuity due is given as;

                         1+i).

                                 =$1200*17.531162.

                                 =$21037.3944.