Issue A
Mary has saved $500 for the past 19 years earning an interest of 5% per annum. The principle is compounded annually and Mary is expected to continue paying the same amount till next year. If Mary closes the account after making the last deposit, the total worth of the account can be computed as follows. The future value of the current payment can be obtained by computing the future value of an ordinary annuity since equal payments are made at the end of the period
FV = P[((1+r) n -1)/r]
= 500[((1.05) 20 -1)/0.05]
=500*1.653/0.05
=826.6488/0.05
=16,532.98
Mary’s account will have $16,532.98 when she closes it after making the last payment in a year’s time. The future value calculated could, however, be different if periodic payments changed or the interest rate change within the 20 years changed.
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Issue B
Mary has been employed by the University for the last 25 years. Her excellent performance has attracted the attention of the board which wants to reward her with a bonus to the retirement package. The board is offering her $75,000 annually for the next 20 years. The package will commence one year after she retires and will be paid for 19 subsequent years after her retirement. Mary is interested in a one-time payment after retirement and the interest rate is 7%. To compute the amount that Mary will receive as a lump sum we determine the lump sum amount which is payable for 19 years and includes the annual pay for the twentieth year
Amount payable for 19 years: P[((1+r)n-1)/r]
= 75000[((1+0.07)^19 -1)/0.07)
=75000[2.6162/0.07]
=75000*37.379
=$2,803,422.36
The total amount for the 20 years therefore will be $2,803,422.36 + $75000 = $2,878,422.36
Issue C
Mary is asked by the company to continue working for the next three years until a suitable replacement is obtained. According to the board, the bonus should remain the same but according to Mary the PV of the bonus will change. The present value of her deferred annuity can be calculated as shown below.
Mary was to receive $2,803,422.36 as the total bonus as computed earlier in issue B. The amount will be received three years later and therefore the current discounted value will be obtained using the formula of present value for a deferred annuity. The total bonus amount to be received will be used as the principal and the interest rate will remain the same. Therefore
PV today = PV in future*[(1/(1+r)^n]
= $2,803,422.36 [1/(1+0.07)^3]
=$2,803,422.36*0.8163
=$2,349,656.2
The present value of the deferred annuity for Mary will, therefore, be $2,349,656.2 as shown in the calculations where the present value of the figure obtainable three years from today should be discounted to establish its present value as of today. If Mary could open an account today that attracts an interest of 7% and deposit the same amount upon signing the annuity program, the figure will grow to =$2,803,422.36.
Issue D
Mary is interested in paying for the fees of her granddaughter Beth’s education. Mary is to pay for half the tuition cost at the university totaling $11,000 each year. The same is expected to rise by 7% each year into the foreseeable future. Beth is currently 12 years and intends to join university when she is 18 years and complete four years later. Mary plans to make a deposit today and the trend will continue until Beth is 18 years. Such deposits will earn an interest of 4% and compounded annually. Mary must, therefore, deposit a constant amount each year in order to pay half of Beth tuition.
The current tuition fee is $11,000 and it is expected to increase by 7% each year. The tuition fee therefore when Beth is 18 years will be 11,000*1.07^6 = 16508.0. Mary, therefore, expects to pay half this amount which will be 8,254.02. The tuition fee will continue growing at 7% per annum for the next four years. Assuming that Mary opens a saving account today for that purpose, the first deposit is therefore, the initial balance in the account. Similarly, it is expected that all the savings will be withdrawn from the account and use to pay the fee.
The total school fees paid by Mary can be computed by compounding the first year school fees as follows 8,254.02*1.07 + 8,254.02*1.07^2+8,254.02*1.07^3
= 8254.02+8831.85+9450.08+10111.53 = $36647.48.
$36,647.48 is the total amount that Mary will pay for Beth in the four years. The figure, however, can change if the expected growth rate of 7% changes.
Mary plans to make a deposit today and will continue to make such deposits for the next six years until Beth starts College. The targeted amount should be $36,647.48 and therefore the annual deposits that earn an interest of 4% for six-year should add up to this amount. We, therefore, find the annuity amount that Mary should deposit each year so that she could raise the above fees for Beth
A= P[((1+r)^n-1)/r]
36647.48 = P[(1.04)^6-1)/r]
36647.48 = 6.633P
P = 5,525.04
Mary should therefore deposit $5,525.04 now and continued doing so each year until Beth is 18 years in order to raise the required amount of $36,647.48 in six years which will be adequate to pay for half of the school fees.
References
Smart, S. (2008). Corporate finance . Cengage Learning.
Time Value of Money | Concept, Explanation & Examples. (2013). Retrieved from https://accounting-simplified.com/management/investment-appraisal/time-value-of-money.html
Vernimmen, P. (2014). Corporate Finance . New York: Wiley-Blackwell.
