Yue Zhao

E-Portfolio TVM Assignment

According to the given information, before John was injured, he was the president of his family’s business, and earned approximately $200,000 per year. Suppose John could work 25 more years before retirement, and he could earn the same amount of $200,000 each year, then the $200,000 could be viewed as an annuity, for the same amount is to be received each period, here annually.

To compensate for John’s loss, they need to compute John’s lost future income into today’s value. Because money has a time value. This means the money invested today will grow to a larger dollar amount in the future. Usually interest rate is used to reflect the time value of money.

Suppose the interest rate would remain constant for the following 25 years, and an annual rate of 7% (it might be an estimate based on average similar investment growth rate) could be reached, John’s equivalent today’s value (or Present Value) of his lost future income then can be computed.

Below is the known information:Annuity Amount = $200,000.Periods Number = 25.Annual Interest Rate = 7%.

We need to calculate equivalent Present Value of the 25 years annuity payments.Present Value (PV) = ?

Suppose John gets the first payment one year from now:

We can turn the above calculations into a table for all 25 years:

Annuity Payment Factor Present Value n

1st Year $200,000 * (1+7%)-1 (0.93458) = $186,916 1

2nd Year 200,000 * (1+7%)-2 (0.87344) = 174,688 2

Current YearPV = ?

1st Year$200,000

PV1 * (1+ Annual Interest Rate) = 200,000PV1 * (1+7%) = 200,000PV1 = 200,000 * (1+7%)-1

PV1 = 200,000 * 0.93458PV1 = 186,916

Current YearPV = ?

2nd Year$200,000

PV2 * (1+ Annual Interest Rate)2 = 200,000PV2 * (1+7%)2 = 200,000PV2 = 200,000 * (1+7%)-2

PV2 = 200,000 * 0.87344PV2 = 174,688

Yue Zhao

The Accumulated Present Value for all 25 years can be found by separately calculating the PV of each of the 25 payments, and then summing these individual present values together. That accumulated present value (lump-sum) would be equal to the 25 years’ annuity. Though this method is easy to understand, but its process is very tedious.

Fortunately we can use a more simple method to calculate the present value of annuity: To use the table of “Present Value of an Ordinary Annuity of $1”.

Below is the PVA of $1 table for interest rate of 7%:

From the above examples of PV1 and PV2, we can see that the present value of the first 2 years’ annuity is PV1 + PV2 = 186,916 + 174,688 = $361,604.

Also, PV1 + PV2 = 200 * (0.93458 + 0.87344) = 200,000 * 1.80802 = $361,604.

We can see the above 1.80802 is just the factor we find from table PVA of $1 using n=2, and i=7%.

Thus we can conclude that the two methods can lead to the same results.

Using table PVA of $1, we calculate the PV of the ordinary annuity (PVA) as following:

PVA = Annuity Amount * PVA Factor

Here PVA factor is the number found on table PVA of $1, where n=25, and i=7%.

Thus, the Present Value for the 25 years annuity payment can be calculated by:PVA = 200,000 * 11.65358 = $2,330,716

This is just the lump-sum settlement amount John was offered in the question.

3rd Year 200,000 * (1+7%)-3 (0.8163) = 174688 3

… … … … … … … …

25th Year 200,000 * (1+7%)-25 (0.18425) = 36850 25

Yue Zhao

As for whether the settlement is fair, it depends on whether the estimations fairly reflects real future.

What we currently get from calculation is based on the estimations of consistent $200,000 earnings per year for 25 years, and the annual interest rate of 7%. Do these estimation really reflects John’s lost future income? It’s hard to say. John could earn more than $200,000 in some year, or he could also earn more less in some other year. If he would use the money to invest, can he maintain an annual growth rate of 7%? If the big economic environment is bad, think about 2008’s financial crisis, no one can guarantee a return rate of 7% for 25 years. But on the other hand, maybe John could get luck and hit a higher rate in some year. So, estimations are only estimations, it’s derived from the best information available, and describes the most likely outcome of the uncertainty.

Thus, according to my understanding, as long as the estimation amounts are reasonable and acceptable to John’s family, the settlement amount got from the estimations then is fair.

Reflection:This question not only asks us to calculate the time value of money, but it also requires us to explain the calculation to a person that has barely knowledge of the time value of money concepts. To explain the calculation, we need to break it into steps, and illustrate it step by step.The process helps us better comprehend TVM, for it needs us fully understand the concepts and able to interpret it in easy-to-understand words.