Simple Interest
1.3 Equivalent or Dated Values for Specified Focal Dates
A. Computing an Equivalent Payment for a Single Payment
Money has time value, implying the potential for it to accrue interest over time. Occasionally, it becomes necessary to modify or reschedule payment due dates for various reasons. Therefore, the payment amount needs to be adjusted. For example, you may need to defer a payment due to financial constraints or opt to make an early payment to reduce interest costs.
When the due date of payment is altered, a new due date, often referred to as the focal date, is established, especially in scenarios involving the rescheduling of multiple payments. The value of the money on this new due date is known as the equivalent value or dated value. If the payment is postponed (shifted to a future date), we calculate its future value. Conversely, if the payment is made earlier (moved to an earlier date), we determine its present value. This concept is illustrated in Figure 1.3.1 below.
Figure 1.3.1 Rescheduling a single payment: Deferred (left) versus earlier payment (right)
Aslan is required to repay a matured loan amount of $1915 in nine months. However, he realizes that he can clear the loan four months earlier. How much will he have to pay to clear the loan four months earlier if the interest charged is 5.2% p.a.?
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Since the payment is rescheduled at an earlier time than the due date, the present value of the payment should be computed.
- The payment: [asciimath]S=$1915[/asciimath]
- Nominal Interest rate per year: [asciimath]r = 5.2%[/asciimath]
- Time period payment is paid earlier: [asciimath]t =[/asciimath] 4 months [asciimath]=4/12[/asciimath] years
Substituting the values into Formula 1.3b, we obtain
[asciimath]P=S/(1+r*t)[/asciimath]
[asciimath]P=1915/(1+(5.2%)(4/12))[/asciimath]
[asciimath]=1882.372215...[/asciimath]
[asciimath]~~ $1882.37[/asciimath] (Rounded to the nearest cents)
Note: The new amount is lower than the original payment as the interest of four months is deducted from the original amount of $1915.
Nancy realizes that she cannot repay a matured loan amount of $3400 that is due today and decides to reschedule the payment. If the interest charged is 4.8% p.a., what equivalent payment is required to repay this amount in 10 months?
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Since the payment is deferred to a later time than the due date, the future value of the payment should be computed.
- The payment: [asciimath]P=$3400[/asciimath]
- Nominal Interest rate per year: [asciimath]r = 4.8%[/asciimath]
- Time period payment is deferred: [asciimath]t =[/asciimath] 10 months [asciimath]=10/12[/asciimath] years
Substituting the values into Formula 1.3a, we obtain
[asciimath]S=P(1+r*t)[/asciimath]
[asciimath]S=3400(1+(4.8%)(10/12))[/asciimath]
[asciimath]=$3536[/asciimath]
Note: The new amount is higher than the original payment as the interest of 10 months is added to the original amount of $3400.
B. Computing an Equivalent Payment for a Series of Payments
In the previous section, we calculated the equivalent payment for a single payment. Sometimes we need to calculate an equivalent payment for a series of payments. Since money has time value, making direct comparisons, equating, or adding/deducting different monetary values at various points in time is not feasible. To make valid comparisons, it is essential to find their equivalent values at a common reference date, also known as the focal date. The focal date allows all different cash flows to be adjusted (either discounted or compounded) to a single point in time, making them comparable or combinable.
How to find a single equivalent payment for a series of payments
1. Define a focal date (usually the date the new payment is rescheduled to).
2. Find the equivalent value of each original payment individually.
3. Add up all the equivalent values calculated for the focal date. The total will be your new single equivalent payment.
A debt can be paid off by a payment of $3500 due in two months and another payment of $7200 due in 9 months. Find a single equivalent payment to be made in four months that can clear the debt if money earns 5.4% p.a.
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1. The focal date is 4 months from now. Therefore, we should find the dated value of each payment on the focal date. Then the sum of the dated values will be the single equivalent payment.
2. The first payment is earlier than the focal date, so it should be brought forward in time by finding its future value ([asciimath]S_1[/asciimath]), while the second payment is in the future relative to the focal date, so it should be brought backward in time by finding its present value ([asciimath]P_2[/asciimath]).
The time period between 2 months and the focal date is [asciimath]t_1=4-2=2[/asciimath] months.
The time period between 9 months and the focal date is [asciimath]t_2=9-4=5[/asciimath] months.
Note: The nominal interest rate is 5.4% per annum and thus we need to convert time to years before substituting in the formula.
- [asciimath]t_1 =9[/asciimath] months [asciimath]= 9/12[/asciimath] years
- [asciimath]t_2 =2[/asciimath] months [asciimath]= 2/12[/asciimath] years
[asciimath]S_1=3500(1+(5.4%)(2/12))[/asciimath] [asciimath]=$3531.50[/asciimath]
[asciimath]P_2=7200/(1+(5.4%)(5/12))[/asciimath] [asciimath]=$7041.56[/asciimath]
3. The sum of the individual dated values is the new single equivalent payment.
Single equivalent payment [asciimath]= S_1+ P_2[/asciimath]
[asciimath]=3531.50+7041.56[/asciimath]
[asciimath]=$10,573.06[/asciimath]
Therefore, the single equivalent payment of $10,573.06 made in four months will clear the debt.
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C. Loan Repayments
The concept of equivalent values is applied in solving loan repayment problems. Loans taken from financial institutions are often repaid through blended payments, which are equal in size and include portions of both principal and interest. For the loan to be fully repaid, the sum of the present values of all these blended payments must be equal to the original amount borrowed (the principal).
How to find the size of the blended payments
1. Let the size of the payment be represented by [asciimath]x[/asciimath].
2. Find the dated value of each blended payment (i.e., the [asciimath]x[/asciimath] payment) on a focal date of “today”. (In this book, the focal date for such problems is always “today”)
3. Equalize the sum of the dated values of the blended payments to the original payment (e.g., the loan amount)
4. Solve for [asciimath]x[/asciimath].
The original payment of $4600 due today is settled by two equal payments due in 3 and 6 months. Assuming the interest rate is 10% p.a., determine the size of the equal payments.
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1. Let the size of the equal payments be represented by [asciimath]x[/asciimath]. Then the sum of the present values of the payments on the focal date of today must be equal to the loan amount of $4600.
The nominal interest rate is 10% per annum and thus we need to convert the time to years before substituting in the formula.
- [asciimath]t_1 =3[/asciimath] months [asciimath]= 3/12=0.25[/asciimath] years
- [asciimath]t_2 =6[/asciimath] months [asciimath]= 6/12=0.5[/asciimath] years
2. Next we find the dated value of each payment in terms of [asciimath]x[/asciimath] :
[asciimath]P_1=x/(1+(10%)(0.25))[/asciimath] [asciimath]=[/asciimath] [asciimath](1/(1+(10%)(0.25)))x[/asciimath] [asciimath]=0.9756098x[/asciimath]
[asciimath]P_2=x/(1+(10%)(0.5))[/asciimath] [asciimath]=[/asciimath] [asciimath](1/(1+(10%)(0.5)))x[/asciimath] [asciimath]=0.9523810x[/asciimath]
3. The sum of the dated values of the blended payments must be equal to the original payment (e.g., the loan amount).
[asciimath]$4600=P_1+P_2[/asciimath]
[asciimath]4600=0.9756098x +0.9523810x[/asciimath]
4. Finally, we solve the equation for [asciimath]x[/asciimath].
[asciimath]4600=1.9279908x[/asciimath]
[asciimath]x=4600/1.9279908[/asciimath] [asciimath]~~$2385.90[/asciimath] (Rounded to the nearest cents)
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Section 1.3. Exercises
- Debt payments of $4,430 and $5,030 are due in 4 months and 11 months, respectively. Assuming an interest rate of 7% p.a., determine a single equivalent payment that settles the two payments today. Use “today” as the focal date.
Show/Hide Answer
Equivalent payment = $9,055.69
- A payment of $6,750 was due 7 months ago and another payment of $1,490 is due in 7 months. Assuming an interest rate of 5.9% p.a., determine a single equivalent payment 3 months from now that settles the two payments. Hint: use 3 months from now as the focal date.
Show/Hide Answer
Equivalent Payment = $8,543.14
- Thiago borrowed $6,630 today and will repay the loan in two equal payments, one in 4 months and the other in 10 months. Assuming an interest rate of 7.9% p.a. on the loan, determine the size of the equal payments if a focal date of “today” is used.
Show/Hide Answer
Size of the equal payments = $3,466.53
