Assume you owe $1,000 today and $1,000 one year from now. You find yourself unable to make that payment today, so you indicate to your creditor that you want to make both payments six months from now instead. Prevailing interest rates are at 6% compounded semi-annually. What single payment six months from now (the proposed payment stream) is equivalent to the two payments (the original payment stream)?

Solution:

Step 1: The timeline illustrates the scenario.

I/Y = 6%; C/Y = semi-annually = 2

Step 2: Apply the fundamental concept of time value of money, moving all of the money to the same date. Since the proposed payment is for six months from now, you choose a focal date of six months.

Step 3: Calculate the periodic interest rate, i.

[latex]i=\frac{I/Y}{C/Y}=\frac{6\%}{2}=3\%[/latex].

Step 4: Calculate the total number of compounds, n.

[latex]n=C/Y \times (\text{Number of Years})=2\times\frac{1}{2}=1[/latex].

Step 5: Perform the appropriate time value calculations.

Moving today’s payment of $1,000 six months into the future, you have

[latex]FV=\$1,\!000(1+0.03)^1=\$1,\!030[/latex].

Moving the future payment of $1,000 six months earlier, you have

[latex]\begin{align} PV=\frac{\$1,\!000}{(1+0.03)^1}=\$970.87\end{align}[/latex].

Step 6: Now that money has been moved to the same focal date you can sum the two totals to determine the equivalent payment, which is $1,030 + $970.87 = $2,000.87. Note that this is financially fair to both parties. For making your $1,000 payment six months late, the creditor is charging you $30 of interest. Also, for making your second $1,000 payment six months early, you are receiving a benefit of $29.13. This leaves both parties compensated equitably: Neither party is financially better or worse off because of the change in the deal.

Johnson’s Garden Centre has recently been unprofitable and concludes that it cannot make two debt payments of $4,500 due today and another $6,300 due in three months. After discussions between Johnson’s Garden Centre and its creditor, the two parties agree that both payments could be made nine months from today, with interest at 8.5% compounded monthly. What total payment does Johnson’s Garden Centre need nine months from now to clear its debt?

Solution:

Step 1: The timeline illustrates the scenario.

PV₁ (today) = $4,500; PV₂ (3 months from now) = $6,300; I/Y = 8.5%; C/Y = monthly = 12

Step 2: Due date for all payments = 9 months from today. This is your focal date.

Step 3: Calculate the periodic interest rate, i.

[latex]i=\frac{I/Y}{C/Y}=\frac{8.5\%}{12}=0.708\overline{3}\%=0.00708\overline{3}[/latex].

Step 4: Calculate the total number of compounds, n.

The first payment moves nine months into the future, or 9/12 of a year.

[latex]n=C/Y \times (\text{Number of Years})=12\times\frac{9}{12}=9[/latex].

The second payment moves six months into the future, or 6/12 of a year.

[latex]n=C/Y \times (\text{Number of Years})=12\times\frac{6}{12}=6[/latex].

Step 5: Perform the appropriate time value calculations.

Moving the first payment of $4,500 nine months into the future, you have

[latex]FV_1=\$4,\!500(1+0.00708\overline{3})^9=\$4,\!795.138902[/latex].

Moving the second payment of $6,300 six months into the future, you have

[latex]FV_2=\$6,\!300(1+0.007083\overline{3})^6=\$6,\!572.536425[/latex].

With interest, the two payments total $11,367.68. This is the $10,800 of the original principal plus $567.68 in interest for making the late payments.

You have three debts to the same creditor: $3,000 due today, $2,500 due in 2¼ years, and $4,250 due in 3 years 11 months. Unable to fulfill this obligation, you arrange with your creditor to make two alternative payments: $3,500 in nine months and a second payment due in two years. You agree upon an interest rate of 9.84% compounded monthly. What is the amount of the second payment?

Solution:

Determine the amount of the second payment that is due two years from today. Apply the fundamental concept of time value of money, moving all money from the original and proposed payment streams to a focal date positioned at the unknown payment. Once all money is moved to this focal date, apply the fundamental concept of equivalence, solving for the unknown payment, or x.

Step 1:

With two payment streams and multiple amounts all on different dates, visualize two timelines. The payment amounts, interest rate, and due dates for both payment streams are known.

I/Y = 9.84%; C/Y = monthly = 12

Step 2: Focal Date = 2 years.

Step 3: Calculate the periodic interest rate, i.

[latex]i=\frac{\text{Nominal Rate (I/Y)}}{\text{Compounds per Year (C/Y)}}=\frac{9.84\%}{12}=0.82\%\;\text{or}\;0.0082[/latex]

Step 4: For each payment, calculate the number of compoundings per year by applying the formula

[latex]n=C/Y \times (\text{Number of Years})[/latex].

Step 5:  For each payment, calculate the appropriate time value calculation. Note that all payments before the two year focal date require you to calculate future values, while all payments after the two-year focal date require you to calculate present values.

Steps 4 and 5: Using the circled number references from the timelines:

① [latex]n=12 \times 2 =24[/latex]; [latex]FV_1=\$3,\!000(1+0.0082)^{24}=\$3,\!649.571607[/latex]

② [latex]n=12 \times \frac{1}{4} =3[/latex]; [latex]\begin{align}PV_1=\frac{\$2,\!500}{1.0082^3}=\$2,\!439.494983\end{align}[/latex]

③ [latex]n=12 \times 1\frac{11}{12} =23[/latex]; [latex]\begin{align}PV_2=\frac{\$4,\!250}{1.0082^{23}}=\$3,\!522.207915\end{align}[/latex]

④ [latex]n=12 \times 1\frac{1}{4} =15[/latex]; [latex]FV_2=\$3,\!500(1+0.0082)^{15}=\$3,\!956.110749[/latex]

Step 6:

[latex]\begin{align} \$3,\!649.571607+\$2,\!439.494983+\$3,\!522.207915&=x+\$3,\!956.110749\\ \$9,\!611.274505&=x+\$3,\!956.110749\\ \$5,\!655.16&=x \end{align}[/latex]