Morgan wants to consolidate a lot of smaller debts into a single three-year loan for $25,000. If the loan is charged interest at 7.8% compounded monthly, what is her payment amount at the end of every month?

Solution:

Step 1: The payments are made at the end of the payment intervals, and the compounding period and payment intervals are the same. Therefore, this is an ordinary simple annuity. Calculate the monthly amount of her loan payment, or PMT.

The timeline for the loan appears below.

Step 2: Given information:

PV = $25,000; I/Y = 7.8%; C/Y = 12; P/Y = 12; Years = 3; FV = $0

Step 3: Calculate the periodic interest rate, i.

[latex]i=\frac{I/Y}{C/Y}=\frac{7.8\%}{12}=0.65\%[/latex]

Step 4: Since FV=$0, skip this step.

Step 5: Use the formula for PV_ORD and solve for PMT.

[latex]n=P/Y \times \text{(Number of Years)}=12 \times 3=36[/latex]

[latex]\begin{align} PV_{ORD}&=PMT \left[\frac{1-(1+i)^{-n}}{i}\right]\\ \$25,\!000&=PMT\left[\frac{1-(1+0.0065)^{-36}}{0.0065}\right]\\ PMT&=\frac{\$25,\!000}{32.005957}=\$781.10 \end{align}[/latex]

Calculator instructions:

The figure shows how much principal and interest make up the payments. To pay off her consolidated loan, Morgan’s month-end payments for the next three years will be $781.10.

Franco has placed $10,000 into an investment fund with the goal of receiving equal amounts at the beginning of every month for the next year while he backpacks across Europe. If the investment fund can earn 5.25% compounded quarterly, how much money can Franco expect to receive each month?

Solution:

Step 1: The payments are at the beginning of the payment intervals, and the compounding period and payment intervals are different. Therefore, this is a general annuity due. Calculate the monthly amount he can receive, or PMT.

The timeline for the vacation money appears below.

Step 2: PV_DUE = $10,000; I/Y = 5.25%; C/Y = 4; P/Y = 12; Years = 1; FV = $0

Step 3: Since P/Y [latex]\ne[/latex] C/Y, calculate the equivalent interest rate (i_eq) that matches the payment interval.

[latex]i=\frac{I/Y}{C/Y}=\frac{5.25\%}{4}=1.3125\%[/latex]

[latex]i_{eq}=(1+i)^{\frac{C/Y}{P/Y}}-1=(1+0.013125)^{\frac{4}{12}}-1=0.004355998\; \text{per month}[/latex]

Step 4: Since FV=$0, skip this step.

Step 5: Calculate the payment amount, PMT, using the formula for PV_DUE.

[latex]n=P/Y \times \text{(Number of Years)}=12 \times 1=12[/latex]

[latex]\begin{align} PV_{DUE}&=PMT \left[\frac{1-(1+i_{eq})^{-n}}{i_{eq}}\right] \times(1+i_{eq})\\ \$10,\!000&=PMT \left[\frac{1-(1+0.004355998)^{-12}}{0.004355998}\right] \times(1+0.004355998)\\ PMT&=\frac{\$10,\!000}{11.717849}=\$853.40 \end{align}[/latex]

Calculator instructions:

The figure shows how much principal and interest make up the payments. While backpacking across Europe, Franco will have his annuity pay him $853.40 at the beginning of every month.

Kingsley’s financial adviser has determined that when he reaches age 65, he will need $1.7 million in his RRSP to fund his retirement. Kingsley is currently 22 years old and has saved up $10,000 already. His adviser thinks that his RRSP will average 9% compounded annually throughout the years. To meet his RRSP goal, how much does Kingsley need to invest every month starting today?

Solution:

Step 1: The payments are made at the beginning of the payment intervals, and the compounding period and payment intervals are different. Therefore, this is a general annuity due. Calculate the monthly amount Kingsley needs to contribute, or PMT.

The timeline for Kingsley’s RRSP contributions appears below.

Step 2: PV = $10,000; FV_DUE = $1,700,000; I/Y = 9%; C/Y = 1; P/Y = 12; Years = 43

Step 3: Calculate the equivalent interest rate (i_eq) that matches the payment interval.

[latex]i=\frac{I/Y}{C/Y}=\frac{9\%}{1}=9\%[/latex]

[latex]i_{eq}=(1+i)^{\frac{C/Y}{P/Y}}-1=(1+0.09)^{\frac{1}{12}}-1=0.007207323\; \text{per month}[/latex]

Step 4: You need to move the present value to Kingsley’s age 65, the same date as FV_DUE. Calculate the future value, FV. The FV is money the annuity does not have to save, so you subtract it from FV_DUE.

[latex]n=P/Y \times \text{(Number of Years)}=12 \times 43=516[/latex]

[latex]FV=PV \times (1+i_{eq})^n=\$10,\!000\times(1+0.007207323)^{516}=\$406,\!761.0984[/latex]

[latex]\text{New}\;FV_{DUE}=\$1,\!700,\!000-\$406,\!761.0984=\$1,\!293,\!238.902[/latex]

Step 5: Calculate the payment amount, PMT, using the formula for FV_DUE.

[latex]\begin{align} FV_{DUE}&=PMT \left[ \frac{(1+i_{eq})^n-1}{i_{eq}}\right]\times(1+i_{eq})\\ \$1,\!293,\!238.902&=PMT \left[ \frac{(1+0.007207323)^{516}-1}{0.007207323}\right]\times(1+0.007207323)\\ PMT&=\frac{\$1,\!293,\!238.902}{5,\!544.647665}=\$233.24 \end{align}[/latex]

Calculator instructions:

The figure shows how much principal and interest make up the final balance. To meet his retirement goals, Kingsley needs to invest $233.24 at the beginning of every month for the next 43 years. In doing so, he will achieve a $1.7 million balance in his account at age 65. (Note: Similar to loan payments, the last payment in actuality is required to be a slightly higher amount since the annuity payment was rounded downwards. However, the last payment is treated equally at this time for purposes of all calculations.)

The production department just informed the finance department that in five years’ time the robotic systems on the production line will need to be replaced. The estimated cost of the replacement is $10 million. To prepare for this purchase, the finance department immediately deposits $1,000,000 into a savings annuity earning 6.15% compounded semi-annually, and it plans to make semi-annual contributions starting in six months. How large do those contributions need to be?

Solution:

Step 1:

The payments are made at the end of the payment intervals, and the compounding period and payment intervals are the same. Therefore, this is an ordinary simple annuity. Calculate the monthly amount the finance department needs to contribute, or PMT.

The timeline for the machinery fund appears below.

Step 2: Give information:

PV = $1,000,000; FV_ORD = $10,000,000; I/Y = 6.15%; C/Y = 2; P/Y = 2; Years = 5

Step 3: Calculate the periodic interest rate, i.

[latex]i=\frac{I/Y}{C/Y}=\frac{6.15\%}{2}=3.075\%[/latex]

Step 4: The present value must be moved to the five-year date, the same date as FV_ORD. Apply Formula 9.2A. This FV is money the annuity does not have to save, so it is subtracted from FV_ORD to arrive at the amount the annuity must generate.

[latex]n=P/Y \times \text{(Number of Years)}=2 \times 5=10[/latex]

[latex]FV=PV \times (1+i)^n=\$1,000,\!000\times(1+0.03075)^{10}=\$1,\!353,\!734.306[/latex]

[latex]\text{New}\;FV_{ORD}=\$10,\!000,\!000-\$1,\!353,\!734.306=\$8,\!646,\!265.694[/latex]

Step 5: Use the formula for FV_ORD and solve for PMT.

[latex]\begin{align} FV_{ORD}&=PMT \left[\frac{(1+i)^n-1}{i}\right]\\ \$8,\!646,\!265.694&=PMT\left[\frac{(1+0.03075)^{10}-1}{0.03075}\right]\\ PMT&=\frac{\$8,\!646,\!265.694}{11.503554}=\$751,\!616.87 \end{align}[/latex]

Calculator instructions:

The figure shows how much principal and interest make up the final balance. To have adequate funding for the production line machinery replacement five years from now, the finance department needs to deposit $751,616.87 into the fund every six months. (Note: Similar to loan payments, the last payment in actuality is required to be a slightly lower amount since the annuity payment was rounded upwards. However, the last payment is treated equally at this time for purposes of all calculations.)