A financial adviser is reviewing one of her client’s accounts. The client has been investing $1,000 at the end of every quarter for the past 11 years in a fund that has averaged 7.3% compounded quarterly. How much money does the client have today in his account?

Solution:

Step 1: The payments are at the end of the payment intervals, and both the compounding period and the payment intervals are the same. This is an ordinary simple annuity. Calculate its value at the end, which is its future value, or FV_ORD.

The timeline shows the client’s account.

Step 2: PV = $0; I/Y = 7.3%; C/Y = 4; PMT = $1,000; P/Y = 4; Years = 11

Step 3: Calculate the periodic interest rate, i.

[latex]i=\frac{I/Y}{C/Y}=\frac{7.3\%}{4}=1.825\%[/latex]

Step 4: Skip this step since PV=$0.

Step 5: Apply Formula 11.2A to calculate the future value, FV_ORD.

n = P/Y  × (Number of Years) = 4 × 11 =44

[latex]\begin{align} FV_{ORD}&=PMT \left[ \frac{(1+i)^n-1}{i}\right]\\ &=\$1,\!000 \left[ \frac{(1+0.01825)^{44}-1}{0.01825}\right]\\ &=\$66,\!637.03 \end{align}[/latex]

Therefore, FV_ORD = FV = $66,637.03

Calculator instructions:

The figure shows how much principal and interest make up the final balance. After 11 years of $1,000 quarterly contributions, the client has $66,637.03 in the account.

A savings annuity already contains $10,000. If an additional $250 is invested at the end of every month at 9% compounded semi-annually for a term of 20 years, what will be the maturity value of the investment?

Solution:

Step 1: The payments are at the end of the payment intervals, and the compounding period and payment intervals are different P/Y [latex]\ne[/latex] C/Y. This is an ordinary general annuity. Calculate its value at the end, which is its future value, or FV_ORD.

The timeline is shown below.

Step 2: PV = 10,000; I/Y = 9%; C/Y = 2; PMT = $250; P/Y = 12; Years = 20

Step 3: Since P/Y [latex]\ne[/latex] C/Y, calculate the equivalent periodic rate, i_eq.

[latex]i=\frac{I/Y}{C/Y}=\frac{9\%}{2}=4.5\%[/latex]

[latex]i{eq}=(1+i)^{\frac{C/Y}{P/Y}}-1=(1+0.045)^{\frac{2}{12}}-1=0.007363123[/latex]

Step 4: Apply Formula 9.2A to calculate the future value, FV₁.

Given information: PV = $10,000; n = P/Y × (Number of Years) = 12 × 20 = 240

[latex]FV_1=\$10,\!000(1+0.007363123)^{240}=\$58,\!163.64538[/latex]

Step 5: Apply Formula 11.2A to calculate the future value of the payments, FV_ORD.

[latex]\begin{align} FV_{ORD}&=PMT \left[ \frac{(1+i_{eq})^n-1}{i_{eq}}\right]\\ &=\$250 \left[ \frac{(1+0.007363123)^{240}-1}{0.007363123}\right]\\ &=\$163,\!529.9486 \end{align}[/latex]

Combine steps 4 and 5 to calculate the total future value, FV.

FV = $58,163.64538 + $163,529.9486 =  $221,693.59

Calculator instructions:

The figure shows how much principal and interest make up the final balance. The savings annuity will have a balance of $221,693.59 after the 20 years.

Genevieve has decided to start saving up for a vacation in two years, when she graduates from university. She already has $1,000 saved today. For the first year, she plans on making end-of-month contributions of $300 and then switching to end-of-quarter contributions of $1,000 in the second year. If the account can earn 5% compounded semi-annually in the first year and 6% compounded quarterly in the second year, how much money will she have saved when she graduates?

Solution:

Step 1: There is a change of variables after one year. As a result, you need a Year 1 time segment and a Year 2 time segment. In both segments, payments are at the end of the period. In Year 1, the compounding period and payment intervals are different. In Year 2, the compounding period and payment intervals are the same. This is an ordinary general annuity followed by an ordinary simple annuity. You aim to calculate the future value, FV_ORD.

The timeline for her vacation saving appears below.

Step 2:

Time Segment 1:  PV = $1,000; I/Y = 5%; C/Y = 2; PMT = $300; P/Y = 12; Years = 1

Time Segment 2: PV = FV₁; I/Y = 6%; C/Y = 4; PMT = $1,000; P/Y = 4; Years = 1

For the first time segment:

Step 3: Since P/Y [latex]\ne[/latex] C/Y, calculate the equivalent periodic rate, i_eq.

[latex]i=\frac{I/Y}{C/Y}=\frac{5\%}{2}=2.5\%[/latex]

[latex]i{eq}=(1+i)^{\frac{C/Y}{P/Y}}-1=(1+0.025)^{\frac{2}{12}}-1=0.004123915[/latex]

Step 4: Apply Formula 9.2A to calculate the future value, FV₍₁₎.

Given information: PV = $1,000; n = P/Y × (Number of Years) = 12 × 1 = 12

[latex]FV_{(1)}=\$1,\!000(1+0.004123915)^{12}=\$1,\!050.625[/latex]

Step 5: Apply Formula 11.2A to calculate the future value of the payments_.

[latex]\begin{align} FV_{ORD_1}&=PMT \left[ \frac{(1+i_{eq})^n-1}{i_{eq}}\right]\\ &=\$300 \left[ \frac{(1+0.004123915)^{12}-1}{0.004123915}\right]\\ &=\$3,\!682.786451 \end{align}[/latex]

Calculate the total future value, FV₁.

FV₁  = $1,050.625 + $3,682.786451 = $4,733.411451

This becomes PV for the second time segment.

For the second time segment:

Step 3: Since P/Y = C/Y, calculate the periodic rate, i.

[latex]i=\frac{I/Y}{C/Y}=\frac{6\%}{4}=1.5\%[/latex]

Step 4: Apply Formula 9.2A to calculate the future value, FV₍₂₎.

PV = $4,733.411442 = FV₁; n = P/Y × (Number of Years) = 4 × 1 = 4  

[latex]FV_{(2)}=PV(1+i)^n=\$4,\!733.411442(1+0.015)^{4}=\$5,\!023.870384[/latex]

Step 5: Apply Formula 11.2A to calculate the future value of the payments.

[latex]\begin{align} FV_{ORD_2}&=PMT \left[ \frac{(1+i)^n-1}{i}\right]\\ &=\$1,\!000 \left[ \frac{(1+0.015)^{4}-1}{0.015}\right]\\ &=\$4,\!090.903375 \end{align}[/latex]

Combine steps 4 and 5 to calculate the total future value, FV₂.

FV₂ = $5,023.870384 + $4,090.903375 = $9,114.77

Therefore, FV₂ = FV = $9,114.77

Calculator instructions:

The figure shows how much principal and interest make up the final balance. When Genevieve graduates she will have saved $9,114.77 toward her vacation.

The Set for Life instant scratch n’ win ticket offers players a chance to win $1,000 per week for the next 25 years starting immediately upon validation. If a winner was to invest all of his money into an account earning 5% compounded annually, how much money would he have at the end of his 25-year term? Assume each year has exactly 52 weeks.

Solution:

Step 1: The payments start immediately, and the compounding period and payment intervals are different. Therefore, this a general annuity due. Calculate its value at the end, which is its future value, or FV_DUE.

The timeline for the lottery savings is below.

Step 2: PV = $0; I/Y = 5%; C/Y = 1; PMT = $1,000; P/Y = 52; Years = 25

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

[latex]i=\frac{I/Y}{C/Y}=\frac{5\%}{1}=5\%[/latex]

[latex]i{eq}=(1+i)^{\frac{C/Y}{P/Y}}-1=(1+0.05)^{\frac{1}{52}}-1=0.000938713\;\text{per week}[/latex]

Step 4: Since there is no PV, skip this step.

Step 5: Apply Formula 11.2B to calculate the future value of the payments, FV_DUE.

n = P /Y × (Number of Years) = 52 × 25 = 1300

[latex]\begin{align} FV_{DUE}&=PMT \left[ \frac{(1+i_{eq})^n-1}{i_{eq}}\right]\times(1+i_{eq})\\ &=\$1.\!000 \left[ \frac{(1+0.000938713)^{1300}-1}{0.000938713}\right](1+0.000938713)\\ &=\$2,\!544,\!543.218 \end{align}[/latex]

Therefore, FV_DUE = FV = $2,544,543.22

Calculator instructions:

The figure shows how much principal and interest make up the final balance. If the winner was to invest all of his lottery prize money, he would have $2,544,543.22 after 25 years.

When Roberto’s son was born, Roberto started making payments of $1,000 at the beginning of every six months to a trust fund earning 5.75% compounded monthly. After five years, he changed his contributions and started depositing $500 at the beginning of every quarter. How much money will be in his son’s trust fund when his son turns 18?

Solution:

Step 1: There is a change of variables after five years. As a result, you need two time segments. In both segments, payments are at the beginning of the period and the compounding periods and payment intervals are different. Therefore, Roberto has two consecutive general annuities due. Combined, calculate the future value, or FV_DUE.

The timeline for the trust fund is shown below.

Step 2:

Time segment 1: PV = $0; I/Y = 5.75%; C/Y = 12; PMT = $1,000; P/Y = 2; Years = 5

Time segment 2: PV = FV₁; I/Y = 5.75%; C/Y = 12; PMT = $500; P/Y = 4; Years = 13

For the first time segment:

Step 3: Since P/Y [latex]\ne[/latex]C/Y, calculate the equivalent periodic rate, i_eq.

[latex]i=\frac{I/Y}{C/Y}=\frac{5.75\%}{12}=0.4719\overline{6}\%[/latex]

[latex]i{eq}=(1+i)^{\frac{C/Y}{P/Y}}-1=(1+0.004719\overline{6})^{\frac{12}{2}}-1=0.029096609[/latex]

Step 4: Since the is no PV skip this step.

Step 5: Apply Formula 11.2B to calculate the future value.

n = P/Y × (Number of Years) = 2 × 5 = 10

[latex]\begin{align} FV_{DUE_1}&=PMT \left[ \frac{(1+i_{eq})^n-1}{i_{eq}}\right]\times(1+i_{eq})\\ &=\$1,\!000 \left[ \frac{(1+0.029096609)^{10}-1}{0.029096609}\right]\times(1+0.029096609)\\ &=\$11,\!748.47466=FV_1 \end{align}[/latex]

This becomes PV for the second time segment.

For the second time segment:

Step 3: Since P/Y [latex]\ne[/latex]C/Y, calculate the equivalent periodic rate, i_eq.

[latex]i\;\text{remains unchanged}\;=0.4719\overline{6}\%[/latex]

[latex]i{eq}=(1+i)^{\frac{C/Y}{P/Y}}-1=(1+0.004719\overline{6})^{\frac{12}{4}}-1=0.01444399[/latex]

Step 4: Apply Formula 9.2A to calculate the future value, FV₍₂₎.

Given information: PV = $11,748.47466 = FV_1;
n = P/Y × (Number of Years) = 4 × 13 = 52

[latex]FV_{(2)}=PV(1+i_{eq})^n=\$11,\!748.47466(1+0.01444399)^{52}=\$24,\!765.17[/latex]

Step 5: Apply Formula 11.2B to calculate the future value of the payments.

[latex]\begin{align} FV_{DUE_2}&=PMT \left[ \frac{(1+i_{eq})^n-1}{i_{eq}}\right]\times(1+i_{eq})\\ &=\$500 \left[ \frac{(1+0.01444399)^{52}-1}{0.01444399}\right]\times(1+0.01444399)\\ &=\$38,\!907.21529 \end{align}[/latex]

Combine steps 4 and 5 to calculate the total future value, FV₂.

FV₂ = $24,765.17 + $38,907.21529 = $63,672.38529

Therefore, FV₂ = FV = $63,672.39

Calculator instructions:

The figure shows how much principal and interest make up the final balance. When Roberto’s son turns 18, the trust fund will have a balance of $63,672.39.