Question

investments involving regular payments (tvm solver) worksheet
name: clasie
according to some students, what is the true purpose of homework?
solve each question below. then find the corresponding answer at the bottom of the page. write the letter of the question above them.
o dave invests $70/month for 10 years with no initial payment, compounded annually at 12%. find the final amount.
n valerie opens an account at the royal bank with her initial amount of $2550.00. she contributes $200.00 a month and plans on taking it out when she retires in 35 years. it has an interest rate of 10.2% and is compounded semi - annually. how much will she have when she takes it out?
l you make an initial investment of $18 000.00, and you contribute $40.00 a week for 10 years. youve invested at an interest rate of 4.5%, compounded monthly. find you final amount.
i you want to purchase a car in 6 years for $30 000.00. you make an initial deposit of $8000.00 in your investment at 9%, compounded annually. how much per month will you have to contribute to buy the car?
a how long will it take for $1415.98 to double if invested at 7.85%, compounded monthly?
u you invest $3000.00 for 3 years at 6.95%, compounded monthly. what is your final amount?
r how much money will you have if you put $800.00 into an rrsp at the age of 18 and contribute $75.00/month until the age of 65, at an interest rate of 5%, compounded monthly?
i a donation of $150 000.00 was made to a university. the board of directors decided to invest the donation at 9%, compounded annually. how much was the final amount of the investment after 6 years?
s you invest $7000.00 for 5 years at 7%, compounded monthly. you contribute $15.00 per week. what is your final amount?
g you invest $500.00 at 7.5%, compounded annually. what is the final amount after 15 years?
n if you invest $10 000 in an rrsp at 3.5% interest, compounded monthly, how long will it take to double?
f determine the future value of weekly payments of $30 into an account that pays 1.75% interest, compounded weekly, for 1 year.
t determine the present value of a 3 - year csb with an interest rate of 3.9%, compounded semi - annually, if the future value if $2000.
f maria wants to buy a motorcycle in 4 years, when she turns 20. she deposits $80 every month in a savings account that earns 1.25%, compounded monthly. how much money will she have to buy her motorcycle when she is 20?
h louise is planning to renovate her house. she intends to spend no more that $30 000. she has $20 000 to invest in an account that pays 4.28%, compounded monthly. how many years will it take louise to meet her goal?
v xiao is investing $10 000. she wants it to grow to $15 000 in 5 years. what annual rate of interest, compounded semi - annually, does xiao need to meet her goal?

Answer:

To solve these problems, we use the Time - Value of Money (TVM) formulas. The two main types of TVM problems here are for compound interest (with or without regular payments) and for finding the time to double an investment (using the rule of 72 or the compound - interest formula).

Problem 1: Dave invests $70/month for 10 years with no initial payment, compounded annually at 12%. Find the final amount.

Step 1: Identify the type of TVM problem

This is a future - value of an ordinary annuity problem. The formula for the future value of an ordinary annuity is $F = A\times\frac{(1 + r)^{n}-1}{r}$, where $A$ is the annual payment, $r$ is the annual interest rate, and $n$ is the number of periods. First, we need to find the annual payment. Since the monthly payment is $70, the annual payment $A=70\times12 = 840$. The annual interest rate $r = 0.12$ and the number of years $n = 10$.

Step 2: Apply the formula

$$F=840\times\frac{(1 + 0.12)^{10}-1}{0.12}$$

First, calculate $(1 + 0.12)^{10}\approx3.105848$. Then, $(1.12)^{10}-1\approx2.105848$. Then, $\frac{2.105848}{0.12}\approx17.54873$. Then, $F = 840\times17.54873\approx14740.93$

Problem 2: Valerie opens an account with an initial amount of $25550.00, contributes $200.00 a month, for 35 years at 10.2% compounded semi - annually. Find the final amount.

Step 1: Separate the problem into two parts

We have a lump - sum (the initial amount) and an annuity (the monthly payments). First, convert the interest rate and the number of periods for the lump - sum and the annuity to match the compounding period (semi - annual for the lump - sum and we can also convert the monthly payments to semi - annual payments).

• For the lump - sum: The formula for compound interest is $A = P(1+\frac{r}{m})^{mt}$, where $P = 25550$, $r=0.102$, $m = 2$ (semi - annual compounding), and $t = 35$. So, $A_{1}=25550\times(1+\frac{0.102}{2})^{2\times35}=25550\times(1 + 0.051)^{70}$. Calculate $(1.051)^{70}\approx20.044$. Then, $A_{1}\approx25550\times20.044\approx512124.2$
• For the annuity: The monthly payment $A_{monthly}=200$, so the semi - annual payment $A_{semi}=200\times6 = 1200$. The interest rate per semi - annual period $i=\frac{0.102}{2}=0.051$, and the number of periods $n = 35\times2=70$. The formula for the future value of an ordinary annuity is $F = A\times\frac{(1 + i)^{n}-1}{i}$. So, $F_{2}=1200\times\frac{(1 + 0.051)^{70}-1}{0.051}$. We know that $(1.051)^{70}\approx20.044$, so $(1.051)^{70}-1\approx19.044$. Then, $\frac{19.044}{0.051}\approx373.41$. Then, $F_{2}=1200\times373.41\approx448092$.
• The total final amount $A = A_{1}+F_{2}\approx512124.2 + 448092=960216.2$

Problem 3: You make an initial investment of $18000.00, contribute $40.00 a week for 10 years at 4.5% compounded monthly. Find the final amount.

Step 1: Separate into lump - sum and annuity

• Lump - sum: $P = 18000$, $r = 0.045$, $m = 12$, $t = 10$. The compound - interest formula $A_{1}=18000\times(1+\frac{0.045}{12})^{12\times10}=18000\times(1 + 0.00375)^{120}$. Calculate $(1.00375)^{120}\approx1.5643$. So, $A_{1}\approx18000\times1.5643 = 28157.4$
• Annuity: Weekly payment $A_{weekly}=40$, annual payment $A_{annual}=40\times52 = 2080$. But the interest is compounded monthly. We can use the formula for the future value of an annuity with a different compounding period. First, find the effective annual rate (EAR) of the monthly - compounded rate. $EAR=(1+\frac{0.045}{12})^{12}-1\approx0.04594$. Then, we can use the future - value of an ordinary annuity formula with $A = 2080$, $r = 0.04594$, $n = 10$. $F_{2}=2080\times\frac{(1 + 0.04594)^{10}-1}{0.04594}$. Calculate $(1.04594)^{10}\approx1.5643$. Then, $(1.04594)^{10}-1\approx0.5643$. $\frac{0.5643}{0.04594}\approx12.28$. Then, $F_{2}=2080\times12.28\approx25542.4$. The total final amount $A=A_{1}+F_{2}\approx28157.4 + 25542.4 = 53699.8$ (Note: There are more accurate ways to handle the annuity with monthly compounding and weekly payments, but this is a simplified approach)

Problem 4: You want to purchase a car in 6 years for $30000.00. Initial deposit of $8000.00 at 9%, compounded annually. How much per month will you have to contribute?

Step 1: Find the future value of the initial deposit

Using the compound - interest formula $A = P(1 + r)^{t}$, where $P = 8000$, $r = 0.09$, $t = 6$. $A_{1}=8000\times(1 + 0.09)^{6}\approx8000\times1.6771\approx13416.8$

Step 2: Find the future value of the monthly payments

Let the monthly payment be $M$. The future value of the monthly payments (an ordinary annuity) compounded annually. First, find the effective monthly rate $i=\frac{0.09}{12}=0.0075$, and the number of periods $n = 6\times12 = 72$. The future value of the annuity $F = M\times\frac{(1 + 0.0075)^{72}-1}{0.0075}$. We know that the total future value needed is $30000$, so $30000=13416.8+M\times\frac{(1.0075)^{72}-1}{0.0075}$. Calculate $(1.0075)^{72}\approx1.71255$. Then, $(1.71255 - 1)=0.71255$. $\frac{0.71255}{0.0075}\approx95.0067$. So, $M\times95.0067=30000 - 13416.8 = 16583.2$. Then, $M=\frac{16583.2}{95.0067}\approx174.55$

Problem 5: How long will it take for $1415.98 to double if invested at 7.85%, compounded monthly?

Step 1: Use the compound - interest formula

We know that $A = P(1+\frac{r}{m})^{mt}$, where $A = 2\times1415.98=2831.96$, $P = 1415.98$, $r = 0.0785$, $m = 12$. So, $2=(1+\frac{0.0785}{12})^{12t}$. Take the natural logarithm of both sides: $\ln(2)=12t\times\ln(1+\frac{0.0785}{12})$. Calculate $\ln(1+\frac{0.0785}{12})=\ln(1.006542)\approx0.00652$. $\ln(2)\approx0.6931$. Then, $t=\frac{0.6931}{12\times0.00652}\approx8.8$ years

Problem 6: You invest $3000.00 for 3 years at 6.95%, compounded monthly. What is your final amount?

Step 1: Apply the compound - interest formula

$A = P(1+\frac{r}{m})^{mt}$, where $P = 3000$, $r = 0.0695$, $m = 12$, $t = 3$. So, $A = 3000\times(1+\frac{0.0695}{12})^{12\times3}=3000\times(1 + 0.00579)^{36}$. Calculate $(1.00579)^{36}\approx1.229$. Then, $A\approx3000\times1.229 = 3687$

Problem 7: How much money will you have if you put $800.00 into an RRSP at age 18 and contribute $75.00/month until age 65, at 5% compounded monthly?

Step 1: Separate into lump - sum and annuity

• Lump - sum: $P = 800$, $r = 0.05$, $m = 12$, $t=65 - 18 = 47$. $A_{1}=800\times(1+\frac{0.05}{12})^{12\times47}$. Calculate $(1+\frac{0.05}{12})^{564}\approx12.612$. So, $A_{1}\approx800\times12.612 = 10089.6$
• Annuity: Monthly payment $M = 75$, $r = 0.05$, $m = 12$, $t = 47$. The future value of the annuity $F = M\times\frac{(1+\frac{r}{m})^{mt}-1}{\frac{r}{m}}=75\times\frac{(1+\frac{0.05}{12})^{564}-1}{\frac{0.05}{12}}$. We know that $(1+\frac{0.05}{12})^{564}\approx12.612$, so $(12.612 - 1)=11.612$. $\frac{11.612}{\frac{0.05}{12}}=\frac{11.612\times12}{0.05}=2786.88$. Then, $F = 75\times2786.88 = 209016$. The total amount $A=A_{1}+F\approx10089.6+209016 = 219105.6$

Problem 8: A donation of $150000.00 was invested at 9%, compounded annually for 6 years. Find the final amount.

Step 1: Apply the compound - interest formula

$A = P(1 + r)^{t}$, where $P = 150000$, $r = 0.09$, $t = 6$. $A=150000\times(1 + 0.09)^{6}\approx150000\times1.6771\approx251565$

Problem 9: You invest $7000.00 for 5 years at 7%, compounded monthly. You contribute $15.00 per week. What is your final amount?

Step 1: Separate into lump - sum and annuity

• Lump - sum: $P = 7000$, $r = 0.07$, $m = 12$, $t = 5$. $A_{1}=7000\times(1+\frac{0.07}{12})^{60}\approx7000\times1.4176\approx9923.2$
• Annuity: Weekly payment $M = 15$, annual payment $A = 15\times52 = 780$. The interest is compounded monthly, so we find the effective annual rate $EAR=(1+\frac{0.07}{12})^{12}-1\approx0.0723$. The future value of the annuity $F = 780\times\frac{(1 + 0.0723)^{5}-1}{0.0723}$. Calculate $(1.0723)^{5}\approx1.4176$. Then, $(1.4176 - 1)=0.4176$. $\frac{0.4176}{0.0723}\approx5.776$. Then, $F = 780\times5.776\approx4505.28$. The total amount $A=A_{1}+F\approx9923.2 + 4505.28 = 14428.48$

Problem 10: You invest $500.00 at 7.5%, compounded annually. What is the final amount after 15 years?

Step 1: Apply the compound - interest formula

$A = P(1 + r)^{t}$, where $P = 500$, $r = 0.075$, $t = 15$. $A=500\times(1 + 0.075)^{15}\approx500\times2.8856\approx1442.8$

Problem 11: If you invest $10000 in an RRSP at 3.5% interest, compounded monthly, how long will it take to double?

Step 1: Use the compound - interest formula

$A = P(1+\frac{r}{m})^{mt}$, where $A = 20000$, $P = 10000$, $r = 0.035$, $m = 12$. So, $2=(1+\frac{0.035}{12})^{12t}$. Take the natural logarithm of both sides: $\ln(2)=12t\times\ln(1+\frac{0.035}{12})$. Calculate $\ln(1+\frac{0.035}{12})=\ln(1.002917)\approx0.002913$. $\ln(2)\approx0.6931$. Then, $t=\frac{0.6931}{12\times0.002913}\approx19.8$ years

Problem 12: Determine the future value of weekly payments of $30 into an account that pays 1.75% interest, compounded weekly, for 1 year.

Step 1: Apply the future - value of an ordinary annuity formula

The formula for the future value of an ordinary annuity is $F = A\times\frac{(1 + i)^{n}-1}{i}$, where $A = 30$, $i=\frac{0.0175}{52}\approx0.0003365$, and $n = 52$. $F = 30\times\frac{(1 + 0.0003365)^{52}-1}{0.0003365}$. Calculate $(1.0003365)^{52}\approx1.01765$. Then, $(1.01765 - 1)=0.01765$. $\frac{0.01765}{0.0003365}\approx52.45$. Then, $F = 30\times52.45\approx1573.5$

Problem 13