# Time Value of Money Four Problems

1/ In fifteen years you need $1,000,000 to move to Fiji. Ten years from now you can invest in a five year investment that pays 12% interest compounded quarterly. Five years from now you can invest in a five year investment that pays 10.5% compounded semiannually. If you are able to earn 8.25% interest compounded monthly for the next five years, how much money do you need to invest each month during that time (the next five years) to meet your future needs? (After completing this problem, take your solution and work it "forward" through the three different investments to see if you end up with $1,000,000).

2. Explain the concept of time value of money. What are the variables involved, how do they relate, and why is the concept important?

3.How much would you pay today for an annuity that pays you $1,000 at the end of year 1, $2,000 at the end of year 2, $3000 at the end of year 3, $4,000 at the end of year 4, and $5,000 at the end of year 5? Use annual compounding. Money is worth 12% to you.

4. What is the monthly payment on a $150,000, 30-year mortgage at a 6.5% interest rate?

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Answers in the attached file

1/ In fifteen years you need $1,000,000 to move to Fiji. Ten years from now you can invest in a five year investment that pays 12% interest compounded quarterly. Five years from now you can invest in a five year investment that pays 10.5% compounded semiannually. If you are able to earn 8.25% interest compounded monthly for the next five years, how much money do you need to invest each month during that time (the next five years) to meet your future needs? (After completing this problem, take your solution and work it "forward" through the three different investments to see if you end up with $1,000,000).

We know that we need $1,000,000 at the end of fifteen years. We have to make the investment over five years now which will be worth $1,000,000 at the end of 15 years. The investment that we make in five years would then grow at different rates during the next 10 years. We know the FV and we need to find the PV.

Let us start with the last five years and find out what amount will grow to 1,000,000 in five years at 12% compounded quarterly or 3% in 20 quarters. We can use the PVIF for 3% and 20 periods, this figure is 0.554 and the PV is 1000000*.554=554,000.

We need this amount at the end of 10 years with the last five years rate being 10.5% semi annual. We know find the PV of 554,000 for 10 periods and interest 5.25%. Since 5.25% will not be in a table we use the compound interest formula as below

554000=P*(1.0525)^10

which gives PV as 332,115.20. This is the amount we need at the end of 5 years and now we need to find the annuity value which will grow to 332,115.20 when compounded monthly at 8.25% p.a. or 0.6875% per month

for 60 months. Since we cannot get this in a table, we need to find the FVIFA factor. We can use excel to do this

and we get 824.24 and we get the annuity value as ...

#### Solution Summary

The solution explains the concept of time value of money as well as has some numerical solutions for calculating annuity amount and monthly payment on mortgage