For each problem, show your work steps. A correct answer with no work shown gets half credit (which means you fail the assignment). An incorrect answer with no work receives 0 credit. With time value of money (TVM) problems, that means showing your inputs for the financial calculator. Make your final answer very clear for the graders. Please don't make them hunt for your answer.

If in doubt, look at my solutions for the practice problems for examples of how to "show your work".

 

For example, on 1a, I would show: 

N = x; I/Y = x; PV = x; PMT = 0, Compute FV 

For uneven cash flows, show CF0 = xx, CF1 = xx F01 = x; CF2 = xx F02 = x, etc.  I/Y = xx, Compute PV 

 

Show all dollar and percentage values to two decimals ($xxx.xx) and percentage values to x.xx% (be sure to have enough decimals showing on your calculator). Rounded answers will receive point deductions. 

 

Do no interim rounding on interest rates. For example, if the interest rate is 4.7% with monthly compounding, enter 4.7 then the divide key, then 12, and the "=" key, then hit the I/Y key. That would be 4.7/12 = 0.39166.... Your calculator will probably show 0.3917 (assuming you are showing 4 decimals, or 0.39 if you only show 2 decimals). If you have followed my instructions, your calculator is holding the complete value, but only showing the number of decimals you have the calculator set to.

 

In some cases, an equation will be easier than using the TVM keys, show your equation if that is the case.

If you use Excel, show what the equation and inputs used.

 

As with all TVM problems, assume annual compounding unless otherwise specified. If you are given and interest rate, assume it is the annual percentage rate (APR).

 

1. (a) If you deposit $24,000 in the bank today, what is its future value at the end of eighteen years if it is invested in an account paying 5.80% interest (annual compounding, or APR)? 

(b) What is the present value of $24,000 to be received in eighteen years if the appropriate interest rate is 5.80% APR?

2. We sometimes need to find how long it will take a sum of money (or anything else) to grow to some specified amount. Note that you should enter PV as a negative and FV as a positive.

(a) For example, if a company's sales are growing at a rate of 5.80% per year, approximately how long will it take sales to triple? Show your answer to 2 decimals (x.xx years). If you are not sure how to work this, go back to this week's practice problems. Do not approximate using the "rule of "72".

 

(b) If you want an investment to double in eighteen years, what interest rate must it earn? Show your answer to 2 decimals (x.xx%).  

 

 

3. (a) You are saving for retirement, and you can afford to save $17,500 every year, starting one year from today. If you invest for 30 years at an annual interest rate of 6.20% per year, how much will you have saved for your retirement? Hint, this is the FV of an annuity. You may want to solve parts (c) and (e), then come back and solve (b) and (d), which are annuities due.

 

 

(b) How much would you have in your retirement account if you began these same 30 annual payments immediately? Hint: This is now the FV of an annuity due.

 

 

(c) Now let's look at things a little differently. Suppose that once you retire, you want to be able to withdraw $85,000 per year (starting one year from your retirement) for a total of 25 years during your retirement. How much would you need to have in your account when you retire to make this work assuming an annual interest rate of 6.20%? Hints: This is the PV of an annuity and remember that with this type of problem, you are withdrawing a set amount every year and at the end of the 25 years, your account has zero dollars in it. 

 

 

 

(d) How much would you need to have in your retirement account if you began these same 25 annual withdrawals immediately? Hint: This is now the PV of an annuity due. Reset your calculator to "END" of period payments when you have finished the annuity due problems.

 

 

(e) Changing the scenario, now let's assume that you want to have $1,250,000 in your retirement account at the end of 30 years. You have now decided that you will deposit funds at the end of every month for 30 years. The interest rate is still 6.20% per year. How much do you need to deposit each month in order to reach your goal in 30 years.?

 

4. Compare the results you got in part a) for future value of a "regular" annuity compare these to the value you got for the annuity due (part b). Now look compare the PV of the regular annuity in part (c) to the PV of an annuity due in part (d). What is the relationship that you see? Using the time value of money concepts, you have learned so far, why does this relationship (FV of regular annuity vs. annuity due and PV of regular annuity vs. annuity due) occur? (4 points)

5. In the spring of 2022, you and your family were looking for the house of your dreams. Given your household income and your expenses, you determined that you could afford to pay $1,200 for a monthly house payment. As of February 2022, it looked like you could get a 30-year mortgage rate of 3.00%.

a) Given the above information, what is the maximum amount you could finance for your dream home?

 

b) You got busy and didn't find the right home and now it is early 2023 and you are looking at homes again. The good news is that there are more homes available for sale, but the bad news is that mortgage rates have gone up because of inflation. Assuming that you can get a rate of 6.70% APR for a 30-year mortgage and further assuming that you can still afford to pay $1,200 per month, what is the highest amount you can now finance on a home?

BACKGROUND AND SETUP for part C: It is generally accepted that a 15% down payment is required to avoid "PMI" or private mortgage insurance. This insurance typically costs between 0.3% to 1.2% of the amount borrowed and depends on your credit score and down payment. That means your PMI would cost between $30 and $70 per month per $100,000 borrowed. Here is an example from Freddie Mac (a publicly traded, government-sponsored entity) that assumes a house purchased for $200,000. Note the differences in down payments and PMI requirements.

Assumptions	5% down payment & PMI	15% down payment & no PMI
Down Payment	$10,000	$40,000
Loan Amount	$190,000	$160,000
Mortgage Type	30-year	30-year
Interest rate	4.10%	4.10%
Mortgage Payment (principal & interest only)	$918.08	$773.12
PMI, assuming 0.55%	$104.50	$0.00
Total monthly payment	$1,022.58	$773.12

With a 15% down payment, PMI is waived.

 

c) For this part of the problem, assume that your grandparents left you $40,000 and it is sitting in a high interest savings account (3% APR with monthly compounding). You need to decide if you will use all or part of this money for a down payment for a $150,000 home. What are your major considerations here? Don't just repeat the numbers in the table but use them to frame your thoughts. What I'm looking for are a few pros and cons of your potential decisions. Some things to possibly consider: get a bigger house? Keep the money in the bank as an emergency fund? Use the money to pay down any other outstanding obligations? There is no one right or wrong answer here, but I want you to discuss some options and their impact on your finances. This question doesn't require a financial analysis, I'm asking for a qualitative answer, covering some of the key things you or another potential homeowner might consider in this decision.

 

 

6. What is the present value of the following uneven cash flow stream? The appropriate interest rate is 12.80%, compounded annually. Note that the final cash flow represents a project where there may be reclamation or other "end of project" costs which are greater than any final income and/or salvage value. Note that the final cash flow represents a project where there may be reclamation or other "end of project" costs which are greater than any final income and/or salvage value. 
 

 

7. What annual interest rate will cause $24,000 to grow to $32,000 in eighteen years (assume annual compounding)? Show your answer to 2 decimals (x.xx%)

 

8.Will the future value be larger or smaller if we compound an initial amount more often than annually—for example, every 3 months (quarterly) — holding the stated interest rate constant? Explain your answer. Zero credit unless there is an explanation.

9 a)  What is the future value of $24,000 (deposited today, no other deposits made) after eighteen years under 5.80% annual rate, with semi-annual compounding?  

b) What is the effective annual rate (EAR) for 5.80% annual interest, with interest compounded on a semiannual basis? Be sure to show your EAR answer to 2 decimals, that is xx.xx%  

 

c) What is the future value of $24,000 (as above) after eighteen years under 5.80% annual rate, with quarterly compounding?  

d) What is the effective annual rate (EAR) for 5.80% annual interest rate with quarterly compounding?  

 

e)  Explain how the effective annual rate changes based on the number of compounding periods per year.  

 

 

f) What is the future value of $24,000 (as above) after eighteen years under 5.80% annual interest, with daily compounding? Assume a 365-day year and do not any interim rounding.

 

g) What is the effective annual rate for 5.80% (APR) annual interest with daily compounding?  

 

10. Will the effective annual rate ever be equal to the simple (quoted) rate? Explain.

 

 

11. Six years ago, you bought a home with a purchase price of $225,000 and you paid 15% of that amount as a down payment and financed the remainder. Your mortgage loan terms are 30 years of monthly payments at an annual rate of 3.40%. Do no interim rounding on the calculated monthly interest rate.

a) How much are your monthly mortgage payments?

 

 

b) Over the life of the original loan, how much would you pay in interest?

 

 

c) Remember that you took this loan out 6 years ago (72 payments). You got a new job and are moving across the country. You will be selling this house and looking at buying something new. Considering the 6 years of payments you have made; how much do you still owe on your home?

 

 

12 a) Suppose on January 1 you deposit $25,000 in a savings account that pays a quoted interest rate of 2.60% (APR), with interest added (compounded) daily. How much will you have in your account on December 1, or after 11 months? (Assume N = 334 days) Recall that the interest rate (I/Y) represents the periodic rate based on how many times per YEAR the interest is compounded, hint, this is 365 times per year. Do no interim rounding on the interest rate. As above, and for all TVM type problems, there should be no interim rounding of the interest rates?

b) Now suppose you leave your money in the bank for 23 months. Thus, on January 1 you deposit $25,000 in an account that pays a 2.60% (APR), compounded daily. How much will be in your account on December 1 the next year? (Assume N = 699 days). Do no interim rounding on the interest rate. Do no interim rounding on the interest rate?