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Basics of Time Value of Money (TVM)

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Financial Ratios and Basics of Time Value of Money (TVM) (110 points)

Complete the following problems:

Problem 4-1: Compound Interest
Problem 4-2: Future value calculations
Problem 4-3: Present value calculations
Problem 4-4: Financial Ratios
Problem 4-5: Present value of annuity calculations
Problem 4-6: Future value of annuity calculations
Problem 4-7: Annuity Payments

Complete the problems in an Excel spreadsheet. Be sure to show all your work on the Excel Spreadsheet to receive credit; no hard keys.

Required

Chapter 5 in Foundations of Finance – The Time Value of Money
Hamza, H., & Ben, J. K. (2017). Money time value and time preference in Islamic perspective. Turkish Journal of Islamic Economics, 4(2), 19-35.
Suharto, U. (2018). Riba and interest in Islamic finance: Semantic and terminological ıssue. International Journal of Islamic and Middle Eastern Finance and Management, 11(1), 131-138.
Saraç, M., & Ülev, S. (2017). Investing in Islamic stocks: A wiser way to achieve genuine interest-free finance. Journal of King Abdulaziz University: Islamic Economics, 30(SI). Retrieved from https://ssrn.com/abstract=3061369

Module4 Critical Thinking Assignment:

Financial Ratios and Basics of Time Value of Money (TVM)

*Complete the problems from Module 3 and Module 4 in an Excel spreadsheet. Be sure to show your work to receive credit, no hard keys.

Problem 4-1: Compound Interest

To what amount will the following investments accumulate?

a. 5,000 SAR invested for 10 years at 10% compounded annually

8,000

SAR invested for 7 years at 8% compounded annually

c. 775 SAR invested for 12 years at 12% compounded annually

d. 21,000 SAR invested for 5 years at 5% compounded annually

Problem 4-2: Future value calculations

You would like to make a single investment and have 2 million SAR at the time your retirement in 35 years. You have found a mutual fund that will earn 4 percent annually. How much will you need to invest today? If you were a finance major and learned how to earn a 14% annual return, how much would you have to invest today?

Problem 4-3: Present value calculations

What is the present value of the following future amounts?

a. 800 SAR to be received 10 years from now discounted back to the present at 10%.

b. 300 SAR to be received 5 years from now discounted back to the present at 5%.

c. 1,000 SAR to be received 8 years from now discounted back to the present at 3%.

d. 1,000 SAR to be received 8 years from now discounted back to the present at 20%.

2,000

8,000

2019

8,000

Problem 4-4 Financial Ratios
The Balance Sheet and the Income Statement for Morris Manufacturing Corporation are as follows:
DATA (All amounts in SAR unless otherwise indicated, all sales are on credit and no hard keys.)
Morris Corporation Balance Sheet
	2019
Cash	500
Accounts receivable		2,000
Inventories	1,000
Other current assets	3,500
Net fixed assets	4,500
Total assets			8,000
Accounts payable	1,100
Accrued expenses	600
Short-term notes payable	300
Current liabilities
Long-term debt	2,000
Owners’ equity	4,000
Total liabilities and equity
Morris Corporation Income Statement
Sales (all credit)
Cost of goods sold	-3,300
Gross profits	4,700
Operating expenses (includes 500 depreciation)	-3,000
Operating profits	1,700
Interest expense	-367
Earnings before taxes	1,333
Income taxes	-280
Net Income	1,053
Calculate the following ratios:
1. Current ratio
2. Times interest earned 3. Inventory control 4. Total asset turnover 5. Operating profit margin 6. Operating return on assets 7. Debt ratio 8. Average collection period 9. Fixed-asset turnover 10. Return on equity

Problem 4-5: Present value of annuity calculations

What is the present value of a 10-year annuity that pays 2,500 SAR, given a 7% discount rate?

Problem 4-6: Future value of annuity calculations

In 10 years, you are planning on retiring and buying a house. The house you are looking at currently costs 100,000 SAR and is expected to increase in value each year at a rate of 5 percent annually. Assuming you can earn 10 percent annually on your investments, how much must you invest at the end of each of the next 10 years to be able to buy your dream home when you retire?

Problem 4-7: Loan Amortization

A man purchased a new house for 80,000 SAR. He paid 20,000 SAR down and agreed to pay the rest over the next 25 years in 25 equal end-of-year payments plus 9% compound interest on the unpaid balance. What will these equal payments be?

Chapter 5

The Time Value
of Money

5-‹#›

Learning Objectives

Explain the mechanics of compounding, and bringing the value of money back to the present.

Understand annuities.

Determine the future or present value of a sum when there are nonannual compounding periods.

Determine the present value of an uneven stream of payments and understand perpetuities.

COMPOUND INTEREST, FUTURE, AND PRESENT VALUE

Using Timelines to
Visualize Cash Flows
Timeline of cash flows

Simple Interest
Interest is earned only on principal.
Example: Compute simple interest on $100 invested at 6% per year for three years.
1st year interest is $6.00
2nd year interest is $6.00
3rd year interest is $6.00
Total interest earned: $18.00

Compound Interest
Compounding is when interest paid on an investment during the first period is added to the principal; then, during the second period, interest is earned on the new sum (that includes the principal and interest earned so far).

Compound Interest
Example: Compute compound interest on $100 invested at 6% for three years with annual compounding.
1st year interest is $6.00 Principal now is $106.00
2nd year interest is $6.36 Principal now is $112.36
3rd year interest is $6.74 Principal now is $119.10
Total interest earned: $19.10

Future Value
Future Value is the amount a sum will grow to in a certain number of years when compounded at a specific rate.
FVN = PV (1 + r)n
FVN = the future of the investment at the end of “n” years
r = the annual interest (or discount) rate
n = number of years
PV = the present value, or original amount invested at the beginning of the first year

Future Value Example
Example: What will be the FV of $100 in 2 years at interest rate of 6%?
FV2 = PV(1 + r)2 = $100 (1 + 0.06)2
= $100 (1.06)2
= $112.36

How to Increase the
Future Value?
Future Value can be increased by:
Increasing number of years of compounding (N)
Increasing the interest or discount rate (r)
Increasing the original investment (PV)
See example on next slide

Changing R, N, and PV
a. You deposit $500 in bank for 2 years. What is the FV at 2%? What is the FV if you change interest rate to 6%?
FV at 2% = 500*(1.02)2 = $520.20
FV at 6% = 500*(1.06)2 = $561.80
b. Continue the same example but change time to 10 years. What is the FV now?
FV = 500*(1.06)10= $895.42
c. Continue the same example but change contribution to $1,500. What is the FV now?
FV = 1,500*(1.06)10 = $2,686.27

Figure 5-2
Figure 5-2 illustrates that we can increase the FV by:
Increasing the number of years for which money is invested; and/or
Investing at a higher interest rate.

Computing Future Values using Calculator or Excel
Review discussion in the text book
Excel Function for FV:
= FV(rate,nper,pmt,pv)

Present Value
Present value reflects the current value of a future payment or receipt.

Present Value
PV = FVn {1/(1 + r)n}
FVn = the future value of the investment at the end of n years
n = number of years until payment is received
r = the interest rate
PV = the present value of the future sum of money

PV example
What will be the present value of $500 to be received 10 years from today if the discount rate is 6%?
PV = $500 {1/(1+0.06)10}
= $500 (1/1.791)
= $500 (0.558)
= $279.00

Figure 5-3
Figure 5-3 illustrates that PV is lower if:
Time period is longer; and/or
Interest rate is higher.

Using Excel
Excel Function for PV:
= PV(rate,nper,pmt,fv)

ANNUITIES

Annuity
An annuity is a series of equal dollar payments for a specified number of years.
Ordinary annuity payments occur at the end of each period.

FV of Annuity
Compound Annuity
Depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow.

FV Annuity – Example
What will be the FV of a 5-year, $500 annuity compounded at 6%?
FV5 = $500 (1 + 0.06)4 + $500 (1 + 0.06)3
+ $500(1 + 0.06)2 + $500 (1 + 0.06) + $500
= $500 (1.262) + $500 (1.191) + $500 (1.124)
+ $500 (1.090) + $500
= $631.00 + $595.50 + $562.00 + $530.00 + $500
= $2,818.50

FV of an Annuity –
Using the Mathematical Formulas
FVn = PMT {(1 + r)n – 1/r}
FV n = the future of an annuity at the end of the nth year
PMT = the annuity payment deposited or received at the end of each year
r = the annual interest (or discount) rate
n = the number of years

FV of an Annuity –
Using the Mathematical Formulas
What will $500 deposited in the bank every year for 5 years at 6% be worth?
FV = PMT ([(1 + r)n – 1]/r)
= $500 (5.637)
= $2,818.50

FV of Annuity:
Changing PMT, N, and r
What will $5,000 deposited annually for 50 years be worth at 7%?
FV = $2,032,644
Contribution = $250,000 (= 5000*50)
Change PMT = $6,000 for 50 years at 7%
FV = $2,439,173
Contribution= $300,000 (= 6000*50)

FV of Annuity:
Changing PMT, N, and r
3. Change time = 60 years, $6,000 at 7%
FV = $4,881,122
Contribution = $360,000 (= 6000*60)
4. Change r = 9%, 60 years, $6,000
FV = $11,668,753
Contribution = $360,000 (= 6000*60)

Present Value of an Annuity
Pensions, insurance obligations, and interest owed on bonds are all annuities. To compare these three types of investments we need to know the present value (PV) of each.

PV of Annuity –
Using the Mathematical Formulas
PV of Annuity = PMT {[1 – (1 + r)–1]}/r
= 500 (4.212)
= $2,106

Annuities Due
Annuities due are ordinary annuities in which all payments have been shifted forward by one time period. Thus, with annuity due, each annuity payment occurs at the beginning of the period rather than at the end of the period.

Annuities Due
Continuing the same example: If we assume that $500 invested every year for 5 years at 6% to be annuity due, the future value will increase due to compounding for one additional year.
FV5 (annuity due) = PMT {[(1 + r)n – 1]/r} (1 + r)
= 500(5.637)(1.06)
= $2,987.61
(versus $2,818.80 for ordinary annuity)

Amortized Loans
Loans paid off in equal installments over time are called amortized loans.
Example: Home mortgages, auto loans.
Reducing the balance of a loan via annuity payments is called amortizing.

Amortized Loans
The periodic payment is fixed. However, different amounts of each payment are applied toward the principal and interest. With each payment, you owe less toward principal. As a result, the amount that goes toward interest declines with every payment (as seen in Figure 5-4).

Amortization Example
Example: If you want to finance a new machinery with a purchase price of $6,000 at an interest rate of 15% over 4 years, what will your annual payments be?

Finding PMT –
Using the Mathematical Formulas
Finding Payment: Payment amount can be found by solving for PMT using PV of annuity formula.
PV of Annuity = PMT {1 – (1 + r)–4}/r
6,000 = PMT {1 – (1 + 0.15)–4}/0.15
6,000 = PMT (2.855)
PMT = 6,000/2.855
= $2,101.59

MAKING INTEREST
RATES COMPARABLE

Making Interest Rates Comparable
We cannot compare rates with different compounding periods. For example, 5% compounded annually is not the same as 5% percent compounded quarterly.
To make the rates comparable, we compute the annual percentage yield (APY) or effective annual rate (EAR).

Quoted Rate versus
Effective Rate
Quoted rate could be very different from the effective rate if compounding is not done annually.
Example: $1 invested at 1% per month will grow to $1.126825 (= $1.00(1.01)12) in one year. Thus even though the interest rate may be quoted as 12% compounded monthly, the effective annual rate or APY is 12.68%.

Quoted Rate versus
Effective Rate
APY = (1 + quoted rate/m)m – 1
Where m = number of compounding periods
= (1 + 0.12/12)12 – 1
= (1.01)12 – 1
= .126825 or 12.6825%

Finding PV and FV with
Nonannual Periods
If interest is not paid annually, we need to change the interest rate and time period to reflect the nonannual periods while computing PV and FV.
r = stated rate/# of compounding periods
N = # of years * # of compounding periods in a year
Example: If your investment earns 10% a year, with quarterly compounding for 10 years, what should we use for “r” and “N”?
r = 0.10/4 = 0.025 or 2.5%
N = 10*4 = 40 periods

THE PRESENT VALUE
OF AN UNEVEN STREAM AND PERPETUITIES

The Present Value of an Uneven Stream
Some cash flow stream may not follow a conventional pattern. For example, the cash flows may be erratic (with some positive cash flows and some negative cash flows) or cash flows may be a combination of single cash flows and annuity (as illustrated in Table 5-5).

Perpetuity
A perpetuity is an annuity that continues forever.
The present value of a perpetuity is given by
PV = PP/r
PV = present value of the perpetuity
PP = constant dollar amount provided by the perpetuity
r = annual interest (or discount) rate

Perpetuity
Example: What is the present value of $2,000 perpetuity discounted back to the present at 10% interest rate?
= 2000/0.10
= $20,000

Key Terms
Amortized loan
Annuity
Annuity due
Annuity future value factor
Annuity present value factor
Compound annuity
Compound interest
Effective annual rate (EAR)
Future value
Future value factor
Ordinary annuity
Present value
Present value factor
Perpetuity
Simple interest

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