The Time Value of Money

Assignment 2 Grading Criteria

Maximum Points

Calculated the compounded interest over 20 years and evaluated the value of the savings account upon closing. (CO 1)

Calculated the bonus payout over 20 years vs. a one time payout with interest and distinguished which bonus option would be better for the client. (CO 1)

Calculated the present value of the bonus and analyzed the difference in bonus for the client. (CO 2)

Analyzed the tuition costs for the client and determined what the future costs will be and determined how these funds can be accumulated over time. (CO 4)

Written Components: Organization, usage and mechanics, APA elements, style

Total:

200

Sources of Short-Term Financing (1 of 2)

Trade Credit

The most common form of financing available to a firm is trade credit or accounts payable. Most suppliers will allow 30–60 days credit (before the bill needs to be paid). Suppliers allowing you to delay payments are a ready source of free financing. Suppliers with many competitors that can provide equal products are often very willing to lock you in as a customer. Your business growth and success may be important to the supplier’s future. An example would be a home insulation supplier. In a local area, they might market to 50–100 contractors. If a supplier started a division dealing with the direct customer, they would become a competitor to the companies that they are supplying. Their product may have little to differentiate it from their competitors. Their advantage into the market may not be a lower price but longer terms. The terms become as important a feature as any product feature. This is true for simple commodity products or high-tech items.

To prevent firms from attempting to “stretch the payment period” (paying late), companies establish clear terms for paying bills. In addition to charging interest on late payments, they offer a discount for prompt payments. The most common purchasing terms today are 2/10 net 30. This means the customer receives a two percent discount off the invoice price if the bill is paid within ten days of the invoice date. Otherwise, payment of the entire bill, or the net, is due within 30 days. The cost of failing to take the discount is 36.72 percent.

Cost of failing to take a cash discount

=

Discount percent
100 percent – Discount percent

x

360
Final due date – Discount period

36.72%

=

2.04% x 18

=

2%
100% – 2%

x

360
(30 – 10)

Implementing these policies can have a major impact on company cash needs. As an example, a company with average daily sales of $5,000 with average collections of 45 days, would have an average daily balance in accounts receivable of $225,000 (45 days x $5,000). Putting in place a 2/10 net 30 policy could possibly reduce the average collection time down to 25 days. This reduces the average daily accounts receivable balance to $125,000 (25 x $5,000). This results in a $100,000 savings ($225,000 – $125,000). If the cost of capital is 10 percent, then the annual saving is $10,000 ($100,000 x 10%). In addition, consider the fact that if all customers were taking the 2 percent discount, over the course of a year you would have discounted $36,000 (2% x $5,000 x 360 days). The net loss in this example is $26,000 ($36,000 – $10,000).

Taking the discount on the accounts payables can influence profitability. Assume daily purchases to be $5,000. The total savings for the year (based on taking the discount) would be $36,000 (2% x $5,000 x 360 days).

 Sources of Short-Term Financing (2 of 2)

Bank Credit

In the 1980s and 1990s, the banking industry was deregulated. This deregulation has increased the size of banks, leading to some of the financial difficulties in late 2008. Deregulation has also increased competition between banks and savings and loans. Other intuitions such as CitiGroup can now provide full-service banking, including investing, commercial banking, life insurance, retail banking, and investment banking.

Most commercial bank loans are intended to be short term in nature. These loans are set up for a 90- to 180-day length of time. Companies have a tendency to roll these over, giving them the appearance of longer-term loans. The short-term nature allows the banks to review the health of the companies and to adjust the terms to the current market rate.

Prime

The prime rate is the lowest interest rate given to the lowest risk borrowers. This would be similar to having different mortgage rates for people who have an excellent credit rating of 800 verses people who have an average rating of 600. Everything is indexed up from the prime rate.

LIBOR

Europe has made the London Interbank Offered Rate (LIBOR) their standard. The international nature of modern business allows companies the choice to borrow dollars from European banks-based LIBOR in competition to borrowing from U.S. banks based upon the prime rate.

Commercial Paper

Commercial paper refers to loans that are made directly from companies to the public. Large companies such as GMAC and GE can issue paper (direct paper). Utility, industrial, and financial companies that are too small to have their own selling network issue paper with intermediate dealers (dealer’s paper). New companies that do not have a history for establishing credit can back the loan with assets (asset-backed commercial paper).

Foreign Borrowing

With the globalization of markets, shopping for a loan at the corner bank is outdated. Global banks market worldwide to find an optimum customer, maximizing their profit. United States’ consumption of foreign goods leaves many U.S. dollars abroad looking for a way home. These foreign banks with surplus dollars can offer loans at rates lower than prime in the U.S. market. This has lead some U.S.-based global companies to hold debt in foreign currencies. McDonald’s holds 70 percent of their $7.1 billion debt in foreign currencies.

 Use of Collateral in Short-Term Financing

A company buys materials or merchandise, adds value to them, and then sells them to customers. The conversion time may be short as in the case of a grocery store, or long in terms of building houses. Additionally, labor that is applied to the product is paid for at the time of application. At issue is the financing of supplies and labor until the final sale is made. The financing costs are just as real as the cost of materials and labor.

Materials and merchandise are held as assets in inventory. In the case of materials, they are converted into saleable merchandise. The bottom line is that both the material and merchandise are eventually purchased. At this point, they may become accounts receivable. 
Each business has unique characteristics regarding the duration of each phase. When possible, businesses can use the inventory and account receivables as collateral for short-term financing. Any loans that are made on these assets will require the bank to value these assets. In the case of inventory, the ability to liquidate that asset would be considered. In the case of accounts receivables, they would want to evaluate the quality of your accounts.

Hedging in the Financial Markets

In order for businesses to price products consistently in a changing financial world, they need to stabilize their costs. With raw materials, this could mean long-term contracts. Many commodity items, from corn to coal, are purchased using futures markets. Similarly, financial products can be purchased in a financial futures market. This would allow a company to factor in the cost of financing at a known price for a certain period. This process, called hedging, allows companies to limit their exposure to future risk.

 Time Value of Money (2 of 8)

We’ll now discuss methods to calculate the present value of future cash flows given current investments by using the inverse of the future value formula.

Future Value

Future Value (FV) refers to the amount of cash to be received or paid at a future date.

Single-period

Potential investments offer a rate of return of 9% per period. Given an initial investment of $2,000 how much cash would be available at the end of one period?

The investment will return the principal $2,000 plus 9% of $2,000 or $2,180 which is calculated as follows: 
FV = (1,000 + 1,000 * 0.09) = 1,000 (1 + .09) = 1,000 (1.09) = 1,090 
The FV for one period is:

FV = initial investment * (1 + interest rate)
The equation is: FV = PV (1 + r)
where, FV = Future value 
PV = Present value 
r = Periodic rate of return

Multi-period

To continue with the above example what will the investment be worth after four periods assuming that the accrued amount can be reinvested at a rate of 9%?

FV = 1,000 * 1.09 * 1.09 * 1.09 * 1.09 
= 1,090 * 1.09 * 1.09 * 1.09 = 1188.10* 1.09 * 1.09 = 1295.03 * 1.09 
= 1411.58

After four periods the value of the investment will be $1411.58. Although the above method correctly calculates the future value, it is inefficient because you need to multiply $1,000 with 1.09 four times. What if the investment was for 50 periods instead of 4?

We can simplify the equation by using exponents, as shown:
FV = 1,000 * (1.09)4 = 1,000 * (1.41158) = 1411.58

To calculate the FV for multiple periods, the equation is given by:
FV = initial investment * (1 + interest rate) time or

FV = PV (1 + r)n

where, n = number of periods

 Time Value of Money (3 of 8)

Example: Albert plans to retire in 15 years. Will he be able to afford a $200,000 condominium when he retires if he invests $100,000 in a 15-year certificate of deposit (CD) that pays 6% interest, compounded annually?

Solution: Yes, he will be able to purchase the condominium because he should have $100,000 (1.06)15 = 239,655.82 when he retires.

You can solve this problem in several ways:

· Using the formula and a scientific calculator

· Using a financial calculator

· Using an Excel spreadsheet

Using a financial calculator:

Input: PV = $100,000
I = 6
N = 15
PMT = 0
Compute: FV = $239,655.82

Note: Financial calculators use built-in sign conventions. So the answer may be $239,655.82. Cash inflows are represented by a plus sign and cash outflows are represented by a minus sign. Albert invests $100,000 in the CD, and this is a cash outflow. The result will be a positive value because Albert will receive that amount.

 Time Value of Money (4 of 8)

Present Value

Present Value (PV) calculations provide the basis for valuation analysis and pricing of financial instruments such as bonds and stocks.

Present Value refers to the current value of money — either paid or received — in the future. It is what investors will pay today for future cash flows.

PV calculations are the inverse of FV calculations. You learned earlier that:

FV = PV (1 + r)n

Dividing both sides by (1 + r)n yields:

FV/(1 + r)n = PV (1 + r)n/(1 + r)n

The interest factors cancel, and the equation is now:

PV = FV/(1 + r)n

Example: What is the present value of receiving $9,000 in seven years if alternative investments yield (opportunity cost is) 6%? 
Solution:

Using the PV formula:

PV = 9,000 * (1/1.5036) = 9,000 * 0.6650 = $5985.51

Using a financial calculator:

Input: FV = 9,000, I = 6, N = 7, PMT = 0
Compute: PV = 5985.51

Sometimes there may be numerous cash flows rather than just one value. The present value of a set of cash flows (PMT) occurring at different points in time is simply the sum of the present values for the cash flows, as shown:

PV = PMT1/(1 +Unit 3: Module 3 – … page 5

 Time Value of Money (5 of 8)

Perpetuity
Perpetuity refers to a stream of equal cash flows beginning from the next year and lasting forever. Calculating the present value of perpetuity is very simple unlike other time value of money calculations, but understanding the calculation requires some serious analysis. 
Obviously it is difficult to calculate the present value of perpetuity by individually discounting each cash flow. This calculation would require an infinite amount of time. You can use a short cut formula such as:
PV = PMT/r
where, PMT = equal cash flow 
r = interest rate
Example: What is the present value of $100 perpetuity at 10% interest rate?
Solution: PV = 100/0.1 = $1000
If you receive $100 every year forever starting one year from today, the sum total is worth $1,000 today.
Note the timing of the cash flow and the present value. The first cash flow arrives at the end of one period, but the present value is calculated as of today. The only financial instrument that typically provides a constant cash flow per period is preferred stock. So the price of preferred stock can be modeled as the next period’s dividend divided by the appropriate rate of return.
Annuity
Annuity refers to a stream of equal cash flows that occur over some finite period of time such as car loans, mortgages, leases, bonds, and interest rate swaps. Annuities are similar to perpetuities, but annuities end after a fixed number of years.
An ordinary annuity is a stream of equal cash flows that begins at the end of one period. An annuity due is a stream of equal cash flows that begins today. So a five-period annuity due will comprise a cash flow today and a four-period ordinary annuity. See the timeline, which shows the four-year ordinary annuity and annuity due for $100. 
Note that the present value of an ordinary annuity is less than the present value of an equivalent annuity due. In addition the future value of an ordinary annuity is less than the future value of an equivalent annuity due.

r)1+ PMT2 /(1 + r)2+ PMT3 /(1 + r)3+ … + PMTt/(1 + r)t

Unit 3: Module 3 – … page 6

 Time Value of Money (6 of 8)

Present Value of an Annuity
The present value of an annuity is the difference in the present values of two perpetuities that begin at different points in time, one in the first year and the other in t + 1 years.
We’ll first consider annuities as summations. Recall that the present value of a cash flow stream can be viewed as: 
PV of 4-year annuity = [100/(0.1)] – [100/{0.1(1.1)4}] = 316.99 
This is the value of the annuity at t = 0.
Formula: PV = (PMT/r) x [ 1 – (1/(1+r)n ]
where, PMT = period cash flow
r = period discount rate
n = number of periods or payments
Example: Martha, a customer, proposes the following deal. Should you accept it?
Martha wants to borrow $15,000,000 and repay it in installments of $2,500,000 every year over the next ten years. If the bank can earn 12% on new loans, should it extend the loan to Martha?
Solution: The present value of the repayment stream equals:
PV = (2,500,000/0.12) x [ 1 – { 1/( 1.12)10} ]
PV = 2,500,000 (5.650223028) 
PV = 14,125,557.57
Using a financial calculator:
FV = 0
PMT = 2,500,000
I = 12; N = 10
Compute: PV = 14,125,557.57
The bank should not approve the loan because the present value of payments is less than the loan amount.

Unit 3: Module 3 – … page 7

 Time Value of Money (7 of 8)

Future Value of an Annuity
FV = PV ( 1 + r )n
FV of an annuity is equal to the present value multiplied by (1 + r)n:
FV = PV ( 1 + r )n = (PMT/r) x [ 1 – (1/(1 + r)n ] ( 1 + r )n
FV = (PMT/r) x [ ( 1 + r )n – 1 ]
Example: Calculate the future value of a $100 four-year annuity at 10% interest rate.
Solution:
FV = (100/0.1) x [ (1.1)4 – 1)] = $464.10
The accumulated amount of the annuity at the end of 4 years is $464.10.
Using a financial calculator:
PMT = 100
N = 4
I = 10
PV = 0 
Compute FV = 464.10
FV = PV ( 1 + r )n = (316.99) x (1.1)4 = $464.10

Unit 3: Module 3 – … page 8

 Time Value of Money (8 of 8)

Compounding Periods
Usually interest rates are quoted on an annual basis (APR) although interest may actually be paid semiannually, quarterly, monthly, daily, or even continuously. 10% compounded monthly or 12% compounded quarterly rates are called stated ratesor nominal rates. They differ from the rate that is effectively earned because of the compounding effect in the annual period. For example 10% compounded semiannually does not provide the same return as 10% compounded annually. 10% compounded semiannually is actually 5% for a six-month period. In this case the periodic rate is 5% and effective annual rate is higher than 10%.
To calculate the periodic rate you need to divide the stated rate by the number of periods per year, as shown:
Periodic rate = [stated rate/Number of periods per year ] = APR/m
To calculate the Effective Annual Rate (EAR) use the following formula:
EAR = (1 + periodic rate)(# of periods/year) – 1
The equation for EAR is:
re = (1 + rp)m -1
where, re represents the effective annual rate, rp represents the periodic rate, and m represents the number of periods per year.
Example: What is the effective yield for a stated rate of 9% when interest is compounded semiannually and monthly?
Solution:
Rate of 9% compounded semiannually provides a return of:
1.045 * 1.045 = 1.092025
So 1.092025 – 1 = 9.2025%
Rate of 9% compounded monthly provides a return of:
(1.0075)12 – 1 = 9.3806898%
Using a financial calculator:
NOM = 9; C/Y = 2 Compute EFF = 9.2025%
NOM = 9; C/Y = 12 Compute EFF = 9.3807%
We will now apply the time value of money concept to real-world financial applications.

Unit 3: Module 3 – Applications of Time Value of Money

 Applications of Time Value of Money (1 of 6)

Real-World Financial Applications
We’ll discuss the following applications of the time value of money concept:
· Pension funds
· Mortgages
· Unequal cash flows
· Annuity due
· Net Present Value (NPV)
Pension Funds
Let’s say that you would like to work for 25-30 years after graduation and then retire. By the time you retire you want to accumulate $2,000,000 in the pension fund and invest the same amount each month till retirement. You expect to receive 12% (APR) of return on invested funds. This is a future value of annuity problem where you need to calculate the monthly amount for investment.
Using a financial calculator:
To be able to retire after thirty years or 360 months of service, the amount you’ll need to invest every month is calculated as shown, assuming monthly compounding:
Input: N = 360
I = 1% (Periodic rate)
FV = 2,000,000
PV = 0
Compute: PMT = 572.25 per month.
Therefore, you need to invest only $572.25 per month for 30 years to accumulate 2,000,000 at 12% per annum APR. The effective rate is 12.6825% per annum.
If you want to retire after 25 years, you will need to invest:
Input: N = 300
I = 1% (Periodic rate)
FV = 2,000,000
PV = 0
Compute: PMT = $1,064.48 per month
Most pension fund plans expect you to invest a reasonable amount per month over a long period of time.

3 – … page 2

 Applications of Time Value of Money (2 of 6)

Mortgages
Let’s say that you want to buy a house, and for this purpose you want to borrow $200,000 with the house as the collateral. The interest rate is 6% (APR) per annum. You plan to pay back the loan in equal monthly installments for 30 years. The monthly payment is:
(Note: This is a problem on present value of annuity.)
Using a financial calculator:
Input: N = 360 months
I = 0.5% per month
PV = 200,000
FV = 0
Compute: PMT = $1199.10 per month
Effective interest rate: 6.17%
If you want to pay off the loan in 15 years:
Input: N = 180 months
I = 0.5% per month
PV = 200,000
FV = 0
Compute: PMT = $1687.71 per month
Usually the interest rate for a fifteen-year loan is less, for example 5%. In this case:
Input: N = 180 months
I = 0.5% per month
PV = 200,000
FV = 0
Compute: PMT = $1581.59 per month
Similarly you can calculate monthly payments for auto loans.
Let’s say that you want to borrow $20,000 to buy a car. The interest rate is 6% APR, and you want to pay off the loan by making 60 equal monthly installments. How much is the monthly payment?

Unit 3: Module 3 – … page 3

 Applications of Time Value of Money (3 of 6)

Using a financial calculator:
Input: N = 60 
I = 0.5% periodic rate
PV = 20,000
FV = 0
Compute: PMT = $386.66 per month
Unequal Cash Flows

Year

1

2

3

4

Cash Flow

$1,000

$2,000

$500

$1500

Given these cash flows calculate the present value and the future value of the cash flows at 10% interest rate.
PV = 1000 / (1.1) + 2000/(1.1)2 + 500/(1.1)3 + 1500/(1.1)4
      = 909.09 + 1,652.89 + 375.66 + 1024. 52
      = $3962.16
FV = PV (1 + r)n = 3962.16 (1.1)4 = $5801
Annuity Due
Earlier you calculated the present value and the future value of an ordinary annuity. Now we’ll calculate the PV and the FV of an annuity due.
PV(annuity due) = [ PV(ordinary annuity) ] x (1+r)
FV(annuity due) = [ FV(ordinary annuity) ] x (1+r)
Example: Calculate the PV and FV of a four-year $100 annuity due.
Note: Refer to the earlier lecture for the formulas for PV and FV:
PV(annuity due) = 316.99 (1.1) = 348.69
FV(annuity due) = 464.10 (1.1) = 510.51
 

Unit 3: Module 3 – … page 4

 Applications of Time Value of Money (4 of 6)

Using a financial calculator:
Note: The calculator should be in Begin or annuity due mode.
Input: N = 4
I = 10
PMT = 100
FV = 0 
Compute: PV = 348.69
Input: N = 4
I = 10
PMT = 100
PV = 0 
Compute: FV = 510.51
Net Present Value
The Net Present Value (NPV) of a cash flow is the present value minus the initial investment or cost.
NPV = PV – Cost
Example: A project requires an initial investment of $3,000. It provides cash flows for four years as shown in the table. Calculate the NPV of the project.

Year

1

2

3

4

Cash Flow

$1,000

$2,000

$500

$1,500

Solution:
NPV = $3962.16 – 3000 = 962.16

Unit 3: Module 3 – … page 5

 Applications of Time Value of Money (5 of 6)

Planning Analysis
In discussing time value of money, we will look at several situations that illustrate why these financial calculations are critical to the planning process.
The first case to consider is the birth of a baby and the proud grandparents who want to set aside an amount of money to pay for a child’s college education. In this case, we will consider that we need to have $100,000 available in 18 years. The question is then assuming five percent interest rate, how much money needs to be deposited at the child’s birth. To find this answer we can use the table in your textbook’s Appendix A: Future Value of $1. Find the five percent column, and then follow it down to the 18 period row. This shows that the present value of each future dollar is $2.407. Dividing this into the $100,000 that we need in 18 years yields $41,545 to put into a savings account today.
In this next situation, the child does not have rich grandparents to set aside a large amount of money. The parents decide to set aside money each year so that at 18 the child would have $100,000. For this problem, we look to your textbook’s Appendix C: Future Value of an Annuity of $1. Again, find the five percent column, and then follow it down to the 18 period row. This shows that if we have an account that pays five percent interest, and we make $1 deposits each year on his birthday, we would have $28.132 for college expenses. Divide this 28.132 into $100,000. We find that each year a deposit of $3,554.67 would yield a $100,000 college fund. 
Now, you know why you are paying for your own college education. Moreover, thanks to the availability of student loans, you have borrowed the $100,000. A year after graduation you need to start making yearly deposits to retire the debt. Luckily, you are only charged five percent on the loan, and again you have 18 years to pay it off. Now, we need to look at your textbook’s Appendix D: Present Value of an Annuity of $1. Again, find the five percent column, and then follow it down to the 18 period row. This shows that for 18 yearly $1 payments, $11.69 cents is paid off our loan. Therefore, $100,000/11.69 leaves you with a yearly payment of $8,554.32.
That is more than twice the amount that the parents would have had to pay. Let us look at the total cost to each possible contributor.

Grandparents

1 x $41,454.00

Total $ 41,545.00

Parents

18 x $3,554.67

Total $ 63,984.06

Student

18 x $8,554.32

Total $153,977.76

This is a substantial difference. The proud grandparents and parents would have earned interest $38,455.00 and $36,015.94 respectively. You, however, borrowed the money. Hence, you will be paying $53,997.76 interest in addition to the principle. This is the difference between saving up for purchase and financing the purchase. Unit 3: Module 3 – … page 6

 Applications of Time Value of Money (6 of 6)

So, let us consider another option. You could win the $10 million lottery. This lottery is paid in 25 payments over 25 years. The first payment is delivered at the time of winning. Additional payments are paid on each anniversary for 24 years ($10,000,000 / 25 = $400,000). The $10 million is received over 25 years.
The second option is to take the lump sum of the present value of the $10 million. To do this, we would use the present value table, Present Value of $1. This time, let us look at the six percent column and down to the 25-year row. Here we see that the present value of an annuity factor is 12.783. To calculate the present value of the annuity, multiply 400,000 by 12.783. The result is the present value of the annuity given six percent over 25 years, equaling $5,113,360. This amount is a little over one-half the $10 million.
Which is better, the annuity or lump-sum payout? That would be your decision to make!