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Chapter 6 - Time Value of Money

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Chapter 6 - Time Value of Money

A sum of money in hand today is worth more than the same sum promised with certainty in the future.

Think in terms of money in the bank

The value today of a sum promised in a year is the amount you'd have to put in the bank today to have that sum in a year.

Example:Future Value (FV) = $1,000

k = 5%

Then Present Value (PV) = $952.38 because

$952.38 x .05 = $47.62

and $952.38 + $47.62 = $1,000.00

- Present Value
- The amount that must be deposited today to have a future sum at a certain interest rate

- Terminology
- The discounted value of a future sum is its present value

- Amounts
- Present value
- Future value

- Annuities
- Present value
- Future value

Four different types of problem

- Develop an equation for each
- Time lines - Graphic portrayals
- Place information on the time line

- How much will a sum deposited at interest rate k grow into over some period of time
If the time period is one year:

FV1 = PV(1 + k)

If leave in bank for a second year:

FV2 = PV(1 + k)(1 ─ k)

FV2 = PV(1 + k)2

Generalized:

FVn = PV(1 + k)n

(1 + k)n depends only on k and n

Define Future Value Factor for k,n as:

FVFk,n = (1 + k)n

Substitute for:

FVn = PV[FVFk,n]

- Problem-Solving Techniques
- All time value equations contain four variables
- In this case PV, FVn, k, and n

- All time value equations contain four variables
- Every problem will give you three and ask for the fourth.

How much will $850 be worth in three years at 5% interest?

Write Equation 6.4 and substitute the amounts given.

FVn = PV [FVFk,n ]

FV3 = $850 [FVF5,3]

Look up FVF5,3 in the three-year row under the 5% column of Table 6-1, getting 1.1576

Substitute the future value factor of 1.1576 for FVF5,3

FV3 = $850 [FVF5,3]

FV3 = $850 [1.1576]

= $983.96

- Work directly with equations
- How to use a typical financial calculator
- Five time value keys
- Use either four or five keys

- Some calculators require inflows and outflows to be of different signs
- If PV is entered as positive the computed FV is negative

- Five time value keys

- Basic Calculator functions

N

1

I/Y

6

FV

5000

PMT

0

PV

Answer

4,716.98

What is the present value of $5,000 to be received in one year if the interest rate is 6%?

Input the following values on the calculator and compute the PV:

Future and present value factors are reciprocals

- Use either equation to solve any amount problems

PV= FVn [PVFk,n ]

Finding the Interest Rate

what interest rate will grow $850 into $983.96 in three years. Here we have FV3, PV, and n, but not k.

Use Equation 6.7

PV= FVn [PVFk,n ]

PV= FVn [PVFk,n ]

Substitute for what’s known

$850= $983.96 [PVFk,n ]

Solve for [PVFk,n ]

[PVFk,n ] = $850/ $983.96

[PVFk,n ] = .8639

Find .8639 in Appendix A (Table A-2). Since n=3 search only row 3, and find the answer to the problem is (5% ) at top of column.

- Annuities
- A finite series of equal payments separated by equal time intervals
- Ordinary annuities
- Annuities due

- A finite series of equal payments separated by equal time intervals

- Future value of an annuity
- The sum, at its end, of all payments and all interest if each payment is deposited when received
- Figure 6-3 Time Line Portrayal of an Ordinary Annuity

FVFAk,n

- Four variables in the future value of an annuity equation
- FVAn future value of the annuity
- PMT payment
- k interest rate
- n number of periods
- Helps to draw a time line

Brock Corp. will receive $100K per year for 10 years and will invest each payment at 7% until the end of the last year.

How much will Brock have after the last payment is received?

- FVAn = PMT[FVFAk,n]
- FVFA 7,10 = 13.8164
- FVA10 = $100,000[13.8164] = $1,381,640

- Companies borrow money by issuing bonds
- No repayment of principal is made during the bond’s life
- Principal is repaid at maturity in a lump sum
- A sinking fund provides cash to pay off principal at maturity
- See Concept Connection Example 6-6

- Compounding
- Earning interest on interest

- Compounding periods
- Interest is usually compounded annually, semiannually, quarterly or monthly

- Effective annual rate (EAR)
- The annually compounded rate that pays the same interest as a lower rate compounded more frequently

Year-end Balances at Various Compounding Periods for $100 Initial Deposit and knom = 12%

Table 6.2

- EAR can be calculated for any compounding period using the formula
- m is number of compounding periods per year

Effect of more frequent compounding is greater at higher interest rates

- The annual percentage rate (APR) associated with credit cards is actually the nominal rate and is less than the EAR

- n must be compounding periods
- k must be the rate for a single
- E.g. for quarterly compounding
- k = knom divided by 4, and
- n = years multiplied by 4

- E.g. for quarterly compounding

Save up to buy a $15,000 car in 2½ years.

Make equal monthly deposits in a bank account which pays 12% compounded monthly

How much must be deposited each month?

A “Save Up” problem

Payments plus interest accumulates to a known amount Save ups are always FVA problems

Calculate k and n for monthly compounding,

and

n = 2.5 years x 12 months/year = 30 months.

Write the future value of an annuity expression and substitute.

FVAN = PMT [FVFAk,n ]

$15,000= PMT [FVFA1,30 ]

From Appendix A (Table A-3) FVFA1,30 = 34.7849 substituting

$15,000= PMT [34.7849]

Solve for PMT

PMT = $431.22

PVFAk,n

- Present value of an annuity
- Sum of the present values of all of the annuity’s payments

There are four variables

- PVA present value of the annuity
- PMT payment
- k interest rate
- n number of periods
- Problems give 3 and ask for the fourth

Shipson Co. will receive $5,000 every six months (semiannually) for 10 years. The firm needs cash now and asks its bank to discount the contract and pay Shipson the present value of the expected annuity. This is a common banking service called discounting.

If the payer has good credit, the bank will discount the contract at the current rate of interest, 14% compounded semiannually and pay Shipson the present value of the annuity of the expected payments. How much should Shipson receive?

Solution:

- An amortized loan’s principal is paid off over its life along with interest
- Constant Payments are made up of a varying mix of principal and interest
- The loan amount is the present value of the annuity of the payments

What are the monthly payments on a $10,000, four year loan at 18% compounded monthly?

Solution:

k= kNom /12 = 18%/12 =1.5%

n = 4 years x 12 months/ year = 48 months

PVA= PMT [PVFAk,n]

$10,000 = PMT [PVFAk,n ]

PVFA15,48 = 34.0426

$10,000 = PMT [34.0426]

PMT = $293.75

A car buyer and can make monthly payments of $500. How much can she borrow with a three-year loan at 12% compounded monthly.

Solution:

k= kNom /12 = 12%/12 =1%

n=3 years x 12 months/year = 36 months

PVA= PMT [PVFAk,n ]

PVA = PMT [PVFA1,36 ]

Appendix A (Table A-4) gives PVFA1,36 = 30.1075

PVA= 500 (30.1075)

PVA =$15,053.75

- Shows interest and principal in each loan payment
- Also shows beginning and ending balances of unpaid principal for each period
- To construct we need to know
- Loan amount (PVA)
- Payment (PMT)
- Periodic interest rate (k)

Develop an amortization schedule for the loan in Example 6 -12

Note that the Interest portion of the payment is decreasing while the Principal portion is increasing.

- Used to buy real estate
- Often the largest financial transaction in a person’s life
- Mortgages are typically amortized loans, compounded monthly over 30 years
- Early years most of payment is interest
- Later on principal is reduced quickly

- Mortgages are typically amortized loans, compounded monthly over 30 years

Calculate interest in the first payment on a 30-year,

$100,000 mortgage at 6%, compounded monthly.

Solution:

n= 30 years x 12 months/year = 360

k=6%/12 months/year = .5%

PVA= PMT [PVFAk,n ]

$100,000 = PMT [PVFA.5,360 ]

$100,000 = PMT [166.792 ]

PMT = $599.55

First month’s interest = $100,000 x .005 = $500

leaving $99.55 to reduce principal.

First payment is 83.4%

Next, solve for the monthly payment

PVA= PMT [PVFAk,n ]

$100,000 = PMT [PVFA.5,360 ]

$100,000 = PMT [166.792 ]

PMT = $599.55

The first month’s interest is .5% of $100,000

$100,000 x .005 = $500

The $500 of the first payment goes to interest, leaving $99.55 to reduce principal. The first payment is 83.4%

- Implications of mortgage payment pattern
- Early mortgage payments provide a large tax savings, reducing the effective cost of borrowing
- Halfway through a mortgage’s life, half of the loan is not yet paid off

- Long-term loans result in large total interest amounts over the life of the loan
- Adjustable rate mortgage (ARM)

- Payments occur at beginning of periods
- The future value of an annuity due
- Each PMT earns interest one period longer
- Formulas adjusted by multiplying by(1+k)
- FVAdn = PMT [FVFAk,n](1+k)
- PVAdn = PMT [PVFAk,n](1+k)

Baxter Corp started 10 years of $50,000 quarterly sinking fund deposits today at 8% compounded quarterly. What will the fund be worth in 10 years?

Solution:

k = 8%/4 = 2%

n = 10 years x 4 quarters/year x 40 quarters

FVAdn = PMT [FVFAk,n](1+k)

FVAd40= $50,000[FVFA2,40](1+.02)

FVAd40= 60.4020 from Appendix A (Table A-3).

FVAd40 = $50,000[60.4020](1.02)

=$3,080,502

- Annuity problems always involve a stream of equal payments with a transaction at either the end or the beginning
- End — future value of an annuity
- Beginning — present value of an annuity

- A stream of regular payments goes on forever
- An infinite annuity

- Future value of a perpetuity
- Makes no sense because there is no end point

- Present value of a perpetuity
- The present value of payments is a diminishing series
- Results in a very simple formula

Longhorn Corp issues a security that pays $5 per quarter indefinitely. Similar issues earn 8% compounded. How much can Longhorn sell this security for?

Solution: Longhorn’s security pays a quarterly perpetuity. It is worth the perpetuity’s present value calculated using the current quarterly interest rate.

k = .08 / 4 = .02

PVP = PMT / k = $5.00/.02 = $250

- Compounding periods can be any length
- As the time periods become infinitesimally short, interest is compounded continuously

- To determine the future value of a continuously compounded value:

First Bank is offering 6½% compounded continuously on savings deposits. If $5,000 is deposited and left for 3½ years, how much will it grow into?

Solution:

- Time value problems are often combined due to the complexity of real situations
- A time line portrayal can be critical to keeping things straight

Exeter Inc. has $75,000 in securities earning 16% compounded quarterly. The company needs $500,000 in two years.

Management will deposit money monthly at 12% compounded monthly to be sure of having the cash.

How much should Exeter deposit each month.

Solution: Calculate the future value of the $75,000 and subtract it from $500,000 to get the contribution required from the deposit annuity.

Then solve a save up problem (future value of an annuity) for the payment required to get that amount.

Find the future value of $75,000 with Equation 6.4.

FVn = PV [PVFk,n ]

FV8 = $75,000 [FVF4,8]

= $75,000 [1.3686]

= $102,645

Then the savings annuity must provide:

$500,000 - $102,645 = $397,355

- Use Equation 6.13 to solve for the required payment.
- FVAn = PMT [FVFA k,n ]$397,355 = PMT [FVFA1,24]
- $397,355 = PMT [26.9735]
- PMT = $14,731

- Many real problems have uneven cash flows
- These are NOT annuities
- For example, determine the present value of the following stream of cash flows

- These are NOT annuities

- Must discount each cash flow individually

Calculate the interest rate at which the present value of the stream of payments shown below is $500.

$100

$200

$300

We’ll start with a guess of 12% and discount each amount separately at that rate.

This value is too low, so we need to select a lower interest rate. Using 11% gives us $471.77. The answer is between 8% and 9%.

- Sometimes uneven streams cash have annuities embedded within them
- Use the annuity formula to calculate the present or future value of that portion of the problem