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### Corporate Finance

FINA1003, Fall 2009

Xiaohui Gao

Faculty of Business and Economics

University of Hong Kong

Corporate Finance, Xiaohui Gao

Time Value of Money(summary of single cash flow)

- Time value of money refers to
- A dollar today is better than a dollar in the future
- To compare money flows, we must convert them to the same time point.
- Basic formula:
- where:
- PV = value of original investment
- FV = future value of investment
- n = number of periods
- r = interest rate per period

Corporate Finance, Xiaohui Gao

1

2

3

$100

$200

$300

$400

PV=?

FV=?

Present Value and Future Value of a Stream of Cash FlowsThe present value or future value of a stream of cash flows is the sum of the PV or FV of each cash flow using given interest rate.

For example: uneven cash flow stream with r = 10%

Corporate Finance, Xiaohui Gao

Multiple Cash Flows – FV

- Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years?
- FV = 500(1.09)2 + 600(1.09) = 1248.05
- How much will you have in 5 years if you make no further deposits?
- First way: FV = 500(1.09)5 + 600(1.09)4 = 1616.26
- Second way – use value at year 2:

FV = 1248.05(1.09)3 = 1616.26

Corporate Finance, Xiaohui Gao

PV of Multiple Cash Flows-cash up front v.s. an installment plan

Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money (or the discount rate) is 8%, which method do you prefer?

Corporate Finance, Xiaohui Gao

PV of Multiple Cash Flows

To calculate the PV for a stream of cash flows, we first calculate the PV for each cash flow, then add them together.

If we want to find the FV at time t for the same stream of cash flows, we can use the basic formula after finding FV.

Corporate Finance, Xiaohui Gao

Using Timeline – interest rate 12%

0

1

2

3

4

200

400

600

800

178.57

318.88

427.07

508.41

FV = 2254.74

1432.93

Corporate Finance, Xiaohui Gao

Multiple Cash Flows Using the Calculator

- Using N, I/Y, PV, and FV for each cash flow, then add up
- Another way is to use the cash flow keys
- Press CF and enter the cash flows beginning with year 0.
- You have to press the “Enter” key for each cash flow
- Use the down arrow key to move to the next cash flow
- The “F” is the number of times a given cash flow occurs in consecutive periods
- Use the NPV key to compute the present value by entering the interest rate for I, pressing the down arrow and then compute
- Clear the cash flow keys by pressing CF and then CLR Work

Corporate Finance, Xiaohui Gao

Annuities and Perpetuities

- Annuity
- A stream of equal (level) cash payments received at regular intervals for a specified number of periods.
- Perpetuity
- A stream of equal (level) cash payments received at regular intervals forever.

Corporate Finance, Xiaohui Gao

Annuity

Annuity - - a stream of equal (level) payments received at regular intervals for a specified number of periods.

Example of an annuity - 48 equal monthly payments on a car loan.

Example of an annuity - 20 equal annual payments from winning the Lottery.

Corporate Finance, Xiaohui Gao

PV of Annuity Formula

PV of Annuity Formula

C = cash payment

r = interest rate

t = number of years cash payment is received

Corporate Finance, Xiaohui Gao

Annuity – Sweepstakes Example

- Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?
- PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29
- Using financial calculator: N = 30; I/Y = 5; PMT = 333,333.33; FV = 0; PV = -5,124,150.29

Corporate Finance, Xiaohui Gao

Present Value of an Annuity

What is the present value of a 10 year annuity of $1,000 a year at a discount rate of 9%?

Use the formula:

Use the financial calculator: PMT = 1,000, n = 10, i = 9% : PV = - 6417.66

Use spreadsheet

Corporate Finance, Xiaohui Gao

Spreadsheet Calculation

Corporate Finance, Xiaohui Gao

Annuity--Value of Payments

You want to buy a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?

Corporate Finance, Xiaohui Gao

Buying a House

You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?

Corporate Finance, Xiaohui Gao

Buying a House - Continued

- Bank loan
- Monthly income = 36,000 / 12 = 3,000
- Maximum payment = .28(3,000) = 840
- PV = 840[1 – 1/1.005360] / .005 = 140,105
- Total Price
- Closing costs = .04(140,105) = 5,604
- Down payment = 20,000 – 5,604 = 14,396
- Total Price = 140,105 + 14,396 = 154,501

Corporate Finance, Xiaohui Gao

Annuity-- Calculation of periodic payments

You purchase a house that costs $125,000. You put up $25,000 of your personal wealth and borrow the remaining $100,000 from a mortgage lender. You repay the loan of $100,000 in 360 monthly installments starting a month from today. What is the value of each monthly installment if the interest rate per month is 1%?

Corporate Finance, Xiaohui Gao

Solution

This is an example of an annuity, whose PV is $100,000

You need to solve for the level cash payment C

Using the formula for PV of an annuity:

or 100,000 = C*97.218. Hence C = $1028.61

This type of a loan in which the monthly payment is fixed over the life of the mortgage is called an amortizing loan.

Using your financial calculator to confirm the answer.

Corporate Finance, Xiaohui Gao

Note on Annuities Due

The annuities we have discussed thus far are called Ordinary Annuitiesbecause the first payment is received one period in the future.

If the first payment is received immediately, the annuity is called an Annuity Due.

This distinction is unnecessary from a valuation standpoint since every n period Annuity Due is equivalent to the sum of the payment amount and an n-1 period Ordinary Annuity.

Corporate Finance, Xiaohui Gao

Annuity Value and Annuity Due Value

Annuity Value. Your landscaping company can lease a truck for $8,000 a year (paid at year end) for 6 years. It can instead buy the truck for $40,000. The truck will be valueless after 6 years. If the interest rate your company can earn on its funds is 7%, is it cheaper to buy or to lease? PV(annuity) = 38,132.3 < 40,000

Annuity Due Value. Reconsider the previous problem. What if the lease payments are an annuity due, so that the first payment comes immediately? Is it cheaper to buy or to lease?

PV(Annuity Due) = 32,805 + 8,000 = 40,805 > 40,000

Corporate Finance, Xiaohui Gao

Future Value of an Annuity

The future value of an annuity can be calculated by computing the Present Value of an annuity and multiplying it by the Future Value Interest Factor.

Corporate Finance, Xiaohui Gao

Future Value of an Annuity

What is the future value of a 10 year annuity of $1,000 a year at an interest rate of 9%?

Using financial calculator -- PMT = 1,000, n = 10, i = 9% : FV =15,192.9297

Corporate Finance, Xiaohui Gao

Future Value of an Annuity

- Retirement planning makes extensive use of this methodology about the future value of an annuity.
- Example

You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?

Corporate Finance, Xiaohui Gao

Another Example

You need to accumulate $1,000,000 by the time you retire 30 years later. You decide to set aside a fixed amount each year for the next 30 years, starting one year from today, to pay for your retirement. Assuming an annual interest rate of 10%, what amount should you set aside each year?

Answer: C = $6079.25

Using both the formula and your financial calculator to confirm the answer.

Corporate Finance, Xiaohui Gao

Retirement Planning– PV and FV together

- A couple will retire in 50 years; they plan to spend about $30,000 a year in retirement, which should last about 25 years. They believe that they can earn 8 percent interest on retirement savings.

a. If they make annual payments into a savings plan, how much will they need to save each year? Assume the first payment comes in 1 year.

b. How would the answer to part (a) change if the couple also realizes that in 50 years, right before they start the retirement, they would spend $60,000 on a beach vacation house?

Corporate Finance, Xiaohui Gao

0

1

2

4

3

- $100

$10

$10

$10

$10

Cashflows

PerpetuityThe value of a perpetuity is the amount you would be willing to pay for the right to receive a constant payment forever.

If I invest $100 in a perpetuity that pays 10%, I should receive $10 each year forever. $100 x .10 = $10

Corporate Finance, Xiaohui Gao

Perpetuity

- Therefore the formula for the present value of a perpetuity is:
- Or
- C = cash payment
- r = interest rate

Corporate Finance, Xiaohui Gao

Valuing Perpetuities

Example

- Suppose you wish to endow a chair at your old university. The aim is to provide $100,000 forever and the interest rate is 10%.
- A donation of $1,000,000 will provide an annual income of .10 x $1,000,000=$100,000 forever.
- Another example: preferred stock.

Corporate Finance, Xiaohui Gao

Delayed Perpetuity

- A perpetuity might make its first payment after t+1 years. This is called a delayed perpetuity.
- How would you value this?
- Suppose you are now at the end of year t
- Now, it is like a normal perpetuity
- So, what is the PV today?
- Hence,

Corporate Finance, Xiaohui Gao

Delayed Perpetuity - Example

Again,you wish to endow a chair at your old university. But now the first perpetuity payment will not be received until four years from today,how much money needs to be set aside today? (Note: t + 1 = 4 means t = 3.)

Corporate Finance, Xiaohui Gao

Growing Annuities and Perpetuities

The value of a growing annuity is

The value of a growing perpetuity is

Corporate Finance, Xiaohui Gao

Table 6.2: summary of annuity and perpetuity

Corporate Finance, Xiaohui Gao

Perpetuity - Example

A local bank advertises the following deal: “Pay us $100 a year for 10 years and then we will pay you (or your beneficiaries) $100 a year forever.” Is this a good deal if the interest rate available on other deposits is 8 percent?

Answer: NO!!

At I = 8%

At year 10, the value of a 10-yr $100 annuity is 1,448

The value of a perpetuity is 1,250 < 1,448!

Corporate Finance, Xiaohui Gao

Periodic Interest Rates

- Interest rates can be quoted for days, months, quarters, or any convenient period other than years. These rates are often multiplied by the number of periods in one year to get the Annual Percentage Rate (APR).
- Example
- A monthly interest rate 1.5% is referred to as an 18% APR because 1.5%*12 = 18%.
- If APR is 20% and the interest is compounded quarterly, then the interest rate per period (one quarter here) is 5%.

Corporate Finance, Xiaohui Gao

Annual Percentage Rate (APR)

- Alternatively, APR is the interest rate that is annualized using simple interest.
- If APR is 18% and interest is compounded monthly, what is the actual equivalent annually compounded rate?
- When compounded, we get the following result:

(1+.18/12)12 = 1.01512 = 1.1956 or 19.56%.

- Therefore this 18% APR is equivalent to an annual rate of 19.56%.

Corporate Finance, Xiaohui Gao

Effective Annual Rates

- Effective Annual Rate (EAR) is the interest rate that is annualized using compound interest. In other words, EAR is the interest rate expressed as if it were compounded once per year.
- Example:
- You invest $100 in a bank that pays a monthly rate of 1%. It will grow to $100(1+.01)12 = $112.68 by the end of the year. So, the effect annual interest rate or EAR is 12.68%.
- APR is simply 12*1% = 12%.

Corporate Finance, Xiaohui Gao

EAR & APR

- When comparing interest rates, it is important to be consistent with the periods and the compounding.
- Most institutions report APRs (like banks).
- Never compare APRs. Compare EARs.
- The relationship between EAR and APR is as follows:

where n is the number of compounding periods per year

- with monthly compounding, n = 12
- with quarterly compounding, n = 4
- and so on…

Corporate Finance, Xiaohui Gao

Decisions, Decisions

- You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?
- First account:
- EAR = (1 + .0525/365)365 – 1 = 5.39%
- Second account:
- EAR = (1 + .053/2)2 – 1 = 5.37%
- Which account should you choose and why?

Corporate Finance, Xiaohui Gao

Things to Remember

- You ALWAYS need to make sure that the interest rate and the time period match.
- If you are looking at annual periods, you need an annual rate.
- If you are looking at monthly periods, you need a monthly rate.
- If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly.

Corporate Finance, Xiaohui Gao

Example

Al’s Autos has a 2.4% APR 48 monthly payment financing option on new $20,000 Corolla sedan. What is the monthly payment?

Solution: 2.4% APR is equivalent to 0.2% monthly interest rate. Hence, r = 0.2%, n = 48, PV = $20,000. Using the formula for PV of an annuity:

20,000 = C*45.72. Hence C = $437.4028

What is the EAR of this car loan? (2.43%)

Corporate Finance, Xiaohui Gao

Future Values with Monthly Compounding

- Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years?
- Monthly rate = .09 / 12 = .0075
- Number of months = 35(12) = 420
- FV = 50[1.0075420 – 1] / .0075 = 147,089.2237

Corporate Finance, Xiaohui Gao

Continuous Compounding

- Sometimes investments or loans are figured based on continuous compounding
- EAR = eq – 1
- The e is a special function on the calculator normally denoted by ex. e is the number 2.71828. The formula can be derived from taking the limit of EAR/APR formula.
- Example: What is the effective annual rate of 7% compounded continuously?
- EAR = e.07 – 1 = .0725 or 7.25%

Corporate Finance, Xiaohui Gao

Example

Savings and loan associations (S&Ls) were restricted to pay at most an APR of 5.5%. What is the rate that S&Ls are effectively paying their depositors?

Corporate Finance, Xiaohui Gao

Loan Types

- Pure Discount Loans: the borrower receives money today and repays a single lump sum at some time in the future. (Example: treasury bills)
- Interest-Only Loans: the borrower pays interest each period and repay the entire principal at some point in the future. (Example: corporate bonds)
- Amortized Loans: the borrower pays off the loan by making regular payment.
- A fixed amount each period
- A fixed amount for principal reduction plus interest for that period

Corporate Finance, Xiaohui Gao

Amortized Loan with Fixed Principal Payment - Example

Consider a $5,000, 5 year loan at 9% interest. The loan agreement requires the firm to pay $1,000 in principal each year plus interest for that year.

Since the principal is decreasing every year, the interest amount paid every year is decreasing as well.

Corporate Finance, Xiaohui Gao

Amortized Loan with Fixed Principal Payment - Example

Consider a $5,000, 5 year loan at 9% interest. The loan agreement requires the firm to pay $1,000 in principal each year plus interest for that year.

Corporate Finance, Xiaohui Gao

Amortized Loan with Fixed Payment - Example

- Each payment covers the interest expense plus reduces principal
- Consider a 4 year loan with annual payments. The interest rate is 10% and the principal amount is $1,000.
- What is the annual payment?
- N = 4
- I/Y = 10
- PV = 1,000
- CPT PMT = -315.47

Corporate Finance, Xiaohui Gao

Amortizing

“Amortizing” means that part of the monthly payment is used to pay interest on the loan and part is used to reduce the amount of the loan. ($1000, 10%, 4 years)

Corporate Finance, Xiaohui Gao

Partial Amortization

You take out a 30-year $100,000 mortgage loan with an APR of 8 percent and monthly payment. In 12 years you decide to sell your house and pay off the mortgage. What is the principal balance on the loan?

Answer: monthly payment is $733.76 and you have to pay $83,861.87 to pay off the mortgage.

$733.76*144 = $105,661.44 > $100,000

Corporate Finance, Xiaohui Gao

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