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Chapter 4 Time Value of Money (cont.) - PowerPoint PPT Presentation

Present value of multiple cash flows Nominal interest rate and real interest rate Effective interest rate. Chapter 4 Time Value of Money (cont.). Multiple Cash Flows.

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Nominal interest rate and real interest rate

Effective interest rate

Chapter 4 Time Value of Money(cont.)

• Usually an investment involve multiple/a stream of (negative/positive) cash flows instead of just one payment and one initial investment.

One term deposit

Several term deposits that end at the same time

Several withdrawals out of one deposit

• The future value of several cash flows paid (or several cash flows received) at a certain point of time can be calculated by adding up the future values of each of the cash flows.

• N specifies how many periods away from now is the FV that we want to calculated.

• Ct denotes the actual cash flow that is paid/received at the end of the tth period.

Example: If you make one term deposit of \$300 now and another 2 of \$200 at the end of each of the following two years, and all the deposit expires at the end of the 4th year from now. Interest rate is 8%. How much will your bank account balance be? (draw a time line and assign values to variables in the formula)

Example: (cont.)

• The present value of several cash flows paid (or several cash flows received) in future can be calculated by adding up the present values of each of the cash flows.

• Ct denotes the actual cash flow that is paid/received at the end of the tth period.

Example: If you need to make 3 payments at different point of time: one of \$250 now, a second payment of \$300 at the end of next year (the first year) and a third one of \$500 at the end of the year after next (the second year) . Interest rate is 8%. How much money should you have in your bank account now so that you would be able to make all the three payments at the specified time? (draw a time line and assign values to variables in the formula)

Example: (cont.)

• Using financial calculators:

• Calculate the FV/PV of each cash flows independently then sum the results together

• Make sure the correct t (i.e. N) is used for each cash flow

• When there are several cash flows paid and also several cash flows received, the formula to be used are the same:

• Make sure the correct sign is given to each cash flow

Perpetuity:

A stream of level cash payments that never ends.

Annuity:

Equally spaced level stream of cash flows for a limited period of time.

Assume:

• Deposit \$100

• Annual interest rate is 8% and it never changes

• Interests are withdrawn at the end of every year but never the principal

Cash flows:

• Pay \$100 now

• Receive \$8 at the end of every year forever

PV of Perpetuity: the value of all future cash flows from a perpetuity in terms of a one time payment now

Formula: for a perpetuity whose cash flows occur at the end of every period starting from now.

C = cash payment

r = interest rate / discount rate

Example - Perpetuity

In order to create an endowment, which pays \$100,000 per year, forever, how much money must be set aside today if the rate of interest is 10%?

Example - continued

If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

• Annuity can be viewed as the difference between two perpetuities

PV of Annuity: the value of all future cash flows from an annuity in terms of a one time payment now

Formula: for an annuity whose cash flows occur at the end of every period starting from now and lasting for t periods.

C = cash payment every period

r = interest rate

t = number of periods cash payment is received

PV Annuity Factor (PVAF) - The present value of \$1 a year for each of t years.

[Table A.3 on page 704 ]

• Find the appropriate PVAF according to the right t and r

Example - Annuity

To purchase a car, you are scheduled to make 3 annual installments of \$4,000 per year starting one year from now. Given a rate of annual interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?

• Example – Annuity (cont.)

• Switch from “End” to “Begin

• The inputs are the same as an ordinary annuity

• Example: start paying the installments right now

• HP

Press {shift} (i.e. the yellow button) and then press {BEG/END}

• TI

• Press {2nd}, then {BGN}

• Press {2nd}, then {SET}

• Press {2nd}, then {QUIT}

• To switch back from “Begin” to “End”, just repeat the procedure

• PV of and annuity due equals the multiple of the PV of the ordinary annuity and (1+r)

• Both annuities have the same annual payment and number of periods

• Example: start paying the installments right now

• Calculate the PV of corresponding ordinary annuity

• Multiply by (1+r)

• Present Value of payments

• Implied interest rate for an annuity

• Calculation of periodic payments

• Mortgage payment

• Annual income from an investment payout

• Future Value of annuity

• Example: In 1992, a nurse in a Reno casino won the biggest jack pot - \$9.3 million. That sum was paid in 20 annual installments of \$465,000. What is the PV? r=10% (draw a time line and assign values to variables in the annuity formula)

• Example:Suppose you are buying a house that costs \$125,000, and you want to put down 20% (\$25,000) in cash. Assume that the mortgage loan lasts 30 years, i.e. 360 months. What will be your monthly payment for each option, if the monthly interest rate is 1%? (draw a time line and assign values to variables in the annuity formula)

Example - Future Value of annual payments

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

Inflation: Rate at which prices as a whole are increasing.

• Consumer price index, CPI

Real Interest Rate: Rate at which the purchasing power of the return of an investment increases.

• Real value of money

Nominal Interest Rate: Rate at which money invested grows.

• Nominal value of money

• The quoted interest rate

• Exact formula

• Approximation formula

• Let r= real interest rate, i=inflation rate, and R= nominal interest rate.

Example

If the interest rate on one year government bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate?

• Effective Annual Interest Rate - Interest rate that is annualized using compound interest.

• Give the actual annual interests

• Annual Percentage Rate - Interest rate that is annualized using simple interest.

• Only a way to quote interest rates

• Imposed by legal requirements

• Example

Given APR of 12% and monthly compounding, what is the Effective Annual Rate(EAR)?

• First, calculate month interest rate

• Then, calculate the annual rate after compounding

• Mortgage Amortization (page 88)

• Periodic Payment = Amortization + Periodic Interest

• Periodic Interest = interest rate * prior period loan balance

Example: pay off 100,000 mortgage loan in 360 months at interest rate of 1% per month

Summary:

• Each periodic payment include amortization and interests due.

• As the loan approaches maturity, the amortizations paid increase every period.

• As the loan approaches maturity, the loan balances and interests due decrease every period.

• The last amortization is just enough to payoff the last part of principal.

Annuity Values

You want to buy a new car, but you can make an initial payment of only \$2,000 and can afford monthly payments of at most \$400.

• If the APR on auto loans is 12% and you finance the purchase over 48 months, what is the max price you can pay for the car?

• How much can you afford if you finance the purchase over 60 months?

• Rate on a Loan

If you take out an \$8,000 car loan that calls for 48 monthly payments of \$240 each, what is the APR of the loan? What is the EAR?

Amortizing Loan

You take out a 30-year \$100,000 mortgage loan with an APR of 6% and monthly payments. In 12 years you decide to sell your house and pay off the mortgage. What is principal balance on the loan