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Chapter 4 Time Value of Money (cont.)

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Present value of multiple cash flows

Nominal interest rate and real interest rate

Effective interest rate

- 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- Interests are withdrawn at the end of every year but never the principal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.)

- Adjust your financial calculator
- Switch from “End” to “Begin
- The inputs are the same as an ordinary annuity

- 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}

- 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

- 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