Managerial Finance Chapter 2 - Time Value of Money

1. Time Lines
   Time lines are used to make the problem clear. You would lay out a line with tic marks on it above which would be the time periods in numerical order. Between the tic marks would be the Interest Rates, (if they do not vary only listing it once would be enough). Below the ticks would be the cash flows, real or calculated.
2. Future Value
   To know what money is going to be worth in the future, we must be able to calculate it. The following are commonly used.
   
   PV is Present value
   
   i is set as interest rate, what the money will earn. It can also be noted as I or r (mostly used in financial literature)
   
   INT is the dollars of Interest earned. INT = Beginning amount X i
   
   FV_n is the amount that you have earned at the end of n amount of periods.
   
   n is the number of periods that we are calculating for.
   
   We could calculate each period by using the formula FV = PV + INT which can be broken down to FV = PV(1+i). By extension, we could say FV_n = PV(I + i)ⁿ
   This process is called compounding.
   
   These and all other problems can be solved by using a regular calculator, a financial calculator or a spreadsheet program. What works best for you is what you should use.
   
3. Present Value
   In previous example we figured out what our amount would be worth in the future. We can also calculate for what the value of a future amount of money would be worth now. This is defined as Present Value (PV). To figure it out you need to follow a procedure called discounting. Like before, a time line would be the best way to see what is going on visually. Using the formula from before we can extrapolate a formula to figure PV
   
   PV = FV_n (1/1+i)ⁿ
   
   Again, several ways to solve, chose your best way.
4. Solving for Interest Rate and Time
   FV_n = PV(I + i)ⁿ is a handy formula to know. If we want to solve for interest (i), we must know the other values, likewise for time (n).
5. Future Value of An Annuity
   An annuity is a series of equal payments made at fixed intervals for a specific number of periods. They can be Ordinary or Due.
   
   1. Ordinary Annuities
      In an ordinary Annuity, the payments are made at the end of a time period. An example would be an deposit into a savings account of 100 each year, what would it be worth at each time point. Unlike the FV problem, We keep putting money into the account on a regular basis. This now creates a series of FV problems that must be added together.
   2. annuities Due
      The difference between this and Ordinary Annuities is that payments are made at the beginning of the periods not at the end. This changes the problem slightly but it is still a series of FV problems.
6. Present Value of an Annuity
   No notes taken from book. Mostly how to do on various methods.
7. annuities: Solving for Interest Rate, Number of Periods, or Payment
   No notes taken from book. Mostly how to do on various methods.
8. perpetuites
   Most annuities call for the payments to be made over time, but, a perpetuities is an amount paid regularly for an indefinite period of time. To find the PV you would use:
   
   PV = PMT/i
   
   This winds up being different at different interest rates.
9. Uneven Cash Flow Streams
   Our book will use PMT (Payment ) for annuity situations and CF (Cash Flow) for uneven cash flows. We will use this for when situations develop that income does not come in as steady as it would in an annuities.
   1. Present Value of an Uneven Cash Flow Stream
      Like an annuity, this becomes a series of PV problems with each amount being calculated back to the 0 year and the sums being added together. Spreadsheets are good for this as are Financial calculators that have a CF register in them.
   2. Future Value of an Uneven Cash Flow Stream
      Live the above, we now calculate the other way and again sum things up.
   3. Solving for i with Uneven Cash Flow Streams
      This will really need to be done with a spreadsheet or financial calculator. Attempting it with a regular calculator is a hit or miss way of doing things.
10. Growing Annuities
    No notes taken from book. Mostly how to do on various methods.
11. Semiannual and Other Compounding Periods
    If an interest rate is compounded once a year that would be called annual compounding, twice a year is semiannual compounding. For problems we do we must know what type of periods we are doing and if the interest is per that period or annual. It may need to be changed to deal with a solution to the problem.
    1. Types of Interest Rates
       
       1. Nominal or quoted rate: Often called the Annual Percentage Rate (APR) is the value that is most often quoted but as noted above, it must be taken into account with the times that the rate will have interest charged.
       2. Periodic rate: This is the rate charged per a period. You would use it with time lines and with calculations.
       3. Effective (or equivalent) annual rate (EAR) - annual rate that produces the same results as if we had compounded at a given periodic rate m times a year.
    2. The Result of Frequent Compounding
       Because you earn interest on interest, frequent compounding will result in increased income.
12. Fractional Time Periods No notes taken from book. Mostly how to do on various methods.
13. Amortized Loans
    amortized loans include interest as well as principal payments back to the lender. It is necessary to us tools to figure out these breakdowns as to what is payment and what is interest.