9
Chapter 3
The Time Value of Money (TVM)
Axiom # 2. A dollar received today is worth more than a dollar received in the future. We can't compare these amounts unless we adjust for the time value of money.
There are very few finance problems that don't involve TVM.
What Types of Problems Will We Solve?
1. I need to borrow $15,000 for a car. What is my monthly payment if I repay over four years and the rate is 12%?
2. I want to buy a house in five years and will need a $25,000 downpayment. How much must I save every month if my bank pays 5% annually with monthly compounding?
3. You just inherited $100,000 from Aunt Mary and want to put this in a savings account next month. How much will you have in five years if the annual rate is 4%? How much more will you have if this rate is compounded quarterly?
Using a Financial Calculator
(has PV, PMT, FV, NPV, IRR keys)
You will need a financial calculator to solve these problems.
Only one supported in class is the TI BAII Plus. There is a mail-in rebate coupon in your book.
You will also need these notes to solve these problems.
It is not a 20 minute job to learn to use a financial calculator. Tips:
· See calculator manual and text's work book
· Adjust the nominal (annual) rate to a per period rate based on frequency of compounding or payments
· On the Texas Instruments BA II Plus, be sure to reset 2nd I/Y to P/Y = 1.00 and 2nd Format to four decimals. (New TI calculators have P/Y set to 12 which is not what you want for this class.)
· One of the two or three cash flows must be a negative number or you get an error message.
· It is safest to enter 0 if there are no payments and you are dealing with lump sums;
· You are responsible for calculating monthly loan payments or how long it take to repay a loan; other nonannual compounding problems may be asked.
Compound Interest and Future Value
Compound interest – interest paid on interest.
You have a bank account. The opening balance is $100 and 6% interest is compounded annually. How much will you have in three years?
Year Beginning Balance Interest Future Value
1 $100.00 $6.00 $106.00
2 106.00 6.36 ← 112.36
3 112.36 6.74 119.10
Why did you earn more in the second year than in the first? You now earn interest on the sum of the original principal (present value) and the interest you earned in the first year. You are earning interest on interest which is the concept of compound interest.
Future value is the sum of the present value and the interest paid.
Represents the value of an investment at some future point in time.
Good illustration in Figure 3.1
Present value is the current value in today's dollars of a sum received sometime in the future.
Inflation is a separate issue which we ignore (even though inflation does affect interest rate levels).
Reinvestment – taking money that you have earned on an investment and plowing it back into that investment and letting interest build up in your account.
This can be generalized to
FVn = PV (1 + i) n
Which says that the future value in period n (usually but not always a year) is the present value times (1 plus the interest rate) raised to the n th power were n equals the number of compounding periods.
What will your bank account total in three years if you deposit $100 today and earn 6% compounded annually?
Keystrokes: 100 +/- PV, 3 N, 6 I/Y, O PMT, CPT FV = 119.10
Each problem has at least two cash flows and the calculator requires one must be negative.
The convention is that you deposit (pay out) money at some point in time (a negative number because it leaves your hands) and at some point you withdraw it (a positive number because it returns to your hands).
The TVM formula underlies all time value calculations and works for any investment that pays a fixed amount of interest for the life of the investment.
Future Value Interest Factors (FVIF i,n ) – Table 3-1. This shows the future value of $1.00 when invested for n periods at a rate of i.
Problems: Tables do not address fractional interest rates (6.5%) or nonannual periods (2.5 years). Learn to use your calculator.
Note that the future value of an investment can be increased by
1. increasing the length of time it is compounded;
2. compounding at a higher rate; and/or
3. compounding more frequently.
Be able to solve the Jeep problems on top of page 64. (Note in one case you have $7,752 and in the other $11,167.)
TVM problems involve four or five variables: the present value, the future value, the interest rate per period, the number of compounding periods (usually years) and sometimes a payment or series of payments.
If you know three of the four variables, you can solve for the fourth;
If you know four of the five, you can solve for the fifth.
Present Value
We have discussed future value: how much will I have in three years if I put $100 in the bank today and it earns 6% (answer: $119.10 if compounded annually).
Now, how much must I put in the bank today if I want to have exactly $119.10 in three years if the rate is 16%?
In other words, what is the value today of money to be received in the future? Here we are solving for PV rather than FV.
119.10 FV, 3 N, 6 I/Y, 0 PMT, CPT PV = 100.00
Another way to look at present value is to think of it as the amount of money, that if invested at a given interest rate will generate a particular set of cash flows.
We are doing nothing more than inverse compounding. The term "discount rate" means the interest rate used to bring future money back to the present- that is, the interest rate to discount the future value back to the present.
For a single or a lump sum (and for lump sums only), the Present Value Interest Factor is the reciprocal of the Future Value Interest Factor:
PV = FV n ( 1 )
( ( 1 + i.)n )
Note that the present value of a sum is inversely related to
1. the number of periods for which it is discounted;
2. the rate at which it is discounted; and/or
3. the frequency with which the discount rate is compounded.
There is really only one TVM equation. Although you solve for different variables, the logic behind both is the same: To adjust for the time value of money we must compare dollar values, present and future adjusted to make them comparable in the same time period.
Because all present values are comparable (measured in dollars of the same time period) you can add and subtract the present values of inflows and outflows to determine the present value of an investment.
Caution: The comment in the "Stop and Think" box on page 72 could be misleading if you read it to say that purchasing power is more important than the interest rate.
The point is that over long periods of time, interest rates have averaged more than inflation rates providing a positive return in real terms.
Note that the compound growth techniques we have just described can also be used to calculate growth rates in prices or inflation rates.
Annuities
An annuity is a series of equal dollar payments coming at the end of each period of time for a specified number of periods (months or years). These are frequently encountered in finance problems.
There is a similar type of problem with payments that occur at the beginning of each period, such as those on rental contracts. These are called "annuities due" but we will ignore these problems.
Back to the basics: If you deposit $100 in a bank account each year beginning one year from now, how much will you have in three years if the interest rate is 6%?
0 = PV (nothing in the bank today), 3 N, 6 I/Y, 100+/- PMT,
CPT FV = 318.36
All we are doing is summing up the future value of each individual deposit
We can also present value an annuity. See Table 3.6. How much do I have to put in the bank today if I want to withdraw $500 at the end of each year for the next five years?
Other annuity problems ask if there is a certain amount in an account, how much can be withdrawn each period if I want nothing left at the end. Here FV is zero and we are solving for the payment that depletes the account while the balance earns a fixed rate.
Alternatively, the same problem could ask how long would my savings last if I withdrew $X every year.
The keystrokes or procedures are essentially the same although you are solving for different variables; the first solves for PMT and the second for N.
Nonannual Compounding
Very important concept but badly under emphasized in this text. (Nonetheless, you are responsible for solving these problems.)
What are we doing if we invest our money for ten years at 10% compounded semi-annually? What if it is compounded quarterly?
We need to examine for how many periods we are investing and what we are earning in each period.
If compounded semiannually, 10% per annum for ten years is really 5% for 20 periods;
If quarterly, the investment is for 40 periods at 2.5% per period.
Caution: if you are solving for a rate, the calculator will tell you the per period rate which you will need to convert to a per annum rate. With quarterly compounding, if I/Y shows 4%, this is 4% per period for four quarters which equals 16% per year.
See impact below. Obviously, the more frequent the compounding, the higher the return. Said another way, the shorter the compounding period, the quicker the investment grows.
Future Value of $100 Invested At
10% with Different Compounding Frequencies
FV in FV in
Compounded One Year Ten Years
Annually $110.00 $259.37
Semiannually 110.25 265.33
Quarterly 110.38 268.51
Monthly 110.47 270.70
Amortized Loans
Loans paid off in equal installments are called amortizing loans. Procedures applicable to annuities can be used to calculate payment amounts or the length of time required to pay off a loan.
Payments go to pay interest and then to reduce the principal. Look at how the payment is allocated:
Loan Amortization
$6,000 loan at 15% repayable annually over four years
The annual payment is $2,101.59
Payment Ending
Year Total To Interest To Principal Principal
1 $2,101.59 $900.00 $1,201.59 $4,798.41
2 2,101.59 719.76 1,381.83 3,416.58
3 2,101.59 512.49 1,598.10 1,827.48
4 2,101.59 274.11 1,827.48 -0-
Although the payment remains level, less and less of it is being allocated to interest because the remaining principal is declining, although slowly at first.
Later, more and more of the payment goes to reduce principal as the loan ages.
Usually, you would be asked to solve for the amount of the monthly payment or the interest rate.
I need to borrow $8,000 which I believe I can repay in monthly installments over three years. The bank tells me the rate is 12%. How much is my monthly payment? There are a number of steps used to solve this type of problem. We need to determine:
1. How many monthly payments? 36
2. If the annual rate is 12%, how much is the monthly rate? 1%
3. What will the value of the loan be when it is paid off? Nothing remains so let FV = 0.
4. What is the present value of the loan? The amount borrowed or $8,000 in this example.
So we have
8,000 PV, 36 N, 1.0 I/Y , and 0 FV; CPT PMT = -265.71
Perpetuities
This is an annuity with equal payments that continue forever (like the Energizer Bunny)
An example would be preferred stock that pays a fixed dividend but is not redeemable.
To calculate the present value of a perpetuity, merely divide the amount of the payment by the discount rate (i):
Present Value = Payment amount
Interest or discount rate