Reality Issues and Practical Applications

1
Reality Issues and Practical Applications
2
Introduction

• What will we learn in the class?
• 1) How to find equivalent present or future
  values when the interest rate is not constant
  over all time periods.
• 2) How inflation affects financial calculations.
• 3) How to calculate the interest and principal or
  equity in each payment in a uniform services
  which is used to repay a loan of P dollars.

3
Changing Interest Rates

• All interest formulas previously used assumed
  that the effective interest rate, i, was constant
  over the entire time span of the cash flows.
• In this section interest rates may change from
  one time period to the next.

4
Changing Interest Rates

• Assume it is the interest rate for the time
  period t, then the future value after n time
  periods , F, of a present sum, P, is obtained
  from Equation 3.1 page 84 as follows
• F P ( 1 i1) ( 1 i2 ) ( 1 i3 ) . . . ( 1
  in ) . . . . . Eq. 3.1
• The present value of a future sum, F, is
• P F ( 1 in )-1 ( 1 in-1 )-1 . . . ( 1 i1
  )-1 . . . . Eq. 3.2

5
Changing Interest Rates

• Suppose that the interest rate for these three
  periods was different.
• How would you compute the future value now?
• F1 P(1 11)1
• F2 F1(1 i2)1 P(1 11) (1 i2)
• F3 F2(1 i3)1 P(1 11) (1 i2) (1 i3)
• Every time the interest rate changes, you have to
  stop and only move the money that far. Why? The
  formula will only work where the interest rate is
  constant.

6
Changing Interest Rates
300
200
200
10
10
8
8
12
0
1
2
3
4
200
Figure 3.2 on p.85
7
Example 3.1

• Find the future value, F, at the end of period 9
  of a present value of 1000. The interest rate is
  8 for periods 1, 2 and 3, 10 in periods 4,5,6
  and 7 and 12 in periods 8 and 9.

8
Example 3.1
F
8
8
8
10
10
10
10
12
12
0
1
2
3
4
5
6
7
8
9
V7
V3
1000
9
Example 3.1
F9 1000 (1.08) (1.08) (1.08) (1.1) (1.1) (1.1)
(1.1) (1.12) (1.12) F9 1000 (1.08)3 (1.1)4
(1.12)2 F9 2313.53
10
Example 3.1
V3 1000(FP, 8, 3) V3 1000(1.2597) V3
1259.70 V7 1259.70 (FP, 10, 4) V7
1259.70(1.4641) V7 1844.33 V9 1844.33
(FP, 12, 2) V9 1844.33 (1.2544) V9
2313.53
11
3.3 Inflation

• In all of our previous calculations, the
  effective interest rate included inflation.
• What is inflation?
• Inflation is an increase in prices which is not
  accompanied by an improvement in the goods or
  services purchased.
• How is inflationary pressure created?
• Inflationary pressure is created when more
  dollars are put into an economy without an
  accompanying increase in goods and services.
• Why is it important to consider inflation when
  dealing with other countries?
• Inflation rates in other countries can be
  radically different

12
3.3 Inflation

• One of the components that makes up the interest
  rate, i, that is used in financial transactions
  is an estimate of future inflation.
• If the symbol, j, represents the inflation rate
  and d represents the "real" interest rate then
  Equation 3.5 gives the effective rate in term of
  j and d.
• (1 i) ( 1 d ) ( 1 j ) 1 d j d j
• i d j d j . . . . Eq. 3.5

13
3.3 Inflation

• One of the common measures of the general
  inflation rate for the U. S. Economy is the
  Consumer Price Index ( CPI ).
• Table 3.1 on page 89 gives the consumer price
  index for years 1967 through 1996.
• A measure of the general inflation rate j (all
  the goods and services that people typically
  purchase) in year t is the consumer price index
  at the end of year t divided by the consumer
  price index at the end of year t 1 minus 1.

14
3.3 Inflation

• For example
• The consumer price index at the end of 1988
  354.3
• The consumer price index at the end of 1987
  340.4
• Therefore, the inflation rate in 1988 is
• J1988 (354.3 / 340.4) 1 0.0408 4.1

15
3.3 Inflation

• Dollar amounts that have inflation included are
  called thencurrent dollars.
• All dollars considered prior to this section of
  the course were then-current dollars.
• When using thencurrent dollars the appropriate
  interest rate is the effective rate, i.
• To obtain thencurrent dollars in some future
  year k, Tk, you multiply then current dollars at
  the present time, T0, by the factor ( 1 j )k
• Tk T0 ( 1 j )K . . . . Eq. 3.7

16
3.3 Inflation

• Constant-worth dollars are then-current dollars
  with inflation removed.
• When using constant-worth dollars the appropriate
  interest rate to use is the "real" rate, d.
• To convert from thencurrent dollars, Tk, to
  constant-worth dollars Ck or visa versa use the
  following formula
• Tk Ck ( 1 j )k . . . . Eq. 3.7(a)

17
3.3 Inflation

• From equation 3.7 and 3.7(a) it is clear that T0
  Ck . Table 3.2 at the top of page 92 says that
  if the cash flow is in then-current dollars then
  use the combined (also called the effective)
  interest rate i.
• If the cash flows are in constant-worth dollars
  then use the "real" rate, d, to do the
  calculations.

18
3.3 Inflation - Problem 3.11, p.106

• 90,000 is invested in a program to reduce the
  material requirements in a production process.
  As a result of the investment, the annual
  material requirement is reduced by 10,000 pounds.
  The present unit cost of the material is 2 per
  pound. The price of a pound of the material is
  expected to increase at an annual rate of 8, due
  to inflation.
• Determine the combined interest rate that equates
  the present worth of the savings to the present
  worth of the investment over a 5-year period
• Based on the results obtained, determine the real
  interest rate that equates the two present
  worths.

19
3.3 Inflation - Problem 3.11, p.106
20,000(1.08)5
20,000(1.08)4
20,000(1.08)3
20,000(1.08)2
20,000(1.08)
0
1
2
3
4
5
90,000
20
3.3 Inflation - Problem 3.11, p.106
Ppaid Psaved 90,000 20,000 ( 1.08 ) ( P/A1 ,
i , 8 , 5 ) Solve for i 90,000 / 21,600 4.167
( P/A1 , i , 8 , 5 ) From the Tables ( P/A1 ,
10 , 8 , 5 ) 4.3831 and ( P/A1 , 15 , 8 , 5 )
3.8498
21
3.3 Inflation - Problem 3.11, p.106
(PA, i, 8,5)
4.3831
4.1670
3.8498
12
5
10
15
i
22
3.3 Inflation - Problem 3.11, p.106
i 10 ( 15 - 10 ) ( 4.383 4.167) /
(4.383 3.850 ) i 10 5 ( 0.216 / 0.533 )
11.9075 (b) d (1i)/(1j) - 1
(1.119075)/(1.08) - 1 0.0362 3.62
23
3.4 Principal and Interest in Loan Payments

• Most loans are repaid with a uniform series.
• However, each payment in the series contains two
  parts, an interest component and a principal or
  equity component.
• The interest amount in each payment decreases as
  the repayment schedule progress because the
  unpaid balance is decreasing.
• The authors present the following example
• P Loan Amount 10,000
• i Interest Rate 10
• n Number of Payments 4
• The payments are obtained as follows
• A P ( A/P , i , n ) 10,000 (A/P , 10 , 4 )
  3155

24
3.4 Principal and Interest in Loan Payments
The first payment in this uniform series of four
payments occurs at the end of period one. The
unpaid balance on the loan during period 1 is the
original loan amount of 10,000. Therefore, the
interest due at the end of period one, I1 P i
n 10,000 (0.1) (1) 1,000. The principal or
equity component of the loan payment is E1 A
I1 3155 1000 2155 The unpaid loan balance
after the first payment, U1, is U1 P E1
10,000 2155 7845
25
3.4 Principal and Interest in Loan Payments

• The interest accrued during period 2, I2, is the
  unpaid balance during period 2, U1 times the
  interest rate.
• I2 U1 i 7845 (0.1) 784.50
• And the equity portion of the second loan payment
  , E2 , is calculated
• E2 A I2 3155 784.50 2370.50
• This information is included in table 3.1 on page
  94.

26
3.4 Principal and Interest in Loan Payments

• Lets recap and generalize the formulas as follows
  
• The amount of a uniform series of payments , A ,
  required to repay a loan of , P , at interest
  rate , i , for n periods is
• A P ( A/P , i , n )
• The unpaid balance on the loan after payment
  number t is the present worth at time t of the (
  n t ) payments yet to be made.
• Ut A ( P/A , i , n-t ) . . . . Eq. 3.8
• The interest portion of payment t , It , is the
  unpaid balance after payment t-1, Ut-1 times the
  interest rate.
• It Ut-1 ( i ) . . . . Eq. 3.11
• The equity or principal portion of payment t , Et
  is the total payment A minus the interest , It
• Et A It . . . . Eq. 3.13

27
Problem 3.14, p.106

• Tom and Dale purchase a boat for 150,000 the
  down payment is 15,000 the balance is financed
  over a 10 year period. Equal monthly payments are
  made. Determine the amount of interest paid the
  first month of the monthly interest rate is 1.
• I1 (150,000-15,000)i (150,000-15,000)(0.01
  ) 1350.00

28
3.5 Corporate Bonds

• Bonds are a form of debt capital. In other words
  they are loans.
• When a corporation issues (sells) bonds they are
  borrowing money from the owners (buyers) of the
  bonds.
• This is in contrast to stocks. When an investor
  buys a corporations stock, the investor becomes
  an owner of the corporation not a lender to the
  corporation. Owners have different rights and
  responsibilities from lenders.
• The issuance and sale of bonds is a mechanism by
  which capital may be raised to finance projects
• Bonds represent investment opportunities for
  individuals

29
3.5 Corporate Bonds

• Owners have a say in how the corporation is
  managed lenders or bond holders generally do not.
  
• However, lenders (bond holders) do have priority
  on the corporations assets in the case of a
  bankruptcy.
• Generally, bonds are considered less risky than
  stocks.
• However, the potential return from bonds is also
  lower than that of stocks. (Risk and return are
  almost always related this way)
• All corporate bonds that are of interest in this
  section have the same structure to their cash
  flow.
• This structure is illustrated in Figure 3.3 on
  page 98 and below in a more general form.

30
3.5 Corporate Bonds
Sale Price
Dividend (A)
0
1
2
3
n
Purchase Price (P)
Initially one purchases the bond. This occurs at
the beginning of the time period of interest and
the symbol P is used to represent the purchase
price.
31
3.5 Corporate Bonds

• All corporate bonds of interest pay regular
  dividends.
• Dividends are paid annually, semi-annually, or
  most likely quarterly.
• The dividends are all the same amount and thus
  they constitute a uniform series .
• A symbol is used to represent the amount of
  each dividend to be received.
• F a future value received for the bond.
• The future value could be the sales price
  available in the market at time n when the bond
  owner wants to sell the bond.

32
3.5 Corporate Bonds

• Bond owners may also elect to keep the bond until
  it matures.
• What is meant by "maturity"?
• The maturity date is the date on which the issuer
  of the bond must pay the face value of the bond
  to the bond owner.
• Most people do not keep a bond until it matures.
  They buy and sell the bonds before maturity.

33
3.5 Corporate Bonds

• V par or face value of a bond
• The face value is generally 1,000 for corporate
  bonds.
• It is the value which is due and payable on the
  bonds maturity date.
• The par value is to be repaid by the issuing
  organization at the end of a specified period of
  time (5, 10, 15, 20 years)
• The issuing unit is obligated to redeem the bond
  for par value at maturity.
• r the bond rate per interest period, also
  referred to as the dividend rate
• A dividend - is the bonds dividend rate times
  face value. If dividends are paid more than once
  per year then the dividend is divided by the
  appropriate number to get the dividend amount.
• A Vr

34
3.5 Corporate Bonds Example 1

• Suppose a bond with a 1,000 face value has a
  dividend rate of 8 per year but the dividends
  are paid quarterly. How much are the quarterly
  dividends?
• A(quarterly) (1,000) (0.08/4) 20.
• The examples and exercise in the book have face
  values which are not 1,000. In practice this is
  rare.
• Most corporate bonds are issued with face values
  of 1,000.
• In fact bond brokers state the price of a bond as
  a percentage of face value under the assumption
  that the face value is 1,000.

35
3.5 Corporate Bonds

• Most corporate bonds are issued for 30 years.
• If you called a bond broker and asked the price
  for bonds issued by XYZ corporation and maturity
  in 2025.
• You might get an answer like 99. This means
  that you would be required to pay 99 of the face
  value or (0.99)(1,000) 990 plus commission.
• A quoted price of 102 would mean that the
  purchase price of the bond is (1.02)(1,000)
  1020.

36
Bond Prices Fluctuate

• Generally speaking bond prices move in the
  opposite direction from interest rates.
• When interest rates are increasing, bond prices
  go down.
• Therefore, one would want to buy bonds when
  interest rates are high and sell bonds when
  interest rates are low.
• The alternative to selling the bond before it
  matures is to hold it until its maturity date.
• Then you are certain about the price because
  bonds are redeemable at the face value on their
  maturity date.
• Most corporate bonds have a maturity date which
  is 30 years from the issue date.
• See page 100 for definitions of terms and symbols
  used to solve bond problems. Equation 3.15 is the
  general equation for most problems.

37
3.5 Corporate Bonds
Ppaid Preceived P Dividend (P/A , i , n ) F
(P/F , i , n) (Equation 3,15) P V r (P/A , i
, n ) F (P/F , i , n) Where P the purchase
price. V the face value. r the annual
dividend rate divided by the appropriate
constant. i the desired rate of return on the
investment (for the period between payment of
dividends). n the number of dividends
(periods). F the sales price for the bond ( F
is V if the bond is held to maturity, otherwise,
it is the market price at the time one wishes to
sell the bond). If r is given as an annual
number, but the dividend is paid quarterly, then
divide the rate by 4.
38
3.5 Corporate Bonds

• Three types of bond problems
• Given P, r, n, V and a desired i, find the sales
  price F
• Given F, r, n, V and a desired i, find the
  purchase price P
• Given P, F, r, n, and V, find the yield rate i
  that has been carried on in the investment

39
Problem 3.16, p.107
Dr. Shultz is considering purchasing a bond
having a face value of 2500 and a bond rate of
10 payable semiannually. The bond has a
remaining life of 8 years. How much should she
pay for the bond in order to earn a return on
investment of 14 compounded semiannually.
Assume the bond will be redeemed for face value.
r .10/2 .05 P V r (P/A , i , n ) F (P/F
, i , n) P 2500(.05) (P/A , i , n ) 2500 (P/F
, i , n) The key to solving this problem now is
determining the value of i and n. i .14/2
.07 n 16 P 2500(.05) (P/A , 7 , 16 ) 2500
(P/F , 7 , 16) P 125(PA,7,16)
2500(PF,7,16) 125(9.4466) 2500(0.3387)
2027.58
40
Problem 3.20, p.107
Dr. Shultz purchases a 10,000 face value bond
for 10,500. It matures in 15 years after paying
1200 at the end of each year. What yield rate
was earned on the investment? P V r (P/A , i ,
n ) F (P/F , i , n) 10500 1200 (P/A , i , 15
) 10000 (P/F , i , 15) 0 -10,500
1200(PA,i,15) 10,000(PF,i,15) Since Dr.
Shultz paid a premium for the bond, meaning he
paid more than 100 of the face value, then the
yield to maturity will be less than the dividend
rate. In a sense, he is taking a loss.
41
Problem 3.20, p.107
What is the dividend rate? r 1200/10000
12/year What is the current yield? Current yield
annual dividend/price Where Price what you
paid for the bond (not face value) In this case,
the price is higher than the face value of the
bond, which means that the current yield will be
less than 12. Current yield 12000/10500
11.4 When the current yield is less than the
dividend rate, the yield to maturity will be less
than the current yield. So, when looking for i, I
will look at values below 11.4.
42
Problem 3.20, p.107
What if I paid 9500 for that bond? Current yield
12000/9500 12.6 The current rate would be
greater than the dividend rate. Now look in the
12 and 10 tables to solve for i. At i 12 0
-10,500 1200(PA,i,15) 10,000(PF,i,15)
0 -10,500 1200(6.8109) 10,000(.1827) 0
-10,500 8173.08 1827 -499.92 At i
10 0 -10,500 1200(PA,i,15)
10,000(PF,i,15) 0 -10,500 1200(7.6061)
10,000(.2394) 0 -10,500 9127.32 2394
1021.32
43
Problem 3.20, p.107
Interpolating
44
3.6 Special and Limiting Cases of Time Value of
Money Factors
F P(1 i)n What happens to this expression
when n goes to infinity? F goes to infinity as
long as i is positive. What happens to this
expression when i goes to 0? F goes to P.
45
Formulas
ieff ( 1 r / m )m - 1