Desiree Chesler



19.a. Interest, Present and Future Value, and Investment Decisions

Future Value and Present Value

Interest is generally defined as the price paid for borrowing and lending money and it is expressed as a percentage rate over a specified period of time. The rate of interest, represented by the term R, can be used to calculate the present value of future payments, as well as the future value of money invested today. Present value (PV), also called the bond price, can most easily be defined as the current value of a future cash payments, discounted at a specific interest rate. Future value (FV), on the other hand, is simply the value of a present amount of money at some specified time and interest rate in the future. When a person buys bonds, he or she is lending money and when a person writes and sells bonds, he or she is borrowing money. The seller of the bond agrees to repay the principal amount at a specified time along with interest to the buyer of that bond.

Below are the basic calculations of PV and FV for four different situations: one period, multiple periods, perpetuities, and annuities, along with some example problems.

One Period

A. A________(_______A (1+R) FV = A (1+R)

0. 1

Ex: You put $100 in the bank for one year at an interest rate of 10%. How much is it worth after one year? It is worth: FV = 100 (1+.10) = $110

B. B/(1+R)____(__ B PV = B/(1+R)

0. 1

Ex: A bond promises to pay $110 in one year at an interest rate of 10%. How much is it worth today? Today it is worth: PV =110/(1 +.10) = $100

Multiple Periods – “n periods”

A. A ( A(1+R) ( A(1+R)2 ( A(1+R)n FV = A(1+R)n

0 1 2 n

Ex: You put $100 in the bank at an interest rate of 10%. How much is it worth after 3 years? It is worth:

FV = 100(1.10)3 = $133

B. B/(1+R)n ( B/(1+R)2 ( B(1+R) ( B PV = B/(1+R)n

0 1 2 n

Ex: I owe you $133 in three years, with an interest rate of 10%. What is the bond price? The bond price is: PV = 133/(1.10)3 = $100

I. Perpetuity (a constant stream of identical cash flows without end)

A. B/R ( B ( B ( B PV = B/R

0 1 2 (

Ex: The holder of a bond receives a payment of $5 per year forever and the interest rate is 5%. How much is the bond worth? PV = 5/.05 = $100.

II. Annuity (a regular periodic payment for a specified period of time)

A. A(1+R)n FV= A(1+R) + A(1+R)2 +…+A(1+R)n

+A(1+R)2

A ( A ( A ( A(1+R)

0 1 2 n

Ex: You want to invest $100 per year for three years with an interest rate of 10%. What will the future value and earnings be? FV = 100(1.1) + 100(1.1)2 + 100(1.1)3 = 110 + 121 + 133 = 364. Earnings = FV – Principal = 364-300 = $64

B(1). B/(1+R)n PV = B/(1+R) + B/(1+R)2 + … + B/(1+R)n

+B/(1+R)2

B/(1+R) ( B ( B ( B

0 1 2 n

(2). Another formula used to calculate PV of annuities: PV – PV(1+R) = B/(1+R)n – B

a) multiply (1+R) by PV = B/(1+R) + B/(1+R)2 + … + B/(1+R)n (

b) PV(1+R) = B + B/1+R +…+B/(1+R)n-1 ( subtract (a) – (b) (

c) PV – PV(1+R) = B/(1+R)n – B

Ex: A person plans to pay you $100 per year for three years, at an interest rate of 10%. How much can you sell the bond for? PV = 100/1.1 +100/(1.1)2 +100/(1.1)3 = 91 +83 +75 = $249

Short Term vs. Long Term Bonds

There is a negative relationship between the interest rate (R) and present value (PV). That is, as R(, PV ( and as R(, PV(. This is becomes obvious once you realize that R lies in denominator of the PV equation. Thus, an increase in R causes an increase in the denominator and a decrease in PV, and vice versa for a decrease in R. The size by which present value increases or decreases, however, depends on whether the bond is short term or long term. With short-term bonds, there is a relatively small change in present value as a result of a change in the interest rate. On the other hand, with long term bonds, present value changes by a large amount if interest changes. This is due to the exponent in the present value equation. With a long term bond, for example 10 years, the denominator (1 + R) is raised to the 10th power, causing PV to change by a lot, compared to a 2 year bond, which would only square the denominator. Thus, there is a complex relationship between the interest rate, present value, and the length of a bond.

❑ R(, PV (

❑ R(, PV(

Uses of Present Value

The two most common uses for determining present value are for evaluating business investments and bond pricing.

1) Evaluating Business Investments

In order to calculate the present value of a business investment, it is necessary to know the cash outflow and inflow. The outflow (a negative amount) is the payment made in a certain investment and the inflow is the present value of the yearly payments received in the investment. Therefore, PV = -outflow + inflow, measuring how much better or worse off one is participating in the investment rather than putting the money in the bank. Thus, the PV is used to determine whether or not to accept the project. If the present value is greater than zero, or positive, one would accept the project. If the present value is negative, than the business investment would be rejected. Lastly, the projects must be mutually exclusive, wherein if you do one, you can’t do the other. With this in mind, it is obvious that one would choose the highest present value between the mutually exclusive projects.

Ex: An investment entails paying $190 and $20 will be paid forever, with an interest rate of 10%. Is this a worthwhile investment? PV = B/R = $200 ; PV of investment = -190 + 200 = 10 . Since PV>0, this is a worthwhile investment and you should accept the project.

2) Bond Pricing

The current price of a bond equals the present value of all future payments plus interest. Moreover, the present value of a bond is also known as the bond price, clearly showing how it is synonymously used as bond pricing. As shown previously, PV can be calculated for one period, multiple periods, perpetuities, and annuities. Therefore, in all of these various cases, the equations used to determine the present value of the bond also determine the bond price. A long time ago, coupon stripping was used, which involved a bond split into a top half, known as the zero coupon bond, and a bottom half, with coupons. Present value can also be used for bond pricing the two parts of the bond, as well as the bond as a whole. The top part of the bond was usually calculated using the present value of multiple periods and the bottom part with coupons is calculated using the present value of annuities. An example of this is shown below.

Ex: R = 5%

I .O .U. $100 in 10 years Top Part: PV=100/(1.05)10

5 5 5 5 5 5 5 5 5 5 Bottom Part: PV = 5/1.05 + 5/(1.05)2 +…+5/(1.05)10

Questions

1) Who lends money and who borrows money when buying and selling bonds?

2) What is the relationship between interest rate, present value, and the length of a bond?

3) What are the two main uses for calculating present value?

4) What are the steps used to derive the 2nd equation for present value of an annuity?

5) What is the difference between annuity and perpetuity?

6) When do you accept a project and when do you reject it?

7) How do you calculate PV and FV for one period?

8) With business investments what must the projects be and how do you choose between them?

9) How do you calculate PV for a perpetuity?

10) What is another name for present value?

Answers

1) The buyer lends money and the writer and seller borrows money.

2) As interests rates increase, present value decreases. As interest rates decrease, present value increases. A short-term bond causes a small change in present value. A long-term bond causes a large change in present value.

3) Present value is used for evaluating business investments and bond pricing

4) 1.Multiply (1+R) by PV = B/(1+R) + B/(1+R)2 + … + B/(1+R)n ( 2. PV(1+R) = B + B/1+R +…+B/(1+R)n-1 ( subtract (1) – (2) ( 3. PV – PV(1+R) = B/(1+R)n – B

5) Perpetuity is a stream of payment forever, or a perpetual annuity, while an annuity is periodic payment for a specified period of time.

6) Accept if PV>0 or positive and reject if PV ................
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