Bond – a fixed (nominal) income asset which has a:
-face value (stated value of the bond)
- coupon interest rate (stated interest rate)
- maturity date (length of time for fixed income payments)
Bonds are debt issued by either corporations, governments, or other entities,. They are obligated to pay the interest periodically until maturity date plus the value of the bond at the time it matures.
Ex: $1,000 bond, coupon rate of 5%, maturity of 5 years
- the fixed income from this bond is $50 per year (=.05*$1,000) for five years plus $1,000 when it is redeemed at maturity
Q: what is this stream of fixed income worth today?
A: It is necessary to consider the present discounted value of this fixed income stream
PRESENT DISCOUNTED VALUE
- think of this as the amount of money you would have to put into the bank today at whatever the interest rate is for the duration under consideration
- discounting, is the exact opposite of compounding
Sum of $ (PV) interest rate (r), duration future value (FV)
Present Value*(1 + interest rate) = Future Value
PV(1+r) = FV1
Solving for PV: PV = FV1/(1+r)
To arrive at the present value of the future sum, we discount it (i.e., divide by (1+r))
This can be extended to many periods. For two periods:
Value after one year gets interest giving the future value after two years:
[PV(1+r)]*(1+r) = FV2
PV(1+r)2 = FV2
Solving for PV: PV= FV2/(1+r)2
Hopefully you see a pattern with the number of years for discounting and the exponent
Extending this to the fixed income stream from the bond discussed earlier, where there is interest income (I) each year and a face value (F) at maturity after 5 years:
PV = 50/(1+r) + 50/(1+r)2 + … + 50/(1+r)5 + 1,000/(1+r)5
WE DO NOT USE THE STATED INTEREST RATE IN THE
DENOMINATORS HERE (we will come back to this shortly)
Note that PV = f(r, n) where n is the number of years with discounting For a given single payment in the future:
- the higher is r, the lower is the present value
- the farther in the future the sum is (larger is n), the lower is its present value
(explain each of these results)
Ex: Present value of $1,000 in the future
(1) If r = 5% and n = 2 years, PV = $1,000/(1.05)2 = $907.03
(2) If r = 10% and n = 2 years, PV = $1,000/(1.1)2 = $826.45
(3) If r = 5% and n = 5 years, PV = $1,000/(1.05)5 = $783.53
#3 shows why investments whose “payoffs” occur far into the future are not as likely to occur as those paying off quicker, sometimes leading to the need for government to subsidize these
APPLICATIONS OF THESE PRINCIPLES
1) Net Present Value (NPV) and Internal Rate of Return (IRR)
Q: If an investment project is expected to generate a revenue stream in future years of R1 in year 1, etc., what is the present value of that profit stream?
A: Its present value is: R1/(1+r) + R2/(1+r)2 + … + Rn/(1+r)n
What is the profitability of this investment project? Assuming costs occur up front (=C):
Profit = Revenue – Cost. In this context, however, it is necessary to discount the revenue stream, which leads to this project’s Net
NPV = R1/(1+r) + R2/(1+r)2 + … + Rn/(1+r)n – C
So, Net Present Value is the present discounted value of future revenues less the costs involved, which is the present discounted value of future profits.
In this equation, we have not solved for the discount factor, r.
Since we know the revenue stream (the R’s, which are marginal revenue) and the costs (C, which is marginal cost), we can solve for the rate of discount, r, that maximizes profit (i.e., causes MR =
- The rate of discount that equates the present discounted value of future revenues with the cost of the project is the Internal Rate of
Firms will often rank the desirability of various projects based on their internal rates of return.
Rule #1: Those projects with the highest IRR’s get preference to those with lower IRR’s.
Rule #2: Firms decide which