CTC 475 Review
Interest/equity breakdown What to do when interest rates change Nominal interest rates Converting nominal interest rates to regular
interest rates Converting nominal interest rates to effective
interest rates
CTC 475
Changing interest rates to match cash flow intervals
Objectives
Know how to change interest rates to match cash flow intervals
Understand continuous compounding
What if the cash flow interval doesn’t match the compounding interval?1. Cash flows occur more frequently than the
compounding interval Compounded quarterly; deposited monthly Compounded yearly; deposited daily
2. Cash flows occur less frequently than the compounding interval
Compounded monthly; deposited quarterly Compounded quarterly; deposited yearly
Cash flows occur more frequently than the compounding interval Use ieff=(1+i)m-1 and solve for i
Note that a nominal interest rate must first be converted into ieff or i before using the above equation
Cash flows occur less frequently than the compounding interval
Use ieff=(1+i)m-1 and solve for ieff
Note that a nominal interest rate must first be converted into ieff or i before using the above equation
Case 1 Example
Cash flows occur more frequently than compounding interval
Solve for i
Example--Cash flows are more frequent than compounding interval (solve for i) 8% per yr compounded qtrly (recognize this as a
nominal interest rate and convert to 2% per quarter compounded quarterly)
Individual makes monthly deposits (cash flows are more frequent than compounding interval)
We want an interest rate of ?/month compounded monthly
Use ieff=(1+i)m-1 and solve for i
Example-Continued Use ieff=(1+i)m-1 and solve for i .02=(1+i)3-1 (m=3; 3 months per quarter) 1.02 =(1+i)3
Raise both sides by 1/3 i=.662% per month compounded monthly
Case 2 Example
Cash flows occur less frequently than compounding interval
Solve for ieff
Example--Cash flows are less frequent than compounding interval (solve for ieff) 8% per yr compounded qtrly (recognize this as a
nominal interest rate and convert to 2% per quarter compounded quarterly)
Individual makes semiannual deposits (cash flows are less frequent than compounding interval)
We want an equivalent interest rate of ?/semi compounded semiannually
Use ieff=(1+i)m-1 and solve for ieff
Example-Continued Use ieff=(1+i)m-1 and solve for ieff
ieff =(1+.02)2-1 (m=2; 2 qtrs. per semi) ieff =4.04% per semi compounded
semiannually
What is Continuous Compounding? Appendix D
Continuous CompoundingNominal Int. Rate Calculation ieff
8%/yr comp yrly (1+.08/1)1-1 8%
8%/yr comp semi (1+.08/2)2-1 8.16%
8%/yr comp qtrly (1+.08/4)4-1 8.24%
8%/yr comp month. (1+.08/12)12-1 8.30%
8%/yr comp daily (1+.08/365) 365-1 8.328%
8%/yr comp hourly (1+.08/8760)8760-1 8.329%
Continuous Compounding
As the time interval gets smaller and smaller (eventually approaching 0) you get the equation:
ieff=er-1
Therefore the effective interest rate for 8% per year compounded continuously = e.08-1=8.3287%
Continuous Compounding
If the interest rate is 12% compounded continuously, what is the effective annual rate?
ieff=er-1
ieff= e.12-1=12.75%
Continuous Compounding
Continuous compounding factors can be found in Appendix D of your book (for r=8,10 and 20%
Equations can be found on page 650 Always assume discrete compounding (use
Appendix C) unless the problem statement specifically states continuous compounding
Continuous Compounding; Single Cash Flow If $2000 is invested in a fund that pays
interest @ a rate of 10% per year compounded continuously, how much will the fund be worth in 5 years?
Method 1-Use book factors F=P(F/Pr,n)=2000(F/P10,5)=2000(1.6487) F=$3,297
Continuous Compounding; Single Cash Flow If $2000 is invested in a fund that pays
interest @ a rate of 10% per year compounded continuously, how much will the fund be worth in 5 years?
Method 2-Use equation F=P*ern=e(.1*5)=2000(1.6487) F=$3,297
Continuous Compounding; Single Cash Flow If $2000 is invested in a fund that pays
interest @ a rate of 10% per year compounded continuously, how much will the fund be worth in 5 years?
Method 3-Find effective interest rate ieff=er-1 = e.10-1 = 10.52% F=P(1+i)5 = 2000(1.1052)5 = $3,298
Continuous Compounding; Uniform Series $1000 is deposited each year for 10 years
into an account that pays 10%/yr compounded continuously. Determine the PW and FW.
P=A(P/A10,10)=1000(6.0104)=$6,010
F=A(F/A10,10)=1000(16.338) =$16,338 Or F=P(F/P10,10)=6010(2.7183)=$16,337
Continuous Compounding
The continuous compounding rate must be consistent with the cash flow intervals (i.e. 12% per year compounded continuously won’t work with semiannual deposits)
Must change r to 6% per semi compounded semiannually
Equation is (r/n) where r is the annual rate and n is the # of intervals in a year
Next lecture
Methods of Comparing Investment Alternatives
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