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