CTC 475 Review Interest/equity breakdown What to do when interest rates change Nominal interest rates (r=12% per year compounded quarterly) Converting nominal interest rates to regular interest rates (i=3% per quarter compounded quarterly) Converting nominal interest rates to effective interest rates (ieff=12.55% per year compounded yearly)

Changing interest rates to match cash flow intervals CTC 475 Changing interest rates to match cash flow intervals

Objectives Define APR and APY Know how to change interest rates to match cash flow intervals Understand continuous compounding

APR and APY APR-annual percentage rate Credit cards, loans, house mortgage Nominal APY-annual percentage yield Investments, CD’s, savings Effective

What if the cash flow interval doesn’t match the compounding interval? Cash flows occur more frequently than the compounding interval Compounded quarterly; deposited monthly Compounded yearly; deposited daily 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 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 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?

Continuous Compounding Nominal 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 Always assume discrete compounding 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? 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 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)

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