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# Sess0304 new - Lecture 3-4 Perpetuities and Annuities...

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Fin3715 – Fall 07 - Kayhan 1 Lecture 3-4: Perpetuities and Annuities Reading: RWJ Chapter 6 Outline: Compounding Frequency Perpetuities and Annuities Applications

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Fin3715 – Fall 07 - Kayhan 2 Compounding Frequency (Intervals) Stated Annual Interest Rate (SAR) – Interest rate that is stated in contracts using simple interest (compounding period must be given). 12% compounded quarterly = 3% per quarter 12% compounded monthly = 1% per month Examples : Loans with monthly payments (e.g. car loans and mortgages) are compounded monthly. Many bonds (e.g., US Gov’t bonds) are compounded semi-annually.
Fin3715 – Fall 07 - Kayhan 3 What’s the FV of \$100 invested for 2 years earning 10% compound semi-annually?

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Fin3715 – Fall 07 - Kayhan 4 What’s the FV of \$100 invested for 2 years earning 10% compound semi-annually? 5 % 100 Period 0 4 3 2 1 Period = half a year # period per year: m = 2 Total number of periods = m*2 = 4 • Interest rate per period: r SAR /m = 10%/2 = 5% FV = 100 (1+5%) 4 = 121.55
Fin3715 – Fall 07 - Kayhan 5 Effective Annual Rate Effective Annual Interest Rate (EAR) – Interest rate that is annualized using compound interest. r EAR = (1 + r SAR /m) m – 1 Remark: The rate over the same period (e.g. EAR) must be used to compare rates of return between two investments with different periods (e.g., monthly versus quarterly). If m > 1, the EAR will always be greater than the SAR.

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Fin3715 – Fall 07 - Kayhan 6 Example 1: Effective Annual Rate Given a monthly rate of 1%, What is the Effective Annual Rate (EAR)? What is the Stated Annual Rate (SAR)?
Fin3715 – Fall 07 - Kayhan 7 Example 1: Effective Annual Rate (Sol’n) EAR = (1+.01) 12 - 1= 12.68% SAR = (.01) (12) = 12%

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Fin3715 – Fall 07 - Kayhan 8 Example 2: Effective Annual Rate You purchase a 30-year \$1,000 CD today. What’s your account worth when it matures, if the interest is 10% compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) daily?
Fin3715 – Fall 07 - Kayhan 9 Example 2: Effective Annual Rate (Sol’n) 1000 (1+.1) 30 = 17,449.4 1000 (1+.1/4) 4*30 = 19,358.1 1000 (1+.1/12) 12*30 = 19,837.4 1000 (1+.1/365) 365*30 = 20,077.3 Remark : More frequent compounding does not add much value beyond the monthly frequency.

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Fin3715 – Fall 07 - Kayhan 10 Example 3: Effective Annual Rate You want to invest \$1,000 in a savings account for two years. After visiting three different banks, you discover the following three options: (i) 12.5% compounded annually; (ii) 11.75 % compounded monthly; (iii) 12% compounded quarterly. Which savings account should you choose?
Fin3715 – Fall 07 - Kayhan 11 Example 3: Effective Annual Rate (Sol’n) (i) EAR = 12.5%; (ii) EAR = (1+.1175/12) 12 – 1 = 12.40% (iii) EAR = (1+.12/4) 4 – 1 = 12.55% Thus, choose 3 rd option.

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Fin3715 – Fall 07 - Kayhan 12 PV of Multiple Cash Flows 0 1 2 T r % CF 0 CF T CF 2 CF 1 Period Remark: To calculate the PV (FV) of a stream of cash flows, ‘move’ all cash flows to a common point in time (e.g., today) and add together.
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