1. Ch 5. Time Value of Money Goal: to learn time value of money and discounted cash flows To understand a tool to value the expected future value in terms of present value. Cash flow: Cash in (inflow) or out (outflow) over times.

2. Why we need this tool? - Mainly for financial decisions: a) Project valuation b) Security valuation – stock and bond

3. I. Time Value of Money: Single Time. 1. Future Value and Compounding Future value: The amount of money an investment will grow to over some period of time. Ex) Investing $200 today and after 2 yrs, the investment will become $400. The $400 is the Future value.

4. 2) FV calculation 1) A single period: FV = Investment * (1+k) Ex) Invest $100 in the saving accounts with the 10% interest per year. FV =100*(1+0.1)=110 Future value is $110

5. 2) More than one period Ex) Invest $100 in the saving account with the 10% interest rate for 2 yrs FV1 = 100*(1+0.1)=110 FV2 = 110*(1+0.1)=121.

6. Here, we reinvest the first interest to get the future value. This is the compounding. That is, compounding the interest means earning interest on interest. The simple interest means no reinvestment on the interest. Ex) invest $100 with 10% with simple interest FV=100+2*0.1*100=120

7. 3) Decomposing FV and Impact of compounding FV = investment + simple interest + compound interest The impact of compounding is small over the short period

8. 2. Present value and discounting - Def: the current value of future cash flows discounted at the appropriate discount rate. In other word, converting FV to PV with discount rate - Why we need PV? We use the PV in evaluating projects or securities with different maturities and FV

9. 1) How to calculate PV Starting from the FV concept

10. (1) Single period case PV =FV/(1+r) Ex) You need $400 to buy text books next year and you can earn 7% on your money How much you have to put up today? PV =400/(1+0.07)=373.83

11. (2) Multi-period Ex) Need $1000 to buy a text book after 2 yrs and you can earn 7% on your money

12. 3. Why we need the FV and PV concept? If you have to pick up one out of three saving accounts with the same maturity but different rates, How do you want to evaluate and compare the accounts? A) $1000, 8% and 3yrs B) $2000, 6% and 3yrs C) $1500, 7% and 3yrs

13. If you have to pick up one out of three saving accounts with the maturities and rates, How do you want to evaluate and compare the accounts? A) $3000, 8% and 1yrs B) $4000, 6% and 2yrs C) $5000, 7% and 3yrs

14. 4. Determining the discount rate How to find k (rate)? (1) Use Future value table (2) Approximation

15. 5. Finding the number of periods Approximation:

16. 6. More about Multiple Periods Until now, we mainly deal with cases with yearly maturities. That is 1 yr, 2 yrs, or 3 yrs What happen if we have to deal with semiannual, quarterly or monthly. Do we have to use the same FV-PV equation

17. Yes! But need some revisions for more compounding. R: annual rate t: years m: revision for different time frame ex) Yearly: m=1 Semiannual : m=2 Quarterly: m=4 Monthly: m=12 Continuous compounding:

18. Ex) Initial investment is $100 and semi- annually compounding for next 2 yrs. And current interest rate is 7%. What is the future value of $100 after 2 yrs?