Presentation on theme: "Ch 4. Time Value of Money Goal:"— Presentation transcript:
1 Ch 4. Time Value of Money Goal: to learn time value of money and discounted cash flowsTo 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 valuationb) Security valuation – stock and bond
3 I. Time Value of Money: Single Time. 1. Future Value and CompoundingFuture 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)=110Future value is $110
5 2) More than one periodEx) Invest $100 in the saving account with the 10% interest rate for 2 yrsFV1 = 100*(1+0.1)=110FV2 = 110*(1+0.1)=121.
6 Here, we reinvest the first interest to get the future value 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 interestFV=100+2*0.1*100=120
7 3) Decomposing FV and Impact of compounding FV = investment + simple interest + compound interestThe 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 PVStarting from the FV concept
10 (1) Single period casePV =FV/(1+r)Ex) You need $400 to buy text books next year and you can earn 7% on your moneyHow much you have to put up today?PV =400/(1+0.07)=373.83
11 (2) Multi-periodEx) Need $1000 to buy a text book after 2 yrs and you can earn 7% on your money
12 (3) PVIF (k,t)As we can see, if we know theWe can calculate the PV easily.This is called PVIF (k,t) , present value interest factor(4) How to use PVIF table
13 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 3yrsB) $2000, 6% and 3yrsC) $1500, 7% and 3yrs
14 If you have to pick up one out of three investment opportunities with the maturities and rates, How do you want to evaluate and compare them?A) $3000, 8% and 1yrsB) $4000, 6% and 2yrsC) $5000, 7% and 3yrs
15 4. Determining the discount rate How to find k (rate)?(1) Use Future value table(2) Approximation
16 5. Finding the number of periods Approximation:
17 6. More about Multiple Periods Until now, we mainly deal with cases with yearly maturities. That is 1 yr, 2 yrs, or 3 yrsWhat happen if we have to deal with semiannual, quarterly or monthly.Do we have to use the same FV-PV equation
18 Yes! But need some revisions for more compounding. R: annual ratet: yearsm: revision for different time frameex) Yearly: m=1Semiannual : m=2Quarterly: m=4Monthly: m=12Continuous compounding:
19 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?
20 II. FV and PV with multiple cash flows FV with multiple cash flows: Two methodsRolling over FV year by yearFV=FV1+FV2+FV3….Ex) Deposit $100 every year for 3 yrs. And 10% interest rate. FV?
21 2) PV with multiple cash flows: Two method Rolling back year by yearPV=PV1+PV2+…..Ex) You are supposed to need $1000 in one year and $2000 in the second year. If you can earn 9% on your money, how much you have to put up today?
22 2. Annuities and Perpetuities Def of Annuity:Constant cash flows for a fixed period of timeEx) car loanEx) Assets with promised to pay $500 at the end of the each of the next three years. What is the price of the asset now?
25 Ex) You stop by a car dealer shop and find a really good car Ex) You stop by a car dealer shop and find a really good car. The sticker price of the car is $ But you don’t have money now. So, want installment payment over 4 yrs. Over conversation, the dealer suggests $632 per month for 48 month at 1% per month. How much is going to be your PV of total installments?
26 2-1) How to use the Annuity table in calculation? Calculating PV2-2) Finding CEx) You stop by a car dealer shop and find a really good car. The sticker price of the car is $ But you don’t have money now. So, if you want installment payment over 4 yrs, how much you have to pay monthly? (Here interest rate is 12%)
27 2-3) Finding rate Ex) an insurance company offers to pay you $1000 per year for 10 years if you pay $6710 up front. What rate is used in this annuity? 3) Def of perpetuities: An annuity in which the cash flow continues forever
28 4) Formula for PV of perpetuities PV=C/k Ex) Preferred stock – promised fixed dividend every period forever.A company want to sell preferred stock at $100 per share. How much of dividend it has to pay. Currently the similar preferred stock is sold at $40 with $1 dividend.
31 Ex) $2000 annuity for 30 years and k= 0. 08 Ex) $2000 annuity for 30 years and k= What is the annuity future value? 6) Annuities due Def: annuity for which the cash flows occur at the beginning of the period Annuity due value = ordinary annuity value * (1+k)
32 7) Uneven Cash Flows;Summing PV and FV of each cash flowsUsing the cash flow patterns to apply formulaEx) If you are supposed to need $100 (1st), $200 (2nd) and 300 (3rd) at the end of each year and your account provides 7% of interest per year, how much do you need to deposit now?
33 Ex) If you are supposed to deposit $100 (1st), $200 (2nd) and 300 (3rd) at the end of each year and your account provides 7% of interest per year, how much your total deposit would be at the end of 3rd year?You are supposed to need $100 (1st), $200 (2nd) and 300 (3rd) at the end of each year, the end of 3rd year. You have $360 now. What interest rate (return) do you need to cover your needs?
34 3. Rate Q1. 10% compounded semi-annually is the same as 10% per year in compounding? No! here, 10% is stated or quoted rate and actually, 10.25% (=(1+0.05)*(1+0.05)-1) is the effective annual rate. To compare to other rates, we need to convert quoted rates into the effective rates
35 EAR is also called EFF %. Ex) Bank A: 15% compounding daily Bank B: 15 EAR is also called EFF %. Ex) Bank A: 15% compounding daily Bank B: 15.5% compounding quarterly Bank C: 16% annually
36 3-1) APRs (Annual Percentage Rate) Def: interest rate charged per period (periodic rate) multiplied by the number of periods per year APR =EAR? No!!!! So, APR is a quoted rate and need to be converted to the EAR
37 Ex) One credit card company selling a card by tele-marketing Ex) One credit card company selling a card by tele-marketing. The company said the card will benefit its cardholders with semi-annual 15%APRs, compared to the other credit card with 16% EAR. Do you agree or not?
38 6. Fraction time periodSuppose you deposited $100 in a bank that pays a normal rate of 10%, compounded, based on a 365 –day year. How much would you have after 9 months?Periodic rate = 0.1/365 per dayFV = 100*(1+0.1/365)^(365*9/12) =
39 7.Loan types and loan Amortization Three types of loans:Pure discount loan:Receive money today and repay a single lump sum in futureInterest only loan:Pay interest each period and repay the entire principal at some point in the futureAmortized loan:Repay parts of the loan amount over time
41 8. growing annuityE.g) suppose a 65-year old is contemplating retirement, expects to live for another 20 years, has a $1 million nest egg, expect the investment to earn a nominal annual rate of 6%, expect inflation to average 3% per year, and wants to withdraw a constant real amount annually over the 20 years so as to maintain a constant standard of living. If the first withdraw is to be made today, What is the amount of that initial withdrawal?
42 (1) step1: real rate calculation Real rate =rr = [(1+rnom)/(1+Inflation)]-1= [1.06/1.03]-1 = %(2) step 2: using the real rate, calculate Annuity due (Payment) – mode: beginning.= Then it grows by 3% (inflation rate) every year.What happen if we want to calculate annuity (payment) at the end of the first year *(1+0.03)=
43 E. g) You need to accumulate $100,000 in 10 years E.g) You need to accumulate $100,000 in 10 years. You plan to make a deposit ina bank now, at Time 0 and then make 9 more deposits at beginning of each of the following 9 years. The bank pays 6% interest, you expect inflation to be 2% per year and you plan to increase your annual deposits at the inflation rate. How much you have to deposit initially?
44 Step 1: calculate real rate = 1.06/1.02 -1 = 0.0392157 Step 2: real value of is /(1+0.02)^10 =Step 3: beginning mode,N=10, I/YR= , PV=0 and FV= , PMT =It means the t=0, deposit is , at t=1, it is *(1+0.02)