2 Chapter Present Value
Road Map Part A Introduction to finance.
Financial markets and financial decisions.
Present value.
Part B Valuation of assets, given discount rates. Part C Determination of discount rates. Part D Introduction to corporate finance.
Main Issues
Present Value Compound Interest Rates Nominal versus Real Cash Flows and Discount Rates Shortcuts to Special Cash Flows
2-2
Present Value
Chapter 2
Contents
1 Valuing Cash Flows . . . . . . . . . . . . . . . . . . . . . . .
1.1 1.2 Future Value (FV) . . . . . . . . . . . . . . . . . . . . . . . . . Present Value (PV) . . . . . . . . . . . . . . . . . . . . . . . . .
2-3
2-4 2-5
2
Compound Interest Rates . . . . . . . . . . . . . . . . . . .
2.1 2.2 2.3 Compound Interest vs. Simple Interest . . . . . . . . . . . . . . . APRs and EARs . . . . . . . . . . . . . . . . . . . . . . . . . . . Compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-7
2-7 2-8 2-9
3
Real vs. Nominal CFs and Rates . . . . . . . . . . . . . . . . 2-10
3.1 3.2 Real vs. Nominal Cash Flows . . . . . . . . . . . . . . . . . . . . 2-10 Real vs. Nominal Interest (Discount) Rates . . . . . . . . . . . . . 2-10
4
Shortcuts to Special Cash Flows . . . . . . . . . . . . . . . . 2-12
4.1 4.2 4.3 4.4 Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12 Annuity with Growth . . . . . . . . . . . . . . . . . . . . . . . . 2-13 Perpetuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14 Perpetuity with Growth . . . . . . . . . . . . . . . . . . . . . . . 2-15
5
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18
15.407 Lecture Notes
Fall 2003
c Jiang Wang
Chapter 2
Present Value
2-3
1
Valuing Cash Flows
CF1
6
"Visualizing" cash flows.
CFT
6 -
t=0
?
t=1
t=T
time
CF0
Example. Drug company develops a flu vaccine.
Strategy A: To bring to market in 1 year, invest $1 B (billion) now and returns $500 M (million), $400 M and $300 M in years 1, 2 and 3 respectively. Strategy B: To bring to market in 2 years, invest $200 M in years 0 and 1. Returns $300 M in years 2 and 3.
Which strategy creates more value?
Problem. How to compare two CF streams. Solution. For each CF stream, find a single number:
Equal in value to the cashflow stream. Valued on the same date (e.g. today) for all cash flows.
Use this number to compare different CF streams.
c Jiang Wang Fall 2003 15.407 Lecture Notes
2-4
Present Value
Chapter 2
1.1
Future Value (FV)
Current interest rate is r. How much will $1 today be worth in one year?
$1 investable at a rate of return r, say, 4%. FV in 1 year is
FV = 1 + r = $1.04.
FV in t years is
FV = $1 (1+r) (1+r) = (1+r)t.
Example. Bank pays an annual interest of 4% on 2-year CDs and you deposit $10,000. What is your balance two years later?
FV = 10, 000 (1 + 0.04)2 = $10, 816.
15.407 Lecture Notes
Fall 2003
c Jiang Wang
Chapter 2
Present Value
2-5
1.2
Present Value (PV)
We can ask the question in reverse. Question. What is the PV of $1 received a year from now?
Consider putting away 1/(1+r) today. A year later receive: 1 (1+r) = 1. 1+r The PV of $1 received a year from now is: 1 . 1+r The present value of F V dollars received t years from now is: FV 1 PV = = FV . (1+r)t (1+r)t
Example. (A) $10 M in 5 years or (B) $15 M in 15 years. Which is better if r = 5%?
PV A = PV B = 10 = 7.84. 1.055 15 = 7.22. 1.0515
c Jiang Wang
Fall 2003
15.407 Lecture Notes
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Present Value
Chapter 2
Solution to Example. Flu Vaccine. Assume that r = 5%. Strategy A:
Time Cash Flow Present Value 0 -1,000 -1,000 1 500.0 476.2 2 400.0 362.8 Total PV 3 300.0 259.2 98.2
Strategy B:
Time Cash Flow Present Value 0 -200 -200 1 -200.0 -190.5 2 300.0 272.1 Total PV 3 300.0 259.2 140.8
Firm should choose strategy B, and its value would increase by $140.8 M.
15.407 Lecture Notes
Fall 2003
c Jiang Wang
Chapter 2
Present Value
2-7
2
2.1
Compound Interest Rates
Compound Interest vs. Simple Interest
Simple interest - earn no interest on interest Compound interest - earn interest on interest.
Example: Compound and simple interest with r = 10%.
Simple Year 1 2 3 . . . 10 . . . 20 . . . 30 Starting Balance 100 110 120 . . . 190 . . . 290 . . . 390 Interest 10 10 10 . . . 10 . . . 10 . . . 10 Ending Balance 110 120 130 . . . 200 . . . 300 . . . 400
Compound Starting Balance Interest 100.00 10.00 110.00 11.00 121.00 12.10 . . . . . . 235.79 23.58 . . . . . . 611.59 . . . 1586.31 61.16 . . . 158.63 Ending Balance 110.00 121.00 133.10 . . . 259.37 . . . 672.75 . . . 1744.94
c Jiang Wang
Fall 2003
15.407 Lecture Notes
2-8
Present Value
Chapter 2
2.2
APRs and EARs
Typically compound interest is quoted using an annual percentage rate (APR) with an associated compounding interval. Example. Fleet Bank's one-year CD offers 6% APR, with semiannual compounding. If you invest $10,000, how much money do you have at the end of one year? What is the actual annual rate of interest you earn?
Quoted APR of rAPR = 6% is not the actual annual rate. It is only used to compute the 6-month interest rate as follows:
(6%)(1/2) = 3%.
Investing $10,000, at the end of one year you have:
10, 000(1 + 0.03)(1 + 0.03) = 10, 000(1.0609) = 10, 609.
In the second 6-month period, you earn interest on interest.
The actual annual rate, the effective annual rate (EAR), is
rEAR = (1 + 0.03)2 - 1 = 6.09%.
Annual rates typically refer to EARs.
15.407 Lecture Notes
Fall 2003
c Jiang Wang
Chapter 2
Present Value
2-9
2.3
Compounding
Let rAPR be the annual percentage rate and k be the number of compounding intervals per year. One dollar invested today yields:
1+ rAPR k
k
dollars in one year. Effective annual rate, rEAR is given by:
rAPR (1 + rEAR ) = 1 + k
k
or
rEAR rAPR = 1+ k
k
- 1.
Example. Suppose =5%:
k rAPR 1 2 12 365 8,760 . . . Value of $1 in a year 1.050000 1.050625 1.051162 1.051268 1.051271 . . . e0.05 = 1.051271 rEAR 5.0000% 5.0625% 5.1162% 5.1267% 5.1271% . . . 5.1271%
Here, e 2.71828.
c Jiang Wang
Fall 2003
15.407 Lecture Notes
2-10
Present Value
Chapter 2
3
3.1
Real vs. Nominal CFs and Rates
Real vs. Nominal Cash Flows
Question: If inflation is 4% per year and you have been promised $1.04 in one year, what is this really worth next year? What can $1.04 buy next year given that prices will rise by 4%? The real or inflation adjusted value of $1.04 in a year is Real Cash Flow = Nominal Cash Flow 1.04 = = $1.00. 1 + inflation 1 + 0.04
In general, at annual inflation rate of i we have
(Real Cash Flow)t = (Nominal Cash Flow)t . (1 + i)t
3.2
Real vs. Nominal Interest (Discount) Rates
Nominal interest rates - typical market rates Real interest rates - interest rates adjusted for inflation.
Example. $1.00 invested at a 6% interest rate grows to $1.06 dollars tomorrow. If inflation is 4% per year, then the real value is $1.06/1.04 = 1.019. The real return is 1.9%.
1 + rreal =
15.407 Lecture Notes
1 + rnominal . 1+i
Fall 2003 c Jiang Wang
Chapter 2
Present Value
2-11
Example. You are forecasting next year's revenues based on this year's sales of $1 M. You expect 2% real growth in your sales (units). You expect inflation to be 4%. Suppose that the appropriate nominal discount rate is 5%. What is the present value of next year's sales revenue?
Next year's nominal sales forecast:
1 1.02 1.04 = 1.0608. PV = 1.0608 = 1.0103. 1.05
Next year's real sales forecast:
1 1.02 = 1.02.
Real discount rate:
rreal = 1 + rn 1.05 -1= - 1 = 0.9615%. 1+i 1.04 1.02 = 1.0103. 1.009615
PV =
Important Rules:
Discount nominal CFs by nominal discount rates. Discount real CFs by real discount rates.
c Jiang Wang
Fall 2003
15.407 Lecture Notes
2-12
Present Value
Chapter 2
4
4.1
Shortcuts to Special Cash Flows
Annuity
A
6
A
6
A
6 -
t=0
1
2
T
time
Today is t = 0 and cash flow starts at t = 1.
PV (Annuity) =
A A A + + + 1+r (1+r)2 1+r)T 1 1 1- . r (1+r)T
= A
Example. Insurance company selling an annuity of $10,000 per year for 20 years. Suppose r = 5%. What should the company sell it for?
P V = 10, 000 1 1 1- 0.05 1.0520
= 10, 000 12.46 = 124, 622.1.
FV (Annuity) = PV (Annuity) (1+r)T .
15.407 Lecture Notes Fall 2003 c Jiang Wang
Chapter 2
Present Value
2-13
4.2
Annuity with Growth
A A(1+g)
6 6
A(1+g)T
6 -
t=0
1
2
T
time
PV (growing annuity)
1 1+g (1+g)T -1 = A + + + 1+r (1+r)2 (1+r)T
= A
1 r-g T 1+r
1-
1+g 1+r
T
if r = g if r = g.
Example. Saving for retirement - Suppose that you are now 30 and would like $2 million at age 65 for your retirement. You would like to save each year an amount that grows by 5% each year. How much should you start saving now, assuming that r=8%?
A= 2, 000, 000 = 6, 472.97. 308.977
c Jiang Wang
Fall 2003
15.407 Lecture Notes
2-14
Present Value
Chapter 2
4.3
Perpetuity
A
6
A
6
A
6
-
t=0
1
2
3
time
A perpetuity is an annuity with infinite maturity.
Example. You just won the lottery and it pays $100,000 a year for 20 years. Are you a millionaire? Suppose that r = 10%.
PV = 100, 000 1 0.10 1- 1 1.1020
= 100, 000 8.514 = 851, 356.
What if the payments last for 50 years?
PV = 100, 000
1 0.10 1- 1 1.1050
= 100, 000 9.915 = 991, 481.
How about forever - a perpetuity?
PV = 100, 000/0.10 = 1, 000, 000.
PV (Perpetuity) =
A . r
Fall 2003 c Jiang Wang
15.407 Lecture Notes
Chapter 2
Present Value
2-15
4.4
Perpetuity with Growth
A A(1+g) A(1+g)2
6 6 6
-
t=0
1
2
3
time
PV (Perpetuity with growth) =
A r-g
where A is the first CF one period from now.
Example. Super Growth Inc. will pay an annual dividend next year of $3. After analyzing the company, you expect the dividend after that to grow at the rate of 5% per year forever. Also for companies of this risk class, the expected return is 10%. What should be Super Growth's price per share?
PV =
= 3(1 + 0.05)2 3(1 + 0.05) 3 + + + 1.10 1.102 1.103 3 0.10 - 0.05
= 60.
c Jiang Wang
Fall 2003
15.407 Lecture Notes
2-16
Present Value
Chapter 2
Example. Mortgage calculation in the U.S.
Pay 20% down payment, and borrow the rest from the bank using the property as collateral Pay a fixed monthly payment for the life of the mortgage Have the option to prepay the mortgage anytime before the maturity date of the mortgage.
Suppose that you bought a house for $500,000 with $100,000 down payment and financed the rest with a thirty-year fixed rate mortgage at 8.5% APR compounded monthly.
The monthly payment M is determined by
360
400, 000 =
t=1
M [1 + (0.085/12)]t 1 [1 + (0.085/12)]360
=
M (0.085/12)
1-
=
M (0.9212) . (0.085/12)
M = $3, 075.65.
Your effective annual interest rate (EAR)
[1 + (0.085/12)]12 - 1 = 1.08839 - 1 = 8.839%.
15.407 Lecture Notes
Fall 2003
c Jiang Wang
Chapter 2
Present Value
2-17
The monthly payments are as follows:
t (month) 1 2 3 . . . 120 121 . . . 240 241 . . . 359 360 Principal 242.37 244.08 245.81 . . . 561.29 565.26 . . . 1309.27 1318.54 . . . 3032.60 3054.07 Interest 2833.33 2831.62 2829.89 . . . 2514.42 2510.44 . . . 1766.43 1757.16 . . . 43.10 21.63 Sum 3075.7 3075.7 3075.7 . . . 3075.7 3075.7 . . . 3075.7 3075.7 . . . 3075.7 3075.7 Remaining P. 399,757.63 399,513.55 399,267.74 . . . 354,415.49 353,850.23 . . . 248,068.95 246,750.41 . . . 3,054.07 0.00
Total monthly payment is the same for each month. The percentage of principal payment increases over time. The percentage of interest payment decreases over time.
c Jiang Wang
Fall 2003
15.407 Lecture Notes
2-18
Present Value
Chapter 2
5
Homework
Readings:
BM Chapter 3.
Assignment:
Problem Set 1.
15.407 Lecture Notes
Fall 2003
c Jiang Wang