Inflation
Valuing real cash payments
Future value
Simple interest: FVsimple=initial investment*(1+r*t) Compound interest: FVcompound=initial investment*(1+r) Present value Discount rate: r Discount factor: DF=
t1 t(1?r)1
(1?r)t Present value: PV=FV*
PV of multiple cash flows CtC1C2PV?(1?1?2?....?r)(1?r)(1?r)t
Ct=the cash flows in year t
Example: Your auto dealer gives you the choice to pay $15,500 cash now or make three
payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money (discount rate) is 8%, which do you prefer?
Initial Payment* 8,000.00
4,000 PVofC1?(1?3,703.70?.08)1
PVofC2?(14,000?3,429.36 ?.08)2
Total PV ? $15,133.06
Perpetuities
PV of perpetuity: PV=C/r
Example: In order to create an endowment, which pays $185,000 per year forever, how much money must be set aside today if the rate of interest is 8%?
PV?185,000 .08?$2,312,500What if the first payment won’t be received until 3 years from today?
PV = 2312,500 / (1 + 0.08) = 1,835,662.5
annuities
Annuities are equally-spaced, level streams of cash flows lasting for a limited period of time. 3 Present value of an annuity: PV=C*[
11-] trr(1?r)The terms within the brackets are collectively called the “annuity factor” PV of multiple cash flows
Future value of annuities
Example: You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, how much will you have saved by the time you retire? 1 FV?$4,000?.10?.10(1?1.10)20??(1?.10)20??
FV?$229,100
Annuities due(即付年金)（与普通年金（即后付年金）的区别仅在于付款时间的不同，一个n期的即付年金相当于一个n-1期的普通年金）(期不等于年) PVannuity FVannuity
due=PVannuity due=FVannuity
(1+r) (1+r)
Example: Suppose you invest $429.59 annually at the beginning of each year at 10% interest. After 50 years, how much would your investment be worth?
FVAD?FVAnnuity?(1?r)
FVAD?($500,000)?(1.10) FVAD?$550,000
EAR & APR
Effective annual interest rate: The period interest rate that is annualized using compound interest.
EAR = (1 + monthly rate) - 1
Annual Percentage Rate: The period interest rate that is annualized using simple interest APR = monthly rate × 12
Example : Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?
EAR?(1.01)12?1?12.68%
APR?(0.01)?(12)?12.00%
Inflation 1+nominal interest rate 1+inflation rate121?real interest rate= Real interest rate?nominal interest rate-inflation rateValuing real cash payments ? Example: You make a loan of $5,000 to Jane who will pay it back in 1 year. The interest
rate is 8%, and the inflation rate is 5% now. What is the present value of Jane?s IOU? Show that you get the same answer when (a) discounting the nominal payment at the nominal rate and (b) discounting the real payment at the real rate. ? (a) 5,000 / (1 + 8%) = $4630
(b) 5,000 / (1.05) = $4762 (real dollar) 4762 / (1.028) = $4630 (2.8% is real interest rate)
不能用实际利率去贴现名义现金流
Chapter 6
bond pricing:
example: For a $1000 face value bond with a bid price of 103:05 and an asked price of 103:06, how much would an investor pay for the bond? ?6?103%??? ? 103.1875% of face value ?32?
1.031875???$1,000? ?$1,031.875?
PV=
couponcouponcoupon?par++…+ 12t(1?r)(1?r)(1?r)PVbond=PVcoupon+PVparvalue
=coupon*(annuity factor)+ par value * (discout factor)
PVBond?PVCoupons?PVParValue
PVBond?coupon?(AnnuityFactor)?parvalue?(DiscountFactor)
1?(1?r)?t
where AnnuityFactor? r1
and DiscountFactor? (1?r)t
Example: What is the value of a 3-year annuity that pays $90 each year and an additional
$1,000 at the date of the final repayment? Assume a discount rate of 4%.
1?(1?.04)?31
PVBond?$90??$1,000? .04(1?.04)3 ?$1,138.75
Warning: bond
The coupon rate IS NOT the discount rate used in the Present Value calculations Example: What is the present value of a 4% coupon bond with face value $1,000 that matures in 3 years? Assume a discount rate of 5%.
Bond yields
Current yield: annual coupon payments divided by bond price
Example:Suppose you spend $1,150 for a $1,000 face value bond that pays a $60 annual coupon payment for 3 years. What is the bond’s current yield?
Yield to maturity:
PV=
couponcouponcoupon?par++…+ 12t(1?r)(1?r)(1?r)Bond rates of return
total incomeRate of return= investment Coupon income+price changeRate of return=investment
Rate of return 只算一年的coupon YTM vs rate of return
YTM ↑ (↓)(unchange) → the price of bond ↓ (↑) (unchange) → the rate of return for that period less (greater)(equal to) than the yield to maturity.
Ytm 通过改变price 去改变p1-p0 从而改变rate of return ，由rate of return 公式得，p1-p0和其成正比，ytm与change in price成反比，故ytm 与其成反比
Chapter 7
Stock market
P/E ratio(本益比): price per share divided by earnings per share Ask price & bid price
Ask price: the price at which current shareholders are willing to sell their shares Bid price: the price at which investors are willing to buy shares Terminology
1,market cap. 2.P/E ratio 3.dividend yield
Example: You are considering investing in a firm whose shares are currently selling for $50 per share with 1,000,000 shares outstanding. Expected dividends are $2/share and earnings are $6/share.
What is the firm?s Market Cap? P/E Ratio? Dividend Yield? MarketCapitalization?$50?1,000,000?$50,000,000 $50P/ERatio??8.33 $6 $2DividendYield??.04?4% $50Measure of value
1. book value 2. liquidation value 3.market value BV= Assets - liabilities
LV = Assets selling price – Liabilities
MV = Tangible & intangible assets + Inv. Opportunities Price and intrinsic value
Vo内在价值
Example: What is the intrinsic value of a share of stock if expected dividends are $2/share
and the expected price in 1 year is $35/share? Assume a discount rate of 10%.
Expected return(ER)
Valuing common stocks
Div1Div2DivH?PHP0???...?
(1?r)1(1?r)2(1?r)H
dividend discount model
consider three simplying cases 1. no growth 2. constant growth 3. noconstant growth