Chapter 4 + 5
Leverage Ratio
= Cost of Investment / Owner's contribution to the purchase = Asset Value / Equity
Sharpe Ratio
=Risk premium/SD of excess returns
What is the present value of $100 promised one year from now at 10% annual interest? A. $89.50 B. $90.00 C. $90.91 D. $91.25
C. $90.91
A monthly growth rate of 0.5% is an annual growth rate of: A. 6.00% B. 5.00% C. 6.17% D. 6.50%
C. 6.17%
Parties to a nominal contract care about inflation:
• Borrowers care about the future real cost to repay. • Lenders care about the future real value of payments.
General formula to find PV of future payment to be received in n years:
PV = FV/(1+i)^n
real interest rate
exact: r=(1+i)/(1+ π) -1 approximate: r= i- π
The higher is n,
the lower is the principal's present value
To estimate risk aversion:
• Use questionnaires • Observe individuals' decisions when confronted with risk • Observe how much people are willing to pay to avoid risk
Risk is measured (quantified):
• over some time horizon for the investment • relative to some benchmark for the payoff
A bond offers a $50 coupon, has a face value of $1,000, and has 10 years to maturity. If the interest rate is 4.0% what is the value of this bond?
$1,081.12
annual to monthly interest rate equation
(1+i)^1/12 then subtract one
monthly to annual interest rate equation
(1+i)^12 then subtract one
real interest rate in dollars
(i-π)*principal loan
measuring distance
(return from outcome-expected return)^2
bond
A promise to make a series of future payments on specific dates. They are legal contracts that: - Require the borrower to make payments to the lender, and - Specify what happens if the borrower fails to pay on the specified date (default).
systemic risk
A threat to the entire financial system, not to a specific person, company or market.
Which of the following best expresses the future value of $100 left in a savings account earning 3.5% for three and a half years? A. $100(1.035)^3.5 B. $100(0.35)^3.5 C. $100 × 3.5 × (1.035) D. $100(1.035)^3/2
A. $100(1.035)^3.5
Which of the following statements is most correct? A. We can always compute the ex post real interest rate but not the ex ante real rate. B. We cannot compute either the ex post or ex ante real interest rates accurately. C. We can accurately compute the ex ante real interest rate but not the ex post real rate. D. None of the statements are correct.
A. We can always compute the ex post real interest rate but not the ex ante real rate.
You have the option to invest in either country A or country B but not both. You carry out some research and conclude that the two countries are similar in every way except that the returns on assets of different classes tend to move together much more in country A- that is, they are more highly correlated in country A than in country B. Which country would choose to invest in and why?
Invest in country B as the benefits from diversification are greater than in country A. Spreading your risk across different asset classes brings greater benefit when the correlation among the returns is lower.
standard deviation
Is a measure of risk because it measures volatility
principle
Final payment on a bond
coupon bond
Most common bond type. It delivers a cash flow.
Face value F
How much the contract is worth at maturity
Coupon rate
Interest payments; coupon payment is C= ic×F
ex ante real interest rates
Interest rates that are adjusted for expected inflation.
Idiosyncratic risk
Uniquely affect individual investors (e.g., an energy price shock): Also called unique risks.
personal discount rate
Value of current consumption relative to future consumption.
spreading risk
finding investments whose payoffs are unrelated.
pure discount bond
has a zero coupon
Two types of diversification
hedging risk spreading risk
higher expected inflation =
higher nominal interest rates
nominal interest rate
i = interest rate expressed in current dollar terms.
the fisher equation-nominal interest rate
i=r+π
Real return
interest payment divided by price level in repayment year. ex: 100 invested 100/1.1 + 10/1.1=100 i.e.=no return
fixed payment loan
loan amount= Fixed payment/(1+R)^n
Probability
measures the likelihood that an event will occur
Financial markets quote
nominal interest rates
Real interest rate
r= inflation-adjusted interest rate.
real rate of return (exact)
r=(1+i)/(1+π)-1
real rate of return (approximate)
r=i-π
calculating standard deviation
radical of the variance divide by initial investment to get percentage
To compare expected outcomes for investments of different sizes consider their
rate of return
expected return
summation: probability of outcome*return from that outcome
The greater the maturity n,
the higher the payments' total value.
The higher the interest rate i,
the lower the present value.
The greater the standard deviation,
the higher the risk.
The value of the coupon bond, PCB, rises when:
- The yearly coupon payments C, rise and - The interest rate i, falls
Consider two scenarios. In the first, the nominal interest rate is 6 percent and the expected rate of inflation is 4 percent. In the second, the nominal interest rate is 5 percent and the expected rate of inflation is 2 percent. In which situation would you rather be a lender? In which would you rather be a borrower?
-As lender want a high real return and so would rather lend with the real interest rate at 3 percent. -As a borrower, you want a low real interest rate and so would rather borrow when the real rate is 2 percent
value at risk
-Estimates a given amount of loss -Tells you what a given loss will be at a given time horizon. -It measures the worst possible loss over a specific horizon at a given probability.
Systematic risk
-Identically affect all investors (e.g., a recession): -Also called common, or economy-wide risks.
If the current interest rate increases, what would you expect to happen to bond prices? Explain.
-Inversely related (bond prices will fall when interest rates increase) -Bond prices are the sum of the present values of the future payments associated with the bond -The higher the interest rate, the lower the present value of these payments.
risk-free asset
-Is an investment whose future payoff is deterministic (non-random) and therefore known with certainty. -Guaranteed and cannot vary. -ex: US T-Bills, money market mutual funds
Leverage
-Is the practice of borrowing to finance part of an investment -Increases the expected return -Increases the standard deviation because it magnifies the effect of price changes. -Have to pay back amount invested, fine if you do well, really bad if you don't
Use the Fisher equation to explain in detail what a borrower is compensating a lender for when he pays her a nominal rate of interest.
-The borrower is compensating the lender for the inflation that is expected over the coming year, as this will reduce the purchasing power of a given number of dollars. -Also paying the lender a real interest rate to compensate the lender for the use of her money. Lender is foregoing the use of her money for the duration of loan -Needs to be compensated for this opportunity cost.
Which would be most affected in the event of an interest rate increase- the price of a five-year coupon bond that paid coupons only in years 3, 4, and 5 or the price of a five-year coupon bond that paid coupons only in years 1, 2, and 3, everything else being equal? Explain.
-The price of the bond with the later payments will fall by relatively more. -change in the interest rate has a greater impact on their present value. -present value formula: a payment made in one year is divided by (1+i) while a payment made in five years time is divided by (1+i)5, so the impact will be bigger in the latter case.
Two types of idiosyncratic risk:
1) (negatively) correlated 2) uncorrelated
solving for IRR (Is revenue enough to make payments)
1) Equate the cost of the machine to the present values of the stream of annual revenues. (cost= annual revenue/(1+i)^n) 2) solve for i 3) if i>interest rate paid on loan} buy, otherwise don't
calculating variance
1) find the expected value 2) subtracted the expected value from each of the possible payoffs and square 3) multiple results of #2 by its corresponding probability 4) add up the quantities
An interest rate defines:
1) the future reward for saving today (& spending tomorrow). 2) the future cost of spending today (& paying tomorrow)
Present value increases when:
1. the future payment FV is higher; 2. the time until payment n gets shorter; 3. the interest rate is lower.
Assuming that the current interest rate is 3 percent, compute the present value of a five-year, 5 percent coupon bond with a face value of $1,000. a. What happens when the interest rate goes to 4 percent? b. What happens when the interest rate goes to 2 percent?
=$50/(1.03) + $50/(1.03)2 + $50/(1.03)3 + $50/(1.03)4 + $1050/(1.03) 5 = $1091.59 a. =$50/(1.04) + $50/(1.04)2 + $50/(1.04)3 + $50/(1.04)4 + $1050/(1.04)5 = $1044.52 The present value falls when the interest rate rises to 4 percent. b. =$50/(1.02) + $50/(1.02)2 + $50/(1.02)3 + $50/(1.02)4 + $1050/(1.02)5 = $1141.40 The present value rises when the interest rate falls to 2 percent.
Suppose Tom receives a one-year loan from ABC Bank for $5,000.00. At the end of the year, Tom repays $5,400.00 to ABC Bank. Assuming the simple calculation of interest, the interest rate on Tom's loan was: A. $400 B. 8.00% C. 7.41% D. 20%
B. 8.00%
If a fair coin is tossed, the probability of coming up with either a head or a tail is: A. ½ or 50 percent. B. Zero. C. 1 or 100 percent. D. Unquantifiable.
C. 1 or 100 percent.
Up to what amount would a risk-neutral gambler pay to enter a game where on the flip of a fair coin, if you call the correct outcome the payoff is $2,000? A. More than $1000 but less than $2000. B. Up to $2,000. C. Up to $1,000. D. More than $1,500.
C. Up to $1,000.
A promise of a $100 payment to be received one year from today is: A. more valuable than receiving the payment today. B. less valuable than receiving the payment two years from now. C. equally valuable as a payment received today if the interest rate is zero. D. not enough information is provided to answer the question.
C. equally valuable as a payment received today if the interest rate is zero.
An investment carrying a current cost of $120,000 is going to generate $50,000 of revenue for each of the next three years. To calculate the internal rate of return we need to: A. calculate the present value of each of the $50,000 payments and multiply these and set this equal to $120,000. B. find the interest rate at which the present value of $150,000 for three years from now equals $120,000. C. find the interest rate at which the sum of the present values of $50,000 for each of the next three years equals $120,000. D. subtract $120,000 from $150,000 and set this difference equal to the interest rate.
C. find the interest rate at which the sum of the present values of $50,000 for each of the next three years equals $120,000.
Which formula below best expresses the real interest rate, (r)? A. i = r - πe B. r = i + πe C. r = i - πe D. πe = i + r
C. r = i - πe
A lender is promised a $100 payment (including interest) one year from today. If the lender has a 6% opportunity cost of money, he/she should be willing to accept what amount today? A. $100.00 B. $106.20 C. $96.40 D. $94.34
D. $94.34
Investing in a mutual fund made up of hundreds of stocks of different companies is an example of all of the following except: A. spreading risk. B. diversifying. C. risk reduction. D. increasing the variance of a portfolio.
D. increasing the variance of a portfolio.
PV of the principal on a bond
F/(1+i)^n
future value equation
FV= PV×(1 + i)^n years in the future
value of the bond
Pcb=C/(1+i)^1+C/(1+i)^2....C+F/(1+i)^n
Correlated
Risks bad for one sector of the economy but good for another. •ex: A rise in oil prices is bad for the car industry but good for the energy industry.
Uncorrelated
Risks specific to one person or company that affect no one else. • A firm-specific shock (e.g., poor maintenance)
Risk premium
The compensation a counter-party requires to hold a risky asset. expected return on risky asset-risk free rate
Excess Return (in a period)
The difference between the realized rate of return on a risky asset and the actual risk-free rate.
Internal Rate of Return (IRR)
The interest rate i that equates the present value of an investment with its cost. If R(interest rate paid on loan) < i, then you should buy the machine.
HOW INFLATION ALTERS RETURNS
The lender's return (in real terms) will be lower. The borrower's cost (in real terms) will be lower.
The more independent sources of risk you hold in your portfolio,
The lower your overall risk.
variance
The mean squared deviation of the possible outcomes from their expected value.
Maturity
The number n of periods until the loan is repaid
Diversification
The principle of holding more than one risk at a time. Reduces idiosyncratic risk an investor bears.
risk-free rate of return
The rate of return on a risk-free asset.
hedging
The strategy of reducing idiosyncratic risk by making two investments with opposing risks.
Price P
The value of the loan, on the market
Future value
The value on a future date of a financial investment made today.
Present value
The value today of a payment that will be received on a future date.
Consider two possible investments whose payoffs are completely independent of one another. Both investments have the same expected value and standard deviation. If you have $1,000 to invest, could you benefit from dividing your funds between these investments?
Yes. Even though the investments have the same standard deviation, by spreading your $1000 across both of them, you reduce your risk. The probability weighted spread of the possible payoffs is smaller. Mathematically, the variance of the payoffs is halved.
future cost
amount today (in dollars) x (1 + % price incrse) ^years
