CFA Level 1. Quantitative Methods
Three ways interest rates can be thought of - answer First, they can be considered
required rates of return—that is, the minimum rate of return an investor must receive in
order to accept the investment.
Second, interest rates can be considered discount rates. In the example above, 5.26
percent is that rate at which we discounted the $10,000 future amount to find its value
today. Thus, we use the terms "interest rate" and "discount rate" almost
interchangeably.
Third, interest rates can be considered opportunity costs. An opportunity cost is the
value that investors forgo by choosing a particular course of action. In the example, if
the party who supplied $9,500 had instead decided to spend it today, he would have
forgone earning 5.26 percent on the money. So we can view 5.26 percent as the
opportunity cost of current consumption.
How are interest rates set in the marketplace? - answer Supply and demand
Interest rate formula - answer=Real risk-free interest rate Inflation premium +Default risk
premium Liquidity premium Maturity premium
real risk-free interest rate - answer the single-period interest rate for a completely risk-
free security if no inflation were expected. In economic theory, the real risk-free rate
reflects the time preferences of individuals for current versus future real consumption.
Inflation Premium - answer compensates investors for expected inflation and reflects
the average inflation rate expected over the maturity of the debt
Nominal Risk Free Interest Rate - answer The sum of the real risk-free interest rate and
the inflation premium. (T-bill).
Liquidity premium - answercompensates investors for the risk of loss relative to an
investment's fair value if the investment needs to be converted to cash quickly.
Maturity Premium - answercompensates investors for the increased sensitivity of the
market value of debt to a change in market interest rates as maturity is extended, in
general (holding all else equal).
Simple Interest - answerInterest rate times the principle.
Is simple or compound interest more important? - answerCompound.
,Important consequence of present value and future value being separated in time? -
answerWe can add amounts of money only if they are indexed at the same point in
time.
EAR formula - answerEAR = (1 + Periodic interest rate)^m - 1
EAR=e^rs−1
EAR = Non EAR - answer0.0816=(1+Periodic rate)^2−1
1.0816=(1+Periodic rate)^2
(1.0816)^1/2−1=Periodic rate
(1.04)−1=Periodic rate
4%=Periodic rate
calculate the continuously compounded rate corresponding to an effective annual rate
of 8.33% - answerFind interest rate that solves:
EAR=e^rs−1
0.0833=e^rs−1
1.0833=e^rs
To solve this equation, we take the natural logarithm of both sides. (Recall that the
natural log of e^rs is ln e^rs=rs .) Therefore, ln 1.0833 = rs, resulting in rs = 8 percent.
We see that a stated annual rate of 8 percent with continuous compounding is
equivalent to an EAR of 8.33 percent.
Difference between ordinary annuity and annuity due? - answerAnnuity due has first
cash flow that occurs immediately while ordinary annuity has first cash flow that occurs
one period from now.
future value annuity factor
Also need to know PV annuity for ordinary annuity and annuity due
perp formula
Solve for interest rate and growth rate
annual compounding periods needed to reach a value
calculate size of payment annuity
calculate amount needed now to fund future annuity
calculate lump sum annuity - answerFV=A[(1+r)^n - 1 / r]
(1+r)^n = FV/PV
N ln(1+r) =ln(2)
N = ln(2) / ln(1+r)
ln(2)/ln(1.07) = 10.24
,Cash flow additivity principle - answerthe idea that amounts of money indexed at the
same point in time are additive
Working capital management - answerManagement of the company's short-term assets
and short-term liabilities.
Steps in computing and applying NPV Rule - answer1) Identify all cash flows associated
with the investment—all inflows and outflows.1
2) Determine the appropriate discount rate or opportunity cost, r, for the investment
project.2
3) Using that discount rate, find the present value of each cash flow. (Inflows have a
positive sign and increase NPV; outflows have a negative sign and decrease NPV.)
4) Sum all present values. The sum of the present values of all cash flows (inflows and
outflows) is the investment's net present value.
5) Apply the NPV rule: If the investment's NPV is positive, an investor should undertake
it; if the NPV is negative, the investor should not undertake it. If an investor has two
candidates for investment but can only invest in one (i.e., mutually exclusive projects),
the investor should choose the candidate with the higher positive NPV.
What is the IRR? - answerThe discount rate that makes the NPV = 0.
What is necessary to recognize a compound rate of return that is equal to the IRR over
the life of the investment? - answerOnly if we can reinvest all interim cash flows at
exactly the IRR.
IRR Rule - answeruses the opportunity cost of capital as a hurdle rate, or rate that a
project's IRR must exceed for the project to be accepted. Note that if the opportunity
cost of capital is equal to the IRR, then the NPV is equal to 0. If the project's opportunity
cost is less than the IRR, the NPV is greater than 0.
IRR Formula
Investment - answerCF/Investment
CF/IRR
What do IRR and NPV rule do when projects are independent? - answerGive same
accept or reject decision.
When does NPV and IRR rank projects differently? - answerWhen they don't have
enough money to do all of them and must rank them in terms of profitability.
The IRR and NPV rank differently when:
1) the size or scale of the projects differ.
2) timing of the projects' cash flows differ.
What do you follow when NPV and IRR are different? - answerNPV
, - it represents addition to shareholder wealth, and we take the basic goal of a company
to be to make shareholders money.
Performance measurement - answerwhich involves calculating returns in a logical and
consistent manner.
MWR
What is IRR equal to? - answerMoney weighted return
Example of an investment measure? - answerTime-weighted rate of return
- measures the compound rate of growth of $1 initially invested in the portfolio over a
stated measurement period
- not affected by money withdrawal/additions.
Time-weighted return steps - answerPrice the portfolio immediately prior to any
significant addition or withdrawal of funds. Break the overall evaluation period into
subperiods based on the dates of cash inflows and outflows.
Calculate the holding period return on the portfolio for each subperiod.
Link or compound holding period returns to obtain an annual rate of return for the year
(the time-weighted rate of return for the year). If the investment is for more than one
year, take the geometric mean of the annual returns to obtain the time-weighted rate of
return over that measurement period.
Pure discount instrument - answerPay less than face value (T-bill).
What does a bank discount basis do?
What is quoted this way? - answerIt's a quoting convention that annualizes, based on a
360 day year, the discount as a percentage of face value.
T-bills.
Why is yield on a bank discount basis not a meaningful measure of investors' return? -
answerFirst, the yield is based on the face value of the bond, not on its purchase price.
Returns from investments should be evaluated relative to the amount that is invested.
Second, the yield is annualized based on a 360-day year rather than a 365-day year.
Third, the bank discount yield annualizes with simple interest, which ignores the
opportunity to earn interest on interest (compound interest).
Effective Annual Yield - answerThe EAY takes the quantity 1 plus the holding period
yield and compounds it forward to one year, then subtracts 1 to recover an annualized
return that accounts for the effect of interest-on-interest.
What does the money market yield do? - answerThis convention makes the quoted
yield on a T-bill comparable to yield quotations on interest-bearing money-market
Three ways interest rates can be thought of - answer First, they can be considered
required rates of return—that is, the minimum rate of return an investor must receive in
order to accept the investment.
Second, interest rates can be considered discount rates. In the example above, 5.26
percent is that rate at which we discounted the $10,000 future amount to find its value
today. Thus, we use the terms "interest rate" and "discount rate" almost
interchangeably.
Third, interest rates can be considered opportunity costs. An opportunity cost is the
value that investors forgo by choosing a particular course of action. In the example, if
the party who supplied $9,500 had instead decided to spend it today, he would have
forgone earning 5.26 percent on the money. So we can view 5.26 percent as the
opportunity cost of current consumption.
How are interest rates set in the marketplace? - answer Supply and demand
Interest rate formula - answer=Real risk-free interest rate Inflation premium +Default risk
premium Liquidity premium Maturity premium
real risk-free interest rate - answer the single-period interest rate for a completely risk-
free security if no inflation were expected. In economic theory, the real risk-free rate
reflects the time preferences of individuals for current versus future real consumption.
Inflation Premium - answer compensates investors for expected inflation and reflects
the average inflation rate expected over the maturity of the debt
Nominal Risk Free Interest Rate - answer The sum of the real risk-free interest rate and
the inflation premium. (T-bill).
Liquidity premium - answercompensates investors for the risk of loss relative to an
investment's fair value if the investment needs to be converted to cash quickly.
Maturity Premium - answercompensates investors for the increased sensitivity of the
market value of debt to a change in market interest rates as maturity is extended, in
general (holding all else equal).
Simple Interest - answerInterest rate times the principle.
Is simple or compound interest more important? - answerCompound.
,Important consequence of present value and future value being separated in time? -
answerWe can add amounts of money only if they are indexed at the same point in
time.
EAR formula - answerEAR = (1 + Periodic interest rate)^m - 1
EAR=e^rs−1
EAR = Non EAR - answer0.0816=(1+Periodic rate)^2−1
1.0816=(1+Periodic rate)^2
(1.0816)^1/2−1=Periodic rate
(1.04)−1=Periodic rate
4%=Periodic rate
calculate the continuously compounded rate corresponding to an effective annual rate
of 8.33% - answerFind interest rate that solves:
EAR=e^rs−1
0.0833=e^rs−1
1.0833=e^rs
To solve this equation, we take the natural logarithm of both sides. (Recall that the
natural log of e^rs is ln e^rs=rs .) Therefore, ln 1.0833 = rs, resulting in rs = 8 percent.
We see that a stated annual rate of 8 percent with continuous compounding is
equivalent to an EAR of 8.33 percent.
Difference between ordinary annuity and annuity due? - answerAnnuity due has first
cash flow that occurs immediately while ordinary annuity has first cash flow that occurs
one period from now.
future value annuity factor
Also need to know PV annuity for ordinary annuity and annuity due
perp formula
Solve for interest rate and growth rate
annual compounding periods needed to reach a value
calculate size of payment annuity
calculate amount needed now to fund future annuity
calculate lump sum annuity - answerFV=A[(1+r)^n - 1 / r]
(1+r)^n = FV/PV
N ln(1+r) =ln(2)
N = ln(2) / ln(1+r)
ln(2)/ln(1.07) = 10.24
,Cash flow additivity principle - answerthe idea that amounts of money indexed at the
same point in time are additive
Working capital management - answerManagement of the company's short-term assets
and short-term liabilities.
Steps in computing and applying NPV Rule - answer1) Identify all cash flows associated
with the investment—all inflows and outflows.1
2) Determine the appropriate discount rate or opportunity cost, r, for the investment
project.2
3) Using that discount rate, find the present value of each cash flow. (Inflows have a
positive sign and increase NPV; outflows have a negative sign and decrease NPV.)
4) Sum all present values. The sum of the present values of all cash flows (inflows and
outflows) is the investment's net present value.
5) Apply the NPV rule: If the investment's NPV is positive, an investor should undertake
it; if the NPV is negative, the investor should not undertake it. If an investor has two
candidates for investment but can only invest in one (i.e., mutually exclusive projects),
the investor should choose the candidate with the higher positive NPV.
What is the IRR? - answerThe discount rate that makes the NPV = 0.
What is necessary to recognize a compound rate of return that is equal to the IRR over
the life of the investment? - answerOnly if we can reinvest all interim cash flows at
exactly the IRR.
IRR Rule - answeruses the opportunity cost of capital as a hurdle rate, or rate that a
project's IRR must exceed for the project to be accepted. Note that if the opportunity
cost of capital is equal to the IRR, then the NPV is equal to 0. If the project's opportunity
cost is less than the IRR, the NPV is greater than 0.
IRR Formula
Investment - answerCF/Investment
CF/IRR
What do IRR and NPV rule do when projects are independent? - answerGive same
accept or reject decision.
When does NPV and IRR rank projects differently? - answerWhen they don't have
enough money to do all of them and must rank them in terms of profitability.
The IRR and NPV rank differently when:
1) the size or scale of the projects differ.
2) timing of the projects' cash flows differ.
What do you follow when NPV and IRR are different? - answerNPV
, - it represents addition to shareholder wealth, and we take the basic goal of a company
to be to make shareholders money.
Performance measurement - answerwhich involves calculating returns in a logical and
consistent manner.
MWR
What is IRR equal to? - answerMoney weighted return
Example of an investment measure? - answerTime-weighted rate of return
- measures the compound rate of growth of $1 initially invested in the portfolio over a
stated measurement period
- not affected by money withdrawal/additions.
Time-weighted return steps - answerPrice the portfolio immediately prior to any
significant addition or withdrawal of funds. Break the overall evaluation period into
subperiods based on the dates of cash inflows and outflows.
Calculate the holding period return on the portfolio for each subperiod.
Link or compound holding period returns to obtain an annual rate of return for the year
(the time-weighted rate of return for the year). If the investment is for more than one
year, take the geometric mean of the annual returns to obtain the time-weighted rate of
return over that measurement period.
Pure discount instrument - answerPay less than face value (T-bill).
What does a bank discount basis do?
What is quoted this way? - answerIt's a quoting convention that annualizes, based on a
360 day year, the discount as a percentage of face value.
T-bills.
Why is yield on a bank discount basis not a meaningful measure of investors' return? -
answerFirst, the yield is based on the face value of the bond, not on its purchase price.
Returns from investments should be evaluated relative to the amount that is invested.
Second, the yield is annualized based on a 360-day year rather than a 365-day year.
Third, the bank discount yield annualizes with simple interest, which ignores the
opportunity to earn interest on interest (compound interest).
Effective Annual Yield - answerThe EAY takes the quantity 1 plus the holding period
yield and compounds it forward to one year, then subtracts 1 to recover an annualized
return that accounts for the effect of interest-on-interest.
What does the money market yield do? - answerThis convention makes the quoted
yield on a T-bill comparable to yield quotations on interest-bearing money-market
