Financial Decision Making, Time Value of Money & Interest Rates—Master Flashcards

200 Q&A flashcards covering key ideas, formulas, applications, and pitfalls for Chapters 3–5 (Valuation Principles, Time Value of Money, Interest Rates).

What is the time value of money?

The economic principle that a dollar received today is worth more than a dollar received in the future because today’s dollar can be invested to earn interest.

Define ‘interest rate’ in the context of time value of money.

The market-determined exchange rate between earlier dollars (present value) and later dollars (future value).

What rule allows you to compare or combine cash flows only when they are at the same point in time?

Rule 1 of time travel: Only values at the same date can be compared or combined.

When moving a cash flow backward in time which operation is required?

Discounting (Rule 3).

Write the formula for future value after n periods when the interest rate is r.

FVₙ = C × (1 + r)ⁿ

Write the formula for present value of a single cash flow C received in n periods at rate r.

PV = C ⁄ (1 + r)ⁿ

What is compound interest?

Earning interest on both the original principal and the accumulated interest from prior periods.

What does EAR stand for?

Effective Annual Rate.

How is an EAR different from an APR?

EAR includes the effect of intra-year compounding; APR states simple interest without compounding within the year.

Give the equation to convert an APR (compounded k times per year) into an EAR.

1 + EAR = (1 + APR ⁄ k)ᵏ

Provide the equation to convert an EAR to an equivalent n-period rate.

Equivalent rate = (1 + EAR)ⁿ – 1

What is a perpetuity?

A stream of equal cash flows that continues forever.

Formula for the present value of a perpetuity with payment C and rate r.

PV = C ⁄ r

Define an annuity.

A stream of equal cash flows paid at regular intervals for a fixed number of periods.

Present value formula for an ordinary annuity.

PV = C × [1 ⁄ r × (1 – 1 ⁄ (1 + r)ᴺ)]

Future value formula of an annuity.

FV = C × [1 ⁄ r × ((1 + r)ᴺ – 1)]

What is an annuity due?

An annuity whose first cash flow occurs immediately (at date 0).

How do you adjust the PV of an ordinary annuity to compute an annuity due?

Multiply the ordinary annuity PV by (1 + r).

Define a growing perpetuity.

A cash-flow stream that lasts forever with payments growing at a constant rate g each period.

Formula for PV of a growing perpetuity.

PV = C ⁄ (r – g) (with r > g).

Define a growing annuity.

A finite stream of cash flows that grow at a constant rate g for N periods.

Formula for PV of a growing annuity.

PV = C × [(1 ⁄ (r – g)) × (1 – ((1 + g)ᵑ ⁄ (1 + r)ᵑ))]

What is the NPV Decision Rule?

Accept projects with positive NPV; they increase firm value.

Explain ‘opportunity cost of capital’.

The best available expected return offered in the market for investments of equivalent risk and term.

Define ‘arbitrage’.

A risk-free profit achieved by simultaneously buying and selling equivalent assets at different prices.

What is the Law of One Price?

Identical goods or securities must have the same price in all competitive markets to avoid arbitrage.

Describe a normal market.

A competitive market in which no arbitrage opportunities exist.

When is the internal rate of return (IRR) easy to compute exactly?

For projects with (1) only two cash flows, or (2) a constant growing perpetuity.

Give the IRR formula when there are only two cash flows: –P now and FV in N years.

IRR = (FV ⁄ P)¹⁄ᴺ – 1

How does risk affect the interest rate a borrower must pay?

Higher default risk leads to a higher required interest rate (risk premium).

Write the equation for an after-tax interest rate when interest is taxable at rate t.

After-tax rate = r × (1 – t)

What is the real interest rate?

The nominal rate adjusted for inflation; approximately r – i.

Provide the exact formula linking real (rᵣ), nominal (r), and inflation (i) rates.

1 + rᵣ = (1 + r) ⁄ (1 + i)

Define ‘yield curve’.

A graph showing the term structure of interest rates, plotting yields against maturities.

What does an inverted yield curve often predict?

A potential economic slowdown or recession.

Explain continuous compounding.

Interest is compounded an infinite number of times per year; EAR = e^APR – 1.

How do you compute the outstanding balance on an amortizing loan?

By taking the PV of the remaining loan payments discounted at the loan’s periodic rate.

What is a teaser rate?

A temporarily low interest rate on a loan that later resets to a higher rate.

Why must the discount rate match the cash flow’s horizon?

Because rates differ by term; using mismatched rates mis-values cash flows.

What is the simple shortcut called ‘Rule of 72’?

Approximate years to double = 72 ⁄ interest rate (%)

How does the PV of an annuity change if the interest rate rises?

It falls, because future payments are discounted more heavily.

Why can’t you discount a perpetuity using an APR?

APR ignores intra-year compounding; use EAR or appropriate periodic rate.

Explain how the Federal Reserve influences very short-term rates.

By setting the federal funds target and controlling bank reserves.

Name two ‘unconventional’ policies used when short rates hit zero.

(1) Large-scale asset purchases (quantitative easing); (2) forward guidance about future rates.

How do taxes influence the effective cost of borrowing for homeowners?

Mortgage interest is tax-deductible, lowering the after-tax borrowing cost.

If a loan’s APR is 9% with monthly payments, what is its monthly periodic rate?

9% ⁄ 12 = 0.75% per month.

State the formula for a loan/annuity payment C with principal P, rate r, periods N.

C = P ⁄ [ (1 ⁄ r) × (1 – 1 ⁄ (1 + r)ᴺ ) ]

What is an amortization schedule?

A table showing each loan payment’s breakdown between interest and principal and the remaining balance.

Why can using the wrong compounding interval lead to errors?

Because EARs change with frequency; mismatched intervals mis-state equivalent rates.

Which compounding frequency produces the highest EAR for a given APR?

Continuous compounding (limit as frequency → ∞).

When discounting semiannual bond coupons, which rate should be used?

The bond’s quoted yield divided by two (the semiannual rate).

Describe ‘value additivity’.

The value of a portfolio equals the sum of the values of its parts in a normal market.

Explain separation principle in corporate finance.

Investment decisions can be evaluated independently from financing decisions in efficient markets.

State the present value of $1 received each quarter forever at a nominal discount rate of 8% EAR.

First convert EAR to quarterly rate: (1.08)¹⁄⁴ – 1 ≈ 1.941%; PV = 1 ⁄ 0.01941 = $51.54

What is the effect of inflation on nominal vs. real returns?

High inflation increases nominal returns but real (purchasing-power) returns may not rise.

If the one-year rate is 3% and the expected one-year rate next year is 5%, approximate the current two-year rate.

r₂ ≈ [(1.03 × 1.05)¹⁄²] – 1 ≈ 3.99%.

What is an ear?

A pun? No: It’s an abbreviation for Effective Annual Rate (EAR), not the thing you hear with.

Why does subprime lending carry higher APRs?

Because lenders require compensation for the higher default risk of subprime borrowers.

How can arbitrage eliminate price differences between New York and London gold markets?

By buying in one market and simultaneously selling in the other until prices converge.

What is the cost of capital for a risk-free three-year project?

The three-year U.S. Treasury yield (EAR) at project inception.

Give a real-life example of a perpetuity.

British consols: Government bonds that promise fixed coupon payments forever.

Why do home equity loans usually have lower rates than credit cards?

They are secured by collateral (your house), reducing default risk and therefore required return.

Define ‘discount factor’.

1 ⁄ (1 + r)ⁿ, the PV today of $1 received n periods in the future.

What is an Exchange-Traded Fund (ETF)?

A security representing a portfolio of assets that trades on an exchange like a single stock.

Explain bid-ask spread.

Difference between the price at which dealers buy (bid) and sell (ask) a security; represents a transaction cost.

How does the presence of transaction costs affect arbitrage?

Arbitrage ensures prices differ, at most, by the size of transaction costs—not disappear.

Why can’t a normal market have negative-NPV voluntary trades?

Because each side can reject trades that reduce its value; trades occur only at zero NPV.

How can continuous compounding be derived from discrete compounding?

By taking the limit as the number of compounding periods per year approaches infinity.

What is the formula for EAR under continuous compounding given an APRc?

EAR = e^APRc – 1

If APRc = 6% with continuous compounding, compute EAR.

EAR = e^0.06 – 1 ≈ 6.1837%.

Explain ‘yield to maturity’ (YTM).

The single discount rate that equates a bond’s price to the present value of its coupon and principal payments.

What does the Fisher equation approximate?

Real rate ≈ nominal rate – inflation rate (r ≈ i + rr).

Why are municipal bond yields lower than corporates with same maturity?

Municipal bond interest is typically exempt from federal (and often state) taxes.

State the effect of leverage on a firm’s equity cost of capital (simplified).

Higher leverage increases equity risk, raising the equity cost of capital (per Modigliani-Miller).

How does a central bank lower short-term rates?

By increasing the money supply / bank reserves via open-market purchases.

What is quantitative easing?

Central bank policy of purchasing long-term securities to lower long-term interest rates.

If monthly mortgage rate is 0.3%, what is its APR?

APR = 0.3% × 12 = 3.6%.

Compute monthly payment on $250 000 30-year mortgage, 3% APR comp’d monthly.

Periodic rate = 0.25%; N = 360; Payment ≈ $1054.01.

Define ‘refinancing’.

Replacing an existing loan with a new one, typically to benefit from lower rates.

Why does PV of a growing annuity converge when g > r?

It doesn’t; formula applies only if r > g.

Give the shortcut for PV of an annuity due.

PV₍due₎ = PV₍ordinary₎ × (1 + r).

What is the NPV of buying a security at fair price in a normal market?

Zero—the cost equals present value of cash flows.

Define ‘portfolio’.

A collection of securities held as a group.

Describe index arbitrage.

Simultaneous trading of an index fund and its component stocks to exploit mispricing.

What is the competitive price of equivalent goods when transaction costs exist?

Unique up to the transaction cost band.

State the Valuation Principle.

Value of a decision is determined by its costs and benefits, using competitive market prices.

How does a firm maximize shareholder value?

By undertaking all projects with positive NPV given correct cost of capital.

What is the present value of $5 000 received semi-annually for 6 years at 6% EAR?

Convert to semi rate: (1.06)¹⁄² – 1 = 2.941%; N = 12; PV = 5000 × [1 ⁄ 0.02941 × (1 – 1⁄(1.02941)¹²)] ≈ $49 517.

Why are Fed funds rates influential for other rates?

They set banks’ marginal cost of short-term funds, influencing loan/deposit rates economy-wide.

What is the effect of risk aversion on risk premia?

More risk-averse investors demand larger risk premia for bearing risk.

Formula linking risk premium, sensitivity to market, and market risk premium (CAPM).

Risk premium = β × (E[Rₘ] – r_f).

Explain short selling.

Selling a security you do not own by borrowing it, hoping to buy it back later at a lower price.

When interest payments are tax deductible, which interest rate do you use for PV?

After-tax cost of debt: r_d × (1 – tax rate).

If real rate is 1% and inflation is 3%, what is nominal rate (approx)?

≈ 1% + 3% = 4% nominal.

Why may nominal 0% rate still mean positive real cost?

If inflation is negative (deflation), real rate = (r – i)/(1 + i) could be positive.

Explain ‘discount factor term structure’.

Sequence of discount factors DFₙ = 1 ⁄ (1 + rₙ)ⁿ for each maturity.

Compute DF for 2-year cash flow if r₂ = 3%.

DF = 1 ⁄ (1.03)² ≈ 0.9426.

What is a shortcut to value equally spaced cash flows starting immediately forever?

Use PV of a perpetuity due: C ⁄ r × (1 + r).

Define ‘simple interest’.

Interest earned only on the original principal, not on accumulated interest.

Provide an example where simple interest is used.

Corporate bonds quoted on a straight-line basis between coupon dates.