{ "deckName": "Exam FM — Determinants of Interest Rates", "examCode": "Exam FM", "cards": [ { "front": "List the **components** that build up a nominal market interest rate $r$ on a debt instrument.", "back": "As an **additive approximation**, $r \\approx r^{*} + IP + DRP + LP + MRP$, where $r^{*}$ is the **real risk-free rate**, $IP$ the **inflation premium**, $DRP$ the **default-risk premium**, $LP$ the **liquidity premium**, and $MRP$ the **maturity (term) risk premium**.\nThe premiums **approximately add** (the exact build-up compounds them multiplicatively, $1+r=(1+r^{*})(1+IP)\\cdots$, but the cross terms are second-order and usually dropped). The first two combine into the **nominal risk-free rate** (e.g. on short Treasuries); the last three are premiums demanded for the specific bond's risks.", "tag": "Rate components" }, { "front": "What is the **real risk-free rate** $r^{*}$, and what mainly determines it?", "back": "It is the interest rate on a riskless security in a world with **no inflation** — the pure time-value-of-money compensation for deferring consumption.\nIt is driven by the economy's underlying **supply of and demand for capital**: households' time preference and willingness to save versus the productivity of available investment opportunities.", "tag": "Rate components" }, { "front": "Distinguish the **default-risk premium**, **liquidity premium**, and **maturity-risk premium** in one line each.", "back": "**Default-risk premium (DRP):** extra yield for the chance the issuer fails to pay coupons or principal — larger for lower credit ratings.\n**Liquidity premium (LP):** extra yield for a bond that is hard to sell quickly near fair value (thinly traded).\n**Maturity-risk premium (MRP):** extra yield for the greater price sensitivity (interest-rate risk) of longer-maturity bonds.", "tag": "Rate components" }, { "front": "A long-term corporate bond yields $7.4\\%$. The real risk-free rate is $1.5\\%$, the inflation premium is $2.3\\%$, the liquidity premium is $0.6\\%$, and the maturity-risk premium is $1.1\\%$. Find the implied **default-risk premium**.", "back": "Use the additive build-up $r \\approx r^{*} + IP + DRP + LP + MRP$ and solve for $DRP$.\n$DRP = r - (r^{*}+IP+LP+MRP)$.\n$DRP = 7.4\\% - (1.5\\% + 2.3\\% + 0.6\\% + 1.1\\%) = 7.4\\% - 5.5\\% = 1.9\\%$.", "tag": "Rate components" }, { "front": "State the **exact Fisher equation** relating the nominal rate $i$, the real rate $r$, and the inflation rate $\\pi$.", "back": "$1 + i = (1 + r)(1 + \\pi)$.\nSolving for the real rate: $1 + r = \\dfrac{1+i}{1+\\pi}$, so $r = \\dfrac{1+i}{1+\\pi} - 1 = \\dfrac{i - \\pi}{1+\\pi}$.\nHere $\\pi$ is the (expected) inflation rate over the period.", "tag": "Fisher equation" }, { "front": "Give the **approximate Fisher equation** and say when it is accurate.", "back": "$i \\approx r + \\pi$ — the nominal rate is roughly the real rate plus inflation.\nExpanding $1+i=(1+r)(1+\\pi)=1+r+\\pi+r\\pi$ shows the exact relation drops the cross term $r\\pi$. The approximation is good when $r$ and $\\pi$ are **small**, since $r\\pi$ is then negligible; it understates $i$ otherwise.", "tag": "Fisher equation" }, { "front": "Numerically compare the **exact vs approximate Fisher equation** when $r = 10\\%$ and $\\pi = 12\\%$. How big is the error?", "back": "**Exact:** $1+i = (1+r)(1+\\pi) = (1.10)(1.12) = 1.232$, so $i = 23.2\\%$.\n**Approximate:** $i \\approx r + \\pi = 10\\% + 12\\% = 22.0\\%$.\nThe gap is exactly the dropped cross term $r\\pi = (0.10)(0.12) = 0.012 = 1.2\\%$, and $23.2\\% - 22.0\\% = 1.2\\%$.\nAt double-digit rates the cross term is no longer negligible, so the approximation materially **understates** the nominal rate — always use the exact form unless told otherwise.", "tag": "Fisher equation" }, { "front": "A bond earns a nominal annual effective rate of $8.5\\%$ in a year when inflation is $3.2\\%$. Find the **exact real rate of return**.", "back": "Use the exact Fisher equation $1+r = \\dfrac{1+i}{1+\\pi}$.\n$1+r = \\dfrac{1.085}{1.032} \\approx 1.051357$.\n$r \\approx 0.051357 = 5.14\\%$.\n(The approximation $i-\\pi = 8.5\\%-3.2\\% = 5.3\\%$ overstates it by about $0.16\\%$.)", "tag": "Fisher equation" }, { "front": "An investor requires a real return of $2.75\\%$ and expects inflation of $4\\%$. What **nominal annual effective rate** must the investment offer (exact Fisher)?", "back": "Use $1+i = (1+r)(1+\\pi)$.\n$1+i = (1.0275)(1.04) = 1.0686$.\n$i = 0.0686 = 6.86\\%$.\n(The approximation $r+\\pi = 2.75\\%+4\\% = 6.75\\%$ understates the required nominal rate.)", "tag": "Fisher equation" }, { "front": "How does a **central bank** influence the general level of interest rates?", "back": "It sets a target for a short-term policy rate and uses tools — open-market operations (buying/selling government securities), the policy/discount rate, and reserve requirements — to expand or contract the money supply.\nEasing (buying securities, lowering the rate) increases the supply of loanable funds and pushes rates **down**; tightening does the reverse, typically to control inflation.", "tag": "Central bank & loanable funds" }, { "front": "In the **supply-and-demand model for loanable funds**, what shifts each curve and how is the equilibrium rate set?", "back": "The **supply** of loanable funds comes from saving (households, firms, government surpluses, foreign capital); the **demand** comes from borrowers funding investment and consumption.\nThe equilibrium interest rate is where supply equals demand. More saving (supply right) or weaker borrowing demand (demand left) **lowers** rates; stronger demand or less saving **raises** them.", "tag": "Central bank & loanable funds" }, { "front": "All else equal, what happens to the **equilibrium interest rate** if expected inflation rises sharply?", "back": "Lenders demand compensation for the eroded purchasing power of future repayments, so the **supply** of loanable funds shifts left (they lend less at any given nominal rate) while borrowers are willing to pay more, shifting **demand** right.\nBoth effects push the equilibrium **nominal rate up** — consistent with the inflation premium in $i \\approx r + \\pi$.", "tag": "Central bank & loanable funds" }, { "front": "Define the **term structure of interest rates** and the **yield curve**.", "back": "The **term structure** is the relationship between the yield (spot rate) on otherwise-identical default-free bonds and their **time to maturity**.\nThe **yield curve** is its graph — yield on the vertical axis, maturity on the horizontal. Its shape (rising, flat, or inverted) reflects the market's expectations and risk premiums across maturities.", "tag": "Term structure & yield curve" }, { "front": "Name and briefly state the four classic **theories of the term structure** of interest rates.", "back": "**Expectations theory:** long rates are the geometric average of current and expected future short rates (no term premium).\n**Liquidity-preference theory:** expectations plus a maturity-risk premium that rises with term, so the curve is biased upward.\n**Market-segmentation theory:** each maturity has its own isolated supply/demand; rates are set independently by maturity.\n**Preferred-habitat theory:** investors favor a maturity range but will move for sufficient extra yield — segmentation softened by premiums.", "tag": "Term structure & yield curve" }, { "front": "Contrast **liquidity-preference**, **market-segmentation**, and **preferred-habitat** theories — what does each add beyond pure expectations?", "back": "**Liquidity-preference:** keeps the expectations backbone but layers on a **maturity-risk premium that grows with term**, so the curve is biased upward even when future short rates are expected flat.\n**Market-segmentation:** rejects the expectations link entirely — each maturity is a **separate market** with its own clienteles (e.g. banks short, pensions/insurers long), and rates at one maturity say nothing about another.\n**Preferred-habitat:** a middle ground — investors have a preferred maturity (their habitat) but will **leave it for a large enough yield pickup**, so premiums can be positive or negative and need not rise monotonically with term.", "tag": "Term structure & yield curve" }, { "front": "Under the **(unbiased) expectations theory**, write the spot–forward relation for a general $n$-year spot rate.", "back": "The $n$-year spot rate compounds the current 1-year rate with the sequence of **expected one-period forward rates**, with **no term premium**:\n$(1+s_{n})^{n} = (1+s_{1})\\displaystyle\\prod_{k=1}^{n-1}\\bigl(1 + E[\\,_{k}f_{1}]\\bigr)$,\nwhere $E[\\,_{k}f_{1}]$ is the expected 1-year rate beginning $k$ years from now.\nSo $1+s_{n}$ is the **geometric mean** of $(1+s_{1})$ and the expected future one-period rates; an upward-sloping curve signals the market expects **rising** future short rates.", "tag": "Term structure & yield curve" }, { "front": "What does each of the three main **yield-curve shapes** typically signal?", "back": "**Normal (upward-sloping):** long yields exceed short yields — the usual state; markets expect stable/rising rates and/or positive term premiums; often signals expansion.\n**Inverted (downward-sloping):** short yields exceed long yields — markets expect rates (and growth) to fall; a classic recession warning.\n**Flat:** little difference across maturities — often a transition between the two regimes.", "tag": "Term structure & yield curve" }, { "front": "Define a **spot rate** $s_{t}$ and a **forward rate**, and give the no-arbitrage link between them.", "back": "A **spot rate** $s_{t}$ is the annual effective yield on a zero-coupon bond maturing at time $t$; the PV of a cash flow at $t$ is $CF_{t}(1+s_{t})^{-t}$.\nA **forward rate** is a rate agreed today for a future period. No-arbitrage requires\n$(1+s_{t+k})^{t+k} = (1+s_{t})^{t}\\,(1+f)^{k}$,\nwhere $f$ is the $k$-year forward rate starting at time $t$.", "tag": "Spot & forward rates" }, { "front": "What is a **bond covenant**, and how do covenants relate to the default-risk premium?", "back": "A covenant is a legally binding clause in the bond indenture that constrains the issuer to protect lenders.\n**Affirmative covenants** require actions (maintain ratios, supply audited statements); **negative covenants** prohibit actions (limit new debt, asset sales, dividends).\nStronger covenants reduce the lender's expected loss, lowering the **default-risk premium** and the bond's required yield.", "tag": "Covenants & default risk" }, { "front": "How do an embedded **call feature** versus a **put feature** or **conversion option** shift a bond's required yield?", "back": "An embedded option is exercisable by whoever it benefits, and the **counterparty is paid for granting it** through the yield.\n**Call feature** (issuer may redeem early, typically when rates fall): hurts the investor (reinvestment risk, capped upside), so investors demand a **higher** yield than an otherwise-identical option-free bond.\n**Put feature** (holder may sell back early) and **convertibility** (holder may convert to equity): both benefit the investor, so the bond can be issued at a **lower** yield. Net effect: $y_{\\text{callable}} > y_{\\text{straight}} > y_{\\text{putable/convertible}}$, all else equal.", "tag": "Covenants & default risk" }, { "front": "How does an issuer's **credit rating** affect the yield investors demand?", "back": "Lower ratings (e.g. moving from investment-grade toward speculative/\"junk\") signal a higher probability of default and larger loss given default, so investors require a larger **default-risk premium** and the yield rises (price falls).\nThe gap between a corporate bond's yield and a same-maturity Treasury is the **credit spread**, which widens as ratings fall and in market stress.", "tag": "Covenants & default risk" }, { "front": "A 1-year zero-coupon bond from a risky issuer has a $4\\%$ probability of default with **zero** recovery. The risk-free rate is $3\\%$. What **promised yield** $y$ makes the investor's expected return equal the risk-free rate?", "back": "Investing $1$ pays $(1+y)$ with probability $0.96$ and $0$ otherwise. Set expected proceeds equal to the risk-free accumulation:\n$0.96\\,(1+y) = 1.03$.\n$1+y = \\dfrac{1.03}{0.96} \\approx 1.072917$, so $y \\approx 7.29\\%$.\nThe **default-risk premium** is about $7.29\\% - 3\\% = 4.29\\%$ — it exceeds the $4\\%$ default probability because the spread must also cover the lost principal on default.", "tag": "Covenants & default risk" }, { "front": "Price a 1-year $\\$1{,}000$-par zero from a risky issuer by **risk-neutral pricing**: default probability $q = 5\\%$, recovery $40\\%$ of par on default, risk-free rate $3\\%$. Find the price and the promised yield.", "back": "Discount the **expected** payoff at the risk-free rate. Payoff is $\\$1{,}000$ if no default ($0.95$) and $0.40 \\times \\$1{,}000 = \\$400$ on default ($0.05$):\n$E[\\text{payoff}] = 0.95(1000) + 0.05(400) = 950 + 20 = \\$970$.\n$P = \\dfrac{970}{1.03} \\approx \\$941.75$.\nPromised yield: $1+y = \\dfrac{1000}{941.75}$, so $y \\approx 6.19\\%$ — a credit spread of about $6.19\\% - 3\\% = 3.19\\%$ over the risk-free rate.", "tag": "Covenants & default risk" }, { "front": "A defaultable 1-year bond promises a yield of $9\\%$. The risk-free rate is $4\\%$ and recovery on default is $30\\%$ of the promised amount. Find the **implied probability of default** $q$.", "back": "Equate expected proceeds to the risk-free accumulation. With promised payoff $1.09$ and recovery $0.30 \\times 1.09 = 0.327$ on default:\n$(1-q)(1.09) + q(0.327) = 1.04$.\n$1.09 - 0.763\\,q = 1.04 \\Rightarrow 0.763\\,q = 0.05$.\n$q = \\dfrac{0.05}{0.763} \\approx 0.0655 = 6.55\\%$.", "tag": "Covenants & default risk" } ] }