Exam FM — Determinants of Interest Rates Flashcards

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**. The 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.

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. It 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.

**Default-risk premium (DRP):** extra yield for the chance the issuer fails to pay coupons or principal — larger for lower credit ratings. **Liquidity premium (LP):** extra yield for a bond that is hard to sell quickly near fair value (thinly traded). **Maturity-risk premium (MRP):** extra yield for the greater price sensitivity (interest-rate risk) of longer-maturity bonds.

Use the additive build-up $r \approx r^{*} + IP + DRP + LP + MRP$ and solve for $DRP$. $DRP = r - (r^{*}+IP+LP+MRP)$. $DRP = 7.4\% - (1.5\% + 2.3\% + 0.6\% + 1.1\%) = 7.4\% - 5.5\% = 1.9\%$.

$1 + i = (1 + r)(1 + \pi)$. Solving for the real rate: $1 + r = \dfrac{1+i}{1+\pi}$, so $r = \dfrac{1+i}{1+\pi} - 1 = \dfrac{i - \pi}{1+\pi}$. Here $\pi$ is the (expected) inflation rate over the period.

$i \approx r + \pi$ — the nominal rate is roughly the real rate plus inflation. Expanding $1+i=(1+r)(1+\pi)=1+r+\pi+r\pi$ 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.

**Exact:** $1+i = (1+r)(1+\pi) = (1.10)(1.12) = 1.232$, so $i = 23.2\%$. **Approximate:** $i \approx r + \pi = 10\% + 12\% = 22.0\%$. The gap is exactly the dropped cross term $r\pi = (0.10)(0.12) = 0.012 = 1.2\%$, and $23.2\% - 22.0\% = 1.2\%$. At 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.

Use the exact Fisher equation $1+r = \dfrac{1+i}{1+\pi}$. $1+r = \dfrac{1.085}{1.032} \approx 1.051357$. $r \approx 0.051357 = 5.14\%$. (The approximation $i-\pi = 8.5\%-3.2\% = 5.3\%$ overstates it by about $0.16\%$.)

Use $1+i = (1+r)(1+\pi)$. $1+i = (1.0275)(1.04) = 1.0686$. $i = 0.0686 = 6.86\%$. (The approximation $r+\pi = 2.75\%+4\% = 6.75\%$ understates the required nominal rate.)

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. Easing (buying securities, lowering the rate) increases the supply of loanable funds and pushes rates **down**; tightening does the reverse, typically to control inflation.

The **supply** of loanable funds comes from saving (households, firms, government surpluses, foreign capital); the **demand** comes from borrowers funding investment and consumption. The 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.

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. Both effects push the equilibrium **nominal rate up** — consistent with the inflation premium in $i \approx r + \pi$.

The **term structure** is the relationship between the yield (spot rate) on otherwise-identical default-free bonds and their **time to maturity**. The **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.

**Expectations theory:** long rates are the geometric average of current and expected future short rates (no term premium). **Liquidity-preference theory:** expectations plus a maturity-risk premium that rises with term, so the curve is biased upward. **Market-segmentation theory:** each maturity has its own isolated supply/demand; rates are set independently by maturity. **Preferred-habitat theory:** investors favor a maturity range but will move for sufficient extra yield — segmentation softened by premiums.

**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. **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. **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.

The $n$-year spot rate compounds the current 1-year rate with the sequence of **expected one-period forward rates**, with **no term premium**: $(1+s_{n})^{n} = (1+s_{1})\displaystyle\prod_{k=1}^{n-1}\bigl(1 + E[\,_{k}f_{1}]\bigr)$, where $E[\,_{k}f_{1}]$ is the expected 1-year rate beginning $k$ years from now. So $1+s_{n}$ 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.

**Normal (upward-sloping):** long yields exceed short yields — the usual state; markets expect stable/rising rates and/or positive term premiums; often signals expansion. **Inverted (downward-sloping):** short yields exceed long yields — markets expect rates (and growth) to fall; a classic recession warning. **Flat:** little difference across maturities — often a transition between the two regimes.

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 $CF_{t}(1+s_{t})^{-t}$. A **forward rate** is a rate agreed today for a future period. No-arbitrage requires $(1+s_{t+k})^{t+k} = (1+s_{t})^{t}\,(1+f)^{k}$, where $f$ is the $k$-year forward rate starting at time $t$.

A covenant is a legally binding clause in the bond indenture that constrains the issuer to protect lenders. **Affirmative covenants** require actions (maintain ratios, supply audited statements); **negative covenants** prohibit actions (limit new debt, asset sales, dividends). Stronger covenants reduce the lender's expected loss, lowering the **default-risk premium** and the bond's required yield.

An embedded option is exercisable by whoever it benefits, and the **counterparty is paid for granting it** through the yield. **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. **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.

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). The 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.

Investing $1$ pays $(1+y)$ with probability $0.96$ and $0$ otherwise. Set expected proceeds equal to the risk-free accumulation: $0.96\,(1+y) = 1.03$. $1+y = \dfrac{1.03}{0.96} \approx 1.072917$, so $y \approx 7.29\%$. The **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.

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 ($0.05$): $E[\text{payoff}] = 0.95(1000) + 0.05(400) = 950 + 20 = \$970$. $P = \dfrac{970}{1.03} \approx \$941.75$. Promised 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.

Equate expected proceeds to the risk-free accumulation. With promised payoff $1.09$ and recovery $0.30 \times 1.09 = 0.327$ on default: $(1-q)(1.09) + q(0.327) = 1.04$. $1.09 - 0.763\,q = 1.04 \Rightarrow 0.763\,q = 0.05$. $q = \dfrac{0.05}{0.763} \approx 0.0655 = 6.55\%$.