05 — Mathematical Finance

Comprehensive study guide for Bocconi Math Module 2 (cod. 30063), General Exam. Source materials: Alice Sicconi's Finance Proofs.pdf, Mathematical Finance.pdf, lect1_calc_fin (2).pdf, General_24524_ENG_SOL.pdf, Final exercises solution.pdf.

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§1. Overview & Exam Relevance

Mathematical Finance is the second block of the second partial and accounts for approximately 14% of the general exam (typically 1 MCQ worth 5 pts and one open-ended question worth up to 20 pts — about 25 pts of the 150-pt exam). It is the topic where a correctly reproduced Lemma 1143 — No arbitrage of the I kind implies the Law of One Price or a clean bond price / duration / volatility calculation yields the highest marginal return on study time.

Topic scope. The exam tests:

• Time value of money: accumulation factor $f(t)$, discount factor $v(t) = 1/f(t)$, capitalization axioms; simple interest $f(t) = 1+it$ vs compound interest $f(t) = (1+i)^t = e^{\delta t}$ (where $\delta = \ln(1+i)$ is the force of interest); decomposability and its characterization via Cauchy's functional equations;
• Interest rates: annual effective rate $i$; equivalent period rate $i_m = (1+i)^{1/m} - 1$ (compound) or $i_m = i/m$ (simple); nominal annual rate convertible $m$ times $j_m = m \cdot i_m$ (an under-approximation of the effective annual rate, applicable only to compound interest);
• Cash flows: investments (outflow then inflows), loans (inflow then outflows), annuities (all same sign, finite), perpetuities (infinite); present value (PV) and final value (FV) formulas;
• Investment appraisal: net present value (NPV) = discounted cash flow function $G(i) = \sum_{s=0}^n a_s (1+i)^{-s}$ evaluated at the opportunity cost $i$; internal rate of return (IRR) $x^*$ is the root of $G(x) = 0$; on an investment cash flow ($a_0<0$, $a_s>0$ for $s\ge 1$), $G$ is strictly decreasing so IRR is unique;
• Annuity formulas: ordinary immediate annuity PV $V_0 = R \cdot a_{\overline{n}|i} = R \cdot \frac{1-(1+i)^{-n}}{i}$, ordinary FV $V_n = R \cdot s_{\overline{n}|i} = R \cdot \frac{(1+i)^n - 1}{i}$; due annuities multiply by $(1+i)$; perpetuity PV $V_0 = R/i$;
• Bonds: zero-coupon $P_0 = N/(1+x^*)^T$; coupon bond $P_0 = \sum_{k=1}^n a_k (1+y)^{-t_k}$; Macaulay duration $D(y) = \frac{1}{P_0}\sum_k t_k \cdot a_k (1+y)^{-t_k}$; bond price volatility $V(y) = P'(y)/P(y) = -D(y)/(1+y)$ (modified duration); first-order price-change estimate $\Delta P \approx -\frac{D(y)}{1+y}\cdot \Delta y \cdot P$;
• Financial markets (L, p): portfolios $\mathbf{x}\in\mathbb{R}^n$, payoff operator $R(\mathbf{x}) = Y\cdot \mathbf{x}$, market value $V(\mathbf{x}) = \mathbf{p}\cdot \mathbf{x}$; law of one price (LOP); arbitrages of I and II kind; pricing rule $f : W\to\mathbb{R}$; Riesz–Markov representation $f(\mathbf{w}) = \boldsymbol{\pi}\cdot\mathbf{w}$; fundamental theorem of finance (no arbitrage $\iff$ strictly positive pricing kernel).

Typical MCQ pattern (from May 2024 General exam).

• MCQ6 (Mode A) / MCQ5 (Mode B): cash flow $(a_0, a_1, a_2, a_3)$ with $a_0<0$ and $a_s>0$ for $s=1,2,3$; discounted cash flow function $G$ satisfies $G(0.03)=5,\;G(0.06)=-3,\;G(0.09)=-7$. Which is true? — Since the cash flow is an investment, $G$ is strictly monotonically decreasing, therefore the unique IRR sits between 3% and 6% (sign change bracketing). The tempting wrong answer "IRR is between 6% and 9%" is ruled out because $G$ is monotonic. Moreover $\lim_{x\to +\infty}G(x) = a_0^+$ (the limit equals $a_0$ from above) so $G(0.09) = -7 > a_0$, i.e. $a_0 < -7$. Correct answer: B ($a_0<-7$).

Typical open-ended patterns (May 2024 Q11 Mode A / Q10 Mode B, and Q12 Mode A / Q11 Mode B).

• Q11 — LOP / no-arbitrage I / proof. (a) State the Law of One Price definition. (b) State the no-arbitrage-of-the-I-kind condition. (c) Prove that no-arbitrage of the I kind implies the LOP. The solution cites textbook definition 1137, section 24.6.6, and lemma 1143. The proof is the highest-yield 20-pt question of this topic — memorize the construction $\mathbf{h} = \mathbf{x} - \mathbf{x}'$ and both sign contradictions.
• Q12 — Bond pricing / duration / volatility. Cash flow 60 / 60 / 1060 at YTM $y=6\%$. (a) Write $P_0 = \frac{60}{1.06} + \frac{60}{1.06^2} + \frac{1060}{1.06^3}$. (b) Write $D(0.06) = \frac{1}{P_0}\!\left(1\cdot\frac{60}{1.06} + 2\cdot\frac{60}{1.06^2} + 3\cdot\frac{1060}{1.06^3}\right)$. (c) Quote $V(y) = -D(y)/(1+y)$ and estimate final price $P_0 - \frac{D(0.06)}{1.06}\cdot (+0.001)\cdot P_0$ — the positive $\Delta y = +0.1\%$ produces a negative $\Delta P$ (price falls when yield rises).

Why this topic is high-leverage for the final partial.

• Bond duration uses weighted averages — the same mechanism as expected value in Probability (§04), only with discounted cash flows as weights.
• The volatility proof relies on the first-order Taylor formula from Differential Calculus (§02).
• The definition of a financial market $(L,\mathbf{p})$ uses matrix operations and the image of a linear map from Linear Algebra (§01); completeness of a market is equivalent to $\operatorname{rank} Y = k$.
• The continuous-compounding relation $M(t) = 1 + \int_0^t f'(s)\,ds$ (Final exercises Q5) uses the First Fundamental Theorem of Calculus (§03, Theorem 1894).

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§2. Definitions

2.1 Capital, accumulation, and the accumulation factor $f(t)$

Let $P$ (or $C$) be the principal / capital invested at time $t=0$, and let $A$ (or $M$) be the accumulated amount (capital + interest) at a future time $t \ge 0$. The accumulation factor $f(t)$ is the value at time $t$ of one euro invested at time $0$: $\boxed{A = P \cdot f(t),\qquad I = A - P = P(f(t) - 1).}$ Conventions (Sicconi's lecture and Marinacci):

• $f(0) = 1$: the value of 1 euro after zero time is still 1.
• $f(t) \ge 1$ for $t \ge 0$ (no loss in a safe investment).
• $f$ is increasing (longer investments never yield less), and if differentiable then $f'(t) \ge 0$.

Capitalization axioms (the four properties of the accumulated-amount function $A(P,t)$):

1. $A = A(P, t)$ — $A$ depends on both principal and time;
2. $A(P_1 + P_2, t) = A(P_1, t) + A(P_2, t)$ — additive in the principal;
3. $A(P, t_1) \le A(P, t_2)$ whenever $t_1 < t_2$ — increasing in time (not necessarily strictly);
4. $A(P, 0) = P$.

2.2 Simple vs compound interest

Simple interest. $f(t) = 1 + it$, linear in $t$ with constant slope $i$ (the annual rate).

Compound interest. $f(t) = (1+i)^t = e^{\delta t}$, convex in $t$, where $\delta = \ln(1+i)$ is the force of interest (also called continuous interest rate or intensity). Compound interest is the unique decomposable continuous accumulation factor (see §3.7).

Comparison. For $t = 1$, simple and compound coincide at $1+i$. For $t < 1$ simple exceeds compound; for $t > 1$ compound exceeds simple. For an investor, compound is more convenient for $t > 1$ (interest earned on interest).

2.3 Discount factor $v(t)$

The discount factor is the reciprocal of the accumulation factor: $v(t) = \frac{1}{f(t)} = \begin{cases}\dfrac{1}{1+it}, & \text{simple discount,}\\[6pt](1+i)^{-t}, & \text{compound discount.}\end{cases}$ It represents the present value at time 0 of one euro received at time $t$. Properties: $v(0) = 1$, $v(t) \le 1$, $v$ is decreasing. $f$ and $v$ are conjugate: $f(t)\cdot v(t) = 1$.

2.4 Annual rate and equivalent rates

Annual interest rate $i$: the interest earned on 1 euro invested for 1 year, $i = f(1) - 1.$

Period rate $i_m$ for $m$ periods per year (so $t = 1/m$):

• simple: $i_m = i/m$;
• compound: $\boxed{i_m = (1+i)^{1/m} - 1}\iff (1+i) = (1+i_m)^m$.

Nominal annual rate convertible $m$ times (TAN — Tasso Annuo Nominale), only defined for compound: $j_m = m \cdot i_m.$ $j_m < i$ always (under-approximation). Never use $j_m$ directly in an accumulation factor — convert to $i_m$ first.

2.5 Decomposability

An accumulation factor $f$ is decomposable if $\forall\,t, h \ge 0,\qquad f(t) \cdot f(h) = f(t+h).$ Intuitively: accumulating for $t+h$ in one step gives the same result as accumulating for $t$, then accumulating the result for another $h$. Compound interest $f(t)=(1+i)^t$ is decomposable; simple interest $f(t)=1+it$ is not (because $(1+it)(1+ih) = 1+i(t+h)+i^2 th \ne 1+i(t+h)$).

2.6 Cash flow $(a_0, a_1, \ldots, a_n)$ — types

A cash flow is a sequence of signed payments $a_0, a_1, \ldots, a_n$ at times $t_0 = 0, t_1, \ldots, t_n$. Sign conventions: $a_s > 0$ means an inflow (money received), $a_s < 0$ means an outflow (money paid). Cash-flow types used in this course:

• Investment: $a_0 < 0$ and $a_s > 0$ for $s = 1,\ldots,n$ (pay up front, receive later).
• Loan: $a_0 > 0$ and $a_s < 0$ for $s = 1,\ldots,n$ (receive up front, repay later).
• Annuity: all $a_s$ have the same sign; finite number of payments.
• Perpetuity: all $a_s$ have the same sign; infinite number of payments.

2.7 Net present value NPV and discounted cash flow function $G$

Given an opportunity cost of capital (compound rate) $i$, the net present value of a cash flow is the sum of discounted payments: $\text{NPV}(i) = \sum_{s=0}^n \frac{a_s}{(1+i)^s} = \sum_{s=0}^n a_s (1+i)^{-s}.$

The discounted cash flow function $G : (-1, +\infty) \to \mathbb{R}$ leaves the rate variable: $\boxed{G(x) = \sum_{s=0}^n a_s (1+x)^{-s}.}$ $\text{NPV}(i) = G(i)$ is $G$ evaluated at the opportunity cost; the internal rate of return $x^*$ is a root of $G(x) = 0$.

Limits of $G$ (investment, $a_0<0,\;a_s>0$):

• $G(0) = \sum_{s=0}^n a_s$ (undiscounted sum);
• $\lim_{x\to -1^+} G(x) = +\infty$ (positive future payoffs blow up);
• $\lim_{x\to +\infty} G(x) = a_0^+$ (all future terms vanish, only $a_0$ remains; the limit equals $a_0$ from above).

2.8 Yield to maturity $y$ and internal rate of return (IRR, $x^*$)

Internal rate of return $x^*$ of a cash flow: the compound rate $x^* \in (-1, +\infty)$ such that $G(x^*) = 0,\qquad \text{i.e.}\quad \sum_{s=0}^n a_s (1+x^*)^{-s} = 0.$ For an investment cash flow, $G$ is strictly monotonically decreasing (see Theorem 3.5), so $x^*$ exists uniquely.

Yield to maturity $y$ of a bond: the IRR of the bond's cash flow viewed from the buyer's side, i.e. the rate $y$ for which the present value of all coupons and the face value equals the bond price $P_0$. Written "YTM" in the lecture.

2.9 NPV decision criterion

For an investment (outflow then inflows), the IRR $x^*$ is a benefit rate (higher is better). Compared to the opportunity cost $i$: $\text{NPV}(i) > 0 \iff x^* > i \iff \text{GOOD investment};$ $\text{NPV}(i) = 0 \iff x^* = i \iff \text{indifferent};$ $\text{NPV}(i) < 0 \iff x^* < i \iff \text{BAD investment}.$ For a loan, signs flip and the effective financing rate $j$ satisfies: NPV$>0 \iff j < i$ (good), NPV$<0 \iff j > i$ (bad).

2.10 Simple (ordinary / due) annuity

A simple annuity has constant installments $R$, constant time periods, and compound interest. Two variants:

| Variant | Payments at | Count | |---|---|---| | Ordinary / posticipated / immediate | $t_1, t_2, \ldots, t_n$ (end of period) | $k = 1, \ldots, n$ | | Due / anticipated | $t_0, t_1, \ldots, t_{n-1}$ (start of period) | $k = 0, \ldots, n-1$ |

Key PV / FV formulas (per-period compound rate $i$; for other periods use $i_m$): $\boxed{V_0 = R\cdot a_{\overline{n}|i} = R\cdot \frac{1 - (1+i)^{-n}}{i}},\qquad V_n = R\cdot s_{\overline{n}|i} = R\cdot \frac{(1+i)^n - 1}{i}.$ For due annuities multiply each value by $(1+i)$: $\ddot V_0 = R\cdot (1+i)\cdot \frac{1-(1+i)^{-n}}{i},\qquad \ddot V_n = R\cdot (1+i)\cdot \frac{(1+i)^n - 1}{i}.$

2.11 Perpetuity

Perpetual simple annuity ($n\to +\infty$) with compound rate $i > 0$: $V_0^{\text{ord}} = \lim_{n\to\infty} R\cdot \frac{1 - (1+i)^{-n}}{i} = \frac{R}{i},\qquad V_0^{\text{due}} = \frac{R}{i}\cdot (1+i).$ No final value exists for a perpetuity (never ends).

Growing perpetuity. With constant growth rate $g < i$, the PV is $V_0 = R/(i-g)$ (Gordon formula, mentioned in the lecture for reference).

2.12 Bond, coupon, face value, bond price $P(y)$

A fixed-income bond pays a stream of cash flows $a_1, a_2, \ldots, a_n$ at times $t_1 < t_2 < \ldots < t_n$ in exchange for a purchase price $P_0$ today. Typical structure:

• coupons $c_1, \ldots, c_{n-1}$ (usually constant $c_k = c$) at times $t_1, \ldots, t_{n-1}$;
• final payment $c_n + R$ at $t_n$, where $R$ is the reimbursement / redemption value (often equal to the face / nominal value $N$).

A zero-coupon bond (ZCB) has no coupons: $a_n = N$ and all intermediate $a_k = 0$.

The bond price at yield $y$ is the present value of all future inflows: $\boxed{P(y) = \sum_{k=1}^n a_k (1+y)^{-t_k}.}$ At $y = y_{\text{YTM}}$, $P(y)$ equals the market price $P_0$ by definition of YTM.

2.13 Macaulay duration $D(y)$ (Definition 59)

The duration of a fixed-income bond is the weighted average of the payment dates, using discounted cash flows as weights: $\boxed{D(y) = \frac{1}{P(y)}\sum_{k=1}^n t_k\cdot a_k (1+y)^{-t_k} = \frac{\sum_k t_k\cdot a_k (1+y)^{-t_k}}{\sum_k a_k (1+y)^{-t_k}}.}$

Properties:

• $D(y)$ is a time (measured in years, not money).
• $t_1 \le D(y) \le t_n$.
• For a ZCB, $D(y) = t_n = T$ (single payment date).
• Only defined for cash flows with one outflow (a typical bond purchase).
• Interpretation: $D(y)$ measures the sensitivity / risk of the bond to a change in $y$ (larger duration ⇒ more volatile).
• Financial-immunization interpretation (see §3.7): $z^* = D(i)$ is the date at which the reinvested-plus-remaining-value profile of the bond is minimized in $y$, hence the bond is immunized against small rate shifts at date $D(i)$.

2.14 Modified duration $D^*(y)$

$D^*(y) = \frac{D(y)}{1+y}.$ The modified duration appears as the sensitivity factor in the price-change formula (§3.8).

2.15 Convexity $C(y)$

Convexity is the second-order sensitivity of the bond price to yield changes: $C(y) = \frac{1}{P(y)}\cdot \frac{d^2 P}{dy^2}.$ It refines the linear first-order approximation to a quadratic: $\frac{\Delta P}{P} \approx -\frac{D(y)}{1+y}\cdot \Delta y + \tfrac{1}{2}C(y)\cdot (\Delta y)^2.$ Convexity is always positive for ordinary bonds and explains why the actual price change is slightly better than the linear approximation.

2.16 Bond price volatility $V(y)$

The (relative) volatility of the bond price is the relative change in $P$ per unit change in $y$: $V(y) \;=\; \lim_{\Delta y \to 0}\frac{\Delta P / P}{\Delta y} \;=\; \frac{P'(y)}{P(y)} \;=\; -\frac{D(y)}{1+y}.$ (See Theorem 3.8 for proof.) Note the negative sign: prices move opposite to yields.

First-order approximation of absolute price change: $\boxed{\Delta P \approx -\frac{D(y)}{1+y}\cdot \Delta y\cdot P.}$

2.17 Financial market $(L, \mathbf{p})$

A financial market $(L, \mathbf{p})$ consists of:

• A finite set $L = \{\mathbf{y}_1, \ldots, \mathbf{y}_n\}\subset \mathbb{R}^k$ of asset payoff vectors, each $\mathbf{y}_j \in \mathbb{R}^k$ listing the payoff in each of the $k$ possible future states of the world; collected as the columns of the payoff matrix $Y = [\mathbf{y}_1 | \cdots | \mathbf{y}_n] \in \mathbb{R}^{k\times n}$.
• A price vector $\mathbf{p} = (p_1, \ldots, p_n) \in \mathbb{R}^n_{>0}$ giving the time-$0$ prices of one unit of each asset.

A portfolio is a vector $\mathbf{x} \in \mathbb{R}^n$ of asset quantities ($x_j > 0$ means long, $x_j < 0$ means short).

The payoff operator $R : \mathbb{R}^n \to \mathbb{R}^k$ is linear: $R(\mathbf{x}) = Y\cdot \mathbf{x} = \sum_{j=1}^n x_j \mathbf{y}_j \in \mathbb{R}^k,$ listing the state-contingent total payoff of the portfolio.

The market value (price) of a portfolio is the linear function $V : \mathbb{R}^n \to \mathbb{R}$, $V(\mathbf{x}) = \mathbf{p}\cdot \mathbf{x} = \sum_{j=1}^n p_j x_j.$ $V(\mathbf{x}) > 0$ is the cost of the portfolio (you pay more than you receive); $V(\mathbf{x}) < 0$ is a benefit.

2.18 Replicable contingent claim, complete market

A vector $\mathbf{w}\in\mathbb{R}^k$ is a replicable contingent claim if there exists a portfolio $\mathbf{x} \in \mathbb{R}^n$ with $R(\mathbf{x}) = \mathbf{w}$. The set of replicable claims is $W = \operatorname{Im} R \subseteq \mathbb{R}^k$.

The market is complete if every claim is replicable, i.e. $W = \mathbb{R}^k$, equivalently $\operatorname{rank} Y = k$ (so $R$ is surjective).

2.19 Law of One Price (LOP) — Definition 1137

Def 1137. The financial market $(L, \mathbf{p})$ satisfies the Law of One Price if for every pair of portfolios $\mathbf{x}, \mathbf{x}' \in \mathbb{R}^n$, $R(\mathbf{x}) = R(\mathbf{x}') \;\Longrightarrow\; V(\mathbf{x}) = V(\mathbf{x}').$

Any two portfolios with the same state-contingent payoff have the same price.

LOP guarantees a well-defined pricing rule $f : W \to \mathbb{R}$ on the space of replicable claims: $f(\mathbf{w}) = V(\mathbf{x})$ for any $\mathbf{x}\in R^{-1}(\mathbf{w})$.

2.20 Arbitrage of the I kind — Section 24.6.6

Def (Section 24.6.6). A portfolio $\mathbf{x} \in \mathbb{R}^n$ is an arbitrage of the I kind (also called "first-kind arbitrage") if $R(\mathbf{x}) \ge \mathbf{0}\quad \text{and}\quad V(\mathbf{x}) < 0.$

You receive a non-negative state-contingent payoff, and you are paid (negative cost) to take the portfolio.

Equivalent phrasing: there exists a portfolio $\mathbf{x}$ such that $\mathbf{p}\cdot \mathbf{x} \le 0$ (or $<0$) yet $R(\mathbf{x})$ is a non-negative and non-zero payoff.

The market has no arbitrages of the I kind if $\forall\,\mathbf{x}\in\mathbb{R}^n,\quad R(\mathbf{x})\ge \mathbf{0}\;\Longrightarrow\; V(\mathbf{x}) \ge 0.$ (Equivalently: $R(\mathbf{x}) = \mathbf{0}\Rightarrow V(\mathbf{x}) = 0$, combined with $R(\mathbf{x})\le\mathbf{0}\Rightarrow V(\mathbf{x})\le 0$.)

2.21 Arbitrage of the II kind

Def. A portfolio $\mathbf{x}$ is an arbitrage of the II kind if $R(\mathbf{x}) > \mathbf{0}\quad \text{and}\quad V(\mathbf{x}) \le 0.$

You receive a strictly positive payoff (with at least one state yielding $>0$), and you pay no more than zero.

The market has no arbitrages of the II kind if $R(\mathbf{x}) > \mathbf{0} \Rightarrow V(\mathbf{x}) > 0$.

Sign conventions used below. For $\mathbf{v}\in\mathbb{R}^k$: $\mathbf{v} \ge \mathbf{0}$ means all components $\ge 0$ (may be all zero); $\mathbf{v} > \mathbf{0}$ means $\mathbf{v}\ge\mathbf{0}$ and at least one component is strictly positive; $\mathbf{v} \gg \mathbf{0}$ (strongly positive) means every component is strictly positive.

2.22 Pricing rule $f$ and pricing kernel $\boldsymbol{\pi}$

When LOP holds, the pricing rule $f : W \to \mathbb{R}$ is well-defined by $f(\mathbf{w}) = V(\mathbf{x})$ for any replicating $\mathbf{x}$. On a complete market with no arbitrages of the I kind, $f$ is linear and increasing, so by Riesz–Markov there exists a unique positive vector $\boldsymbol{\pi} \in \mathbb{R}^k$, $\boldsymbol{\pi} \ge \mathbf{0}$, with $f(\mathbf{w}) = \boldsymbol{\pi}\cdot \mathbf{w}\qquad \forall\,\mathbf{w}\in\mathbb{R}^k.$ $\boldsymbol{\pi}$ is the pricing kernel; component $\pi_s$ is the price of the state-$s$ Arrow claim $\mathbf{e}_s$ (payoff 1 in state $s$, 0 elsewhere). The relation to the price vector is $\mathbf{p} = \boldsymbol{\pi}\cdot Y$ (row vector times payoff matrix).

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§3. Theorems, Propositions & Proofs

3.1 Characterization of decomposable financial laws

Statement. Suppose $f(t)$ is a continuous accumulation factor with $t \ge 0$. Then $f$ is decomposable if and only if $f$ is the compound accumulation factor, i.e. $f(t) = (1+i)^t = e^{\delta t}\qquad\text{for some $i \ge 0$ (with $\delta = \ln(1+i)$).}$

Source: Finance Proofs.pdf p.35.

Proof.

($\Rightarrow$) Suppose $f$ is decomposable. Then by definition $f(t+h) = f(t)\cdot f(h)\qquad \forall\,t,h\ge 0.$ $f$ is continuous and strictly positive (an accumulation factor). By the Cauchy functional-equation theorem for multiplicative functions, every continuous positive solution of $f(x+y) = f(x)\cdot f(y)$ is of the form $f(t) = e^{mt}$ for some $m \in \mathbb{R}$. Setting $m = \delta = \ln(1+i)$, we obtain $f(t) = e^{\delta t} = (1+i)^t$.

($\Leftarrow$) Suppose $f(t) = (1+i)^t = e^{\delta t}$. Then $f(t)\cdot f(h) = (1+i)^t\cdot (1+i)^h = (1+i)^{t+h} = f(t+h),$ so $f$ is decomposable. $\blacksquare$

Remark. The supporting theorem (Cauchy's functional equation) says: if $f$ is continuous at at least one point $x_0 \in \mathbb{R}$ and satisfies $f(x+y) = f(x) + f(y)$, then $f(x) = ax$ for some $a$; if $f$ is strictly positive and continuous at at least one $x_0$ and satisfies $f(x+y) = f(x)\cdot f(y)$, then $f(x) = e^{mx}$ for some $m$.

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3.2 Formulae for ordinary simple annuities

Statement. For an ordinary (posticipated) annuity with constant installments $R$ and compound interest $i$: $V_0 = R\cdot \frac{1-(1+i)^{-n}}{i} = R\cdot a_{\overline{n}|i},\qquad V_n = R\cdot \frac{(1+i)^n - 1}{i} = R\cdot s_{\overline{n}|i}.$

Source: Finance Proofs.pdf p.42.

Proof of the present-value formula $V_0$.

The PV is the sum of the discounted payments at $t_1 = 1, t_2 = 2, \ldots, t_n = n$ (each payment of $R$): $V_0 = \frac{R}{(1+i)^1} + \frac{R}{(1+i)^2} + \cdots + \frac{R}{(1+i)^n} = R\sum_{s=1}^n (1+i)^{-s}.$ The inner sum is a geometric partial sum with first term $(1+i)^{-1}$ and common ratio $(1+i)^{-1}$: $\sum_{s=1}^n (1+i)^{-s} = (1+i)^{-1}\cdot \frac{1 - (1+i)^{-n}}{1 - (1+i)^{-1}} = \frac{1 - (1+i)^{-n}}{(1+i) - 1} = \frac{1 - (1+i)^{-n}}{i}.$ Hence $V_0 = R\cdot \dfrac{1 - (1+i)^{-n}}{i} = R\cdot a_{\overline{n}|i}$.

Proof of the final-value formula $V_n$. The FV is the sum of the accumulated payments: $V_n = R(1+i)^{n-1} + R(1+i)^{n-2} + \cdots + R = R\sum_{s=1}^n (1+i)^{n-s} = R(1+i)^n \sum_{s=1}^n (1+i)^{-s}.$ Using the result above, $V_n = R(1+i)^n\cdot \frac{1 - (1+i)^{-n}}{i} = R\cdot \frac{(1+i)^n - 1}{i} = R\cdot s_{\overline{n}|i}.\qquad \blacksquare$

Corollary (due annuity). For a due (anticipated) annuity, each installment is paid one period earlier, so both values multiply by $(1+i)$: $\ddot V_0 = (1+i)\cdot V_0,\qquad \ddot V_n = (1+i)\cdot V_n.$ (Source: Finance Proofs.pdf p.46, same geometric-sum trick with the extra factor $(1+i)$ factored out.)

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3.3 IRR uniqueness on an investment cash flow

Statement. Let $(a_0, a_1, \ldots, a_n)$ be an investment cash flow with $a_0 < 0$ and $a_s > 0$ for $s = 1, \ldots, n$. Then the discounted cash flow function $G(x) = \sum_{s=0}^n a_s (1+x)^{-s}$ is strictly monotonically decreasing on $(-1, +\infty)$, hence the internal rate of return $x^*$ (the unique root $G(x^*) = 0$, if it exists) is unique.

Source: Mathematical Finance.pdf and General_24524_ENG_SOL.pdf p.4 (Solution to MCQ6).

Proof. Differentiate $G$ term by term (each summand is a smooth function of $x \in (-1, +\infty)$): $G'(x) = \sum_{s=0}^n a_s\cdot (-s)(1+x)^{-s-1} = -\sum_{s=1}^n s\cdot a_s\cdot (1+x)^{-s-1}.$ (The $s=0$ term dies because of the factor $s$.) For an investment, $a_s > 0$ and $s\ge 1$ for each summand, and $(1+x)^{-s-1} > 0$ for $x > -1$. Hence every term in the sum is strictly positive and $G'(x) = -\sum_{s=1}^n\underbrace{s\cdot a_s\cdot (1+x)^{-s-1}}_{>0} \;<\; 0\qquad \forall x\in(-1,+\infty).$ So $G$ is strictly decreasing; a strictly monotonic continuous function has at most one zero, proving uniqueness of $x^*$. $\blacksquare$

Corollary (sign-bracketing of the IRR). If $G(i_1) > 0 > G(i_2)$ for some $i_1 < i_2$, then by the intermediate-value theorem plus strict monotonicity, $x^* \in (i_1, i_2)$.

3.4 Asymptotic behavior of $G$

Statement. For an investment cash flow, $\lim_{x\to +\infty}G(x) = a_0^+$: the limit equals $a_0$ and is approached from above.

Source: General_24524_ENG_SOL.pdf p.4 (MCQ6 solution cites this limit).

Proof. As $x\to +\infty$, $(1+x)^{-s} \to 0$ for every $s \ge 1$, so each of the $n$ positive terms $a_s (1+x)^{-s}$ with $s\ge 1$ tends to $0$. Meanwhile the $s=0$ term is $a_0 (1+x)^0 = a_0$ (constant). Hence $\lim_{x\to +\infty} G(x) = a_0 + 0 + \cdots + 0 = a_0.$ Moreover, since $G$ is strictly decreasing and tends to $a_0$ at infinity, $G(x) > a_0$ for all finite $x$; the notation $a_0^+$ records that the limit is reached from above. $\blacksquare$

Consequence for May 2024 MCQ6. $G(0.09) = -7 > a_0$ gives $a_0 < -7$.

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3.5 Bond pricing formula

Statement. The no-arbitrage price of a coupon bond with cash flow $(a_1, \ldots, a_n)$ at dates $(t_1, \ldots, t_n)$ and yield to maturity $y$ is $\boxed{P(y) = \sum_{k=1}^n a_k (1+y)^{-t_k}.}$

Source: Mathematical Finance.pdf p.26 (Fixed Income Bonds) and General_24524_ENG_SOL.pdf p.8 (May 2024 Q12a).

Justification. The yield to maturity is by definition the compound rate $y$ that equates the bond price to the PV of all future inflows. The formula just writes out that PV. Alternatively, no-arbitrage arguments (applied to a replicating portfolio of zero-coupon bonds) force the price to equal the sum of discounted cash flows.

Worked case. For the May 2024 Q12 bond 60/60/1060 at $y=6\%$: $P_0 = \frac{60}{1.06} + \frac{60}{1.06^2} + \frac{1060}{1.06^3}.$

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3.6 Macaulay Duration — Definition 59

Statement. For a bond with cash flow $(a_1, \ldots, a_n)$ at dates $(t_1, \ldots, t_n)$ and price $P(y)$, the Macaulay duration at yield $y$ is $D(y) = \frac{1}{P(y)}\sum_{k=1}^n t_k\cdot a_k\cdot (1+y)^{-t_k}.$

Source: Mathematical Finance.pdf pp.28–29 (Duration); General_24524_ENG_SOL.pdf p.8 cites "additional textbook on Elementary financial calculus: definition 59".

Interpretation. $D(y)$ is the weighted average of the payment dates, with weights $w_k = a_k (1+y)^{-t_k}/P(y)$ summing to 1. It is a time expressed in the same unit as the $t_k$ (usually years) and satisfies $t_1 \le D(y) \le t_n$.

Worked case. For the May 2024 Q12 bond 60/60/1060 at $y=6\%$: $D(0.06) = \frac{1}{P_0}\left(1\cdot\frac{60}{1.06} + 2\cdot\frac{60}{1.06^2} + 3\cdot\frac{1060}{1.06^3}\right).$

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3.7 Duration as the date of financial immunization

Statement. Let a fixed-income bond have cash flow $(a_1, \ldots, a_n)$ and YTM $i$ at $t=0$. Suppose the rate varies once from $i$ to $i + \Delta i$ at some date $z < t_1$ and then remains constant. Then the bond is financially immunized against the interest-rate risk at the date $z^* = \frac{\sum_{s=1}^n t_s\cdot a_s (1+i)^{-t_s}}{\sum_{s=1}^n a_s (1+i)^{-t_s}} = D(i).$

Source: Finance Proofs.pdf p.62 (Duration as a date of financial immunization).

Proof sketch. Consider the total value at time $z$, as a function of the after-shift rate $x$: $f(x) = V(z, x) = \sum_{k=1}^n a_k (1+x)^{z - t_k}.$

Step I: Find the stationary point in $x$. Differentiate in $x$: $f'(x) = \sum_{k=1}^n a_k (z - t_k)(1+x)^{z - t_k - 1} = (1+x)^{z-1}\left[z\sum_k a_k (1+x)^{-t_k} - \sum_k a_k t_k (1+x)^{-t_k}\right].$ Setting $f'(i) = 0$ (necessary condition for a local extremum at the status-quo rate $x=i$), and noting $(1+i)^{z-1} > 0$, gives $z^* = \frac{\sum_k a_k t_k (1+i)^{-t_k}}{\sum_k a_k (1+i)^{-t_k}} = D(i).$

Step II: Second-order test — $z^* = D(i)$ is a local minimizer. Compute $f''(x)$; at $x = i$ most terms cancel (because $f'(i) = 0$), leaving $f''(i) = \sum_{k=1}^n a_k (z^* - t_k)^2 (1+i)^{z^* - t_k - 2} > 0$ (each summand is a non-negative number, and at least one is strictly positive when $n > 1$). So $z^* = D(i)$ is a local minimum of $f$ in $x$; for small rate shifts, $f(i+\Delta i) \ge f(i)$, i.e. the bond's value at date $D(i)$ is no worse after the shift than before. This is the immunization property. $\blacksquare$

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3.8 Bond price volatility

Statement. Consider a fixed-income bond with price $P(y) = \sum_{s=1}^n a_s (1+y)^{-t_s}$ (viewed as a function of the yield $y$). Then: (1) $\displaystyle \frac{P'(y)}{P(y)} = -\frac{D(y)}{1+y}$, and (2) a linear approximation of the relative price change is $\frac{\Delta P(y)}{P(y)} \approx -\frac{D(y)}{1+y}\cdot \Delta y.$

Source: Finance Proofs.pdf p.64 (Bond Price Volatility); cited in General_24524_ENG_SOL.pdf p.8 as "Elementary financial calculus: section 7.2".

Proof of (1). $P(y) > 0$ (all $a_s, t_s > 0$) and $P$ is differentiable in $y$. Differentiating term by term: $P'(y) = \sum_{s=1}^n a_s \cdot (-t_s)(1+y)^{-t_s - 1} = -\frac{1}{1+y}\sum_{s=1}^n t_s\cdot a_s\cdot (1+y)^{-t_s}.$ Divide by $P(y)$: $\frac{P'(y)}{P(y)} = -\frac{1}{1+y}\cdot \frac{\sum_{s=1}^n t_s\cdot a_s\cdot (1+y)^{-t_s}}{\sum_{s=1}^n a_s\cdot (1+y)^{-t_s}} = -\frac{1}{1+y}\cdot D(y) = -\frac{D(y)}{1+y}.$

Proof of (2). Since $P$ is differentiable on $(-1, +\infty)$, the first-order Taylor formula (§02, Theorem 1481) gives $P(y + \Delta y) - P(y) = P'(y)\cdot \Delta y + o(\Delta y)\qquad \text{as }\Delta y \to 0.$ Divide both sides by $P(y)$: $\frac{\Delta P(y)}{P(y)} = \frac{P'(y)}{P(y)}\cdot \Delta y + \frac{o(\Delta y)}{P(y)}.$ The remainder $o(\Delta y)/P(y)$ is negligible as $\Delta y \to 0$, so $\frac{\Delta P(y)}{P(y)} \approx \frac{P'(y)}{P(y)}\cdot \Delta y = -\frac{D(y)}{1+y}\cdot \Delta y.\qquad \blacksquare$

Practical corollary. Solving for $\Delta P$: $\Delta P \approx -\frac{D(y)}{1+y}\cdot \Delta y\cdot P(y).$ Equivalently, the approximated new price is $P(y + \Delta y) \approx P(y) - \frac{D(y)}{1+y}\cdot \Delta y\cdot P(y).$

Sign discipline. $\Delta y > 0$ (yield rises) $\Rightarrow \Delta P < 0$ (price falls); $\Delta y < 0 \Rightarrow \Delta P > 0$. Memorize: prices and yields move in opposite directions.

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3.9 LOP ⇒ linearity of the pricing rule

Statement. If the financial market $(L, \mathbf{p})$ satisfies the Law of One Price, then the pricing rule $f : W \to \mathbb{R}$ (defined on the space of replicable claims $W = \operatorname{Im} R$) is linear.

Source: Finance Proofs.pdf p.1154 (Linearity of the pricing rule).

Proof. By the LOP, for every $\mathbf{x}\in\mathbb{R}^n$ the price $V(\mathbf{x}) = \mathbf{p}\cdot\mathbf{x}$ depends only on $R(\mathbf{x})$, so the function $f : W\to\mathbb{R}$ defined by $f(\mathbf{w}) = V(\mathbf{x})$ for any $\mathbf{x}$ with $R(\mathbf{x}) = \mathbf{w}$ is well-posed.

Take $\mathbf{w}, \mathbf{w}' \in W$ and scalars $\alpha, \beta \in \mathbb{R}$. Since $\mathbf{w}, \mathbf{w}'\in\operatorname{Im} R$, choose $\mathbf{x}, \mathbf{x}'\in\mathbb{R}^n$ with $R(\mathbf{x}) = \mathbf{w}$, $R(\mathbf{x}') = \mathbf{w}'$, so $V(\mathbf{x}) = f(\mathbf{w})$ and $V(\mathbf{x}') = f(\mathbf{w}')$.

Now compute $f(\alpha\mathbf{w} + \beta\mathbf{w}')$ using linearity of $R$ and $V$: $f(\alpha\mathbf{w} + \beta\mathbf{w}') = f\bigl(\alpha R(\mathbf{x}) + \beta R(\mathbf{x}')\bigr) = f\bigl(R(\alpha\mathbf{x} + \beta\mathbf{x}')\bigr) = V(\alpha\mathbf{x} + \beta\mathbf{x}') = \alpha V(\mathbf{x}) + \beta V(\mathbf{x}') = \alpha f(\mathbf{w}) + \beta f(\mathbf{w}').$ (The first equality uses $\mathbf{w} = R(\mathbf{x})$; the second uses linearity of $R$; the third uses the definition of $f$; the fourth uses linearity of $V = \mathbf{p}\cdot(\cdot)$.) $\blacksquare$

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3.10 Lemma 1143 — No arbitrage of the I kind ⇒ LOP

Lemma 1143. A financial market $(L, \mathbf{p})$ that has no arbitrages of the I kind satisfies the Law of One Price.

Source: Finance Proofs.pdf p.1157. Cited on General_24524_ENG_SOL.pdf p.7 as the proof required for May 2024 Q11c.

This is the flagship finance proof for the exam. Memorize the construction $\mathbf{h} = \mathbf{x} - \mathbf{x}'$ and both sign steps.

Proof. By hypothesis the market has no arbitrage of the I kind, so for every portfolio $\mathbf{z} \in \mathbb{R}^n$, R(\mathbf{z}) \ge \mathbf{0}\;\Longrightarrow\; V(\mathbf{z}) \ge 0.\tag{NAI} Applying (NAI) to $-\mathbf{z}$ (which has $R(-\mathbf{z}) = -R(\mathbf{z})$ and $V(-\mathbf{z}) = -V(\mathbf{z})$ by linearity) gives the dual: R(\mathbf{z}) \le \mathbf{0}\;\Longrightarrow\; V(\mathbf{z}) \le 0.\tag{NAI$'$} Combining both directions at $\mathbf{z} = \mathbf{0}$ (or for any $\mathbf{z}$ with $R(\mathbf{z}) = \mathbf{0}$): R(\mathbf{z}) = \mathbf{0}\;\Longrightarrow\; V(\mathbf{z}) = 0.\tag{$\star$}

Now take two portfolios $\mathbf{x}, \mathbf{x}' \in \mathbb{R}^n$ with the same payoff $R(\mathbf{x}) = R(\mathbf{x}')$. Set $\mathbf{h} = \mathbf{x} - \mathbf{x}'.$ By linearity of $R$, $R(\mathbf{h}) = R(\mathbf{x} - \mathbf{x}') = R(\mathbf{x}) - R(\mathbf{x}') = \mathbf{0}.$ Applying $(\star)$ with $\mathbf{z} = \mathbf{h}$: $V(\mathbf{h}) = 0.$ By linearity of $V$, $V(\mathbf{h}) = V(\mathbf{x} - \mathbf{x}') = V(\mathbf{x}) - V(\mathbf{x}') = 0,$ hence $V(\mathbf{x}) = V(\mathbf{x}')$. This is exactly the Law of One Price. $\blacksquare$

Alternative phrasing (the "two-sign contradiction" version). Suppose $\mathbf{x}, \mathbf{x}'$ have $R(\mathbf{x}) = R(\mathbf{x}')$; set $\mathbf{h} = \mathbf{x}-\mathbf{x}'$, so $R(\mathbf{h}) = \mathbf{0}$. If $V(\mathbf{h}) > 0$ then $-\mathbf{h}$ has $R(-\mathbf{h}) = \mathbf{0} \ge \mathbf{0}$ and $V(-\mathbf{h}) = -V(\mathbf{h}) < 0$, a first-kind arbitrage, contradicting the hypothesis. If $V(\mathbf{h}) < 0$ then $\mathbf{h}$ itself has $R(\mathbf{h}) = \mathbf{0} \ge \mathbf{0}$ and $V(\mathbf{h}) < 0$, again a first-kind arbitrage. So $V(\mathbf{h}) = 0$, giving LOP.

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3.11 Linear and increasing pricing rule (complete + no-arbitrage-I)

Statement. Let $(L, \mathbf{p})$ be a complete financial market with $\mathbf{p} \ne \mathbf{0}$. Then $(L,\mathbf{p})$ has no arbitrages of the I kind if and only if the pricing rule $f$ is linear and increasing, equivalently there exists a unique positive vector $\boldsymbol{\pi} \in \mathbb{R}^k$, $\boldsymbol{\pi} \ge \mathbf{0}$, with $f(\mathbf{w}) = \boldsymbol{\pi}\cdot \mathbf{w}\qquad \forall\,\mathbf{w}\in\mathbb{R}^k.$

Source: Finance Proofs.pdf p.1158 (Linear and increasing pricing rule).

Proof.

($\Leftarrow$) Suppose $f(\mathbf{w}) = \boldsymbol{\pi}\cdot\mathbf{w}$ with $\boldsymbol{\pi}\ge\mathbf{0}$. If $R(\mathbf{x})\ge\mathbf{0}$ then $V(\mathbf{x}) = f(R(\mathbf{x})) = \boldsymbol{\pi}\cdot R(\mathbf{x})\ge 0$ (sum of non-negatives), ruling out arbitrage of the I kind.

($\Rightarrow$) Suppose no arbitrages of the I kind. By Lemma 1143, LOP holds; by Theorem 3.9, $f$ is linear. By completeness, $W = \operatorname{Im} R = \mathbb{R}^k$, so $f$ is defined on all of $\mathbb{R}^k$. Take any $\mathbf{w} \ge \mathbf{0}$; since $W = \mathbb{R}^k$, there exists $\mathbf{x}'$ with $R(\mathbf{x}') = \mathbf{w}\ge\mathbf{0}$. By no-arbitrage-I, $V(\mathbf{x}')\ge 0$, hence $f(\mathbf{w}) = V(\mathbf{x}')\ge 0$. Thus $f$ is positive on non-negative inputs, i.e. increasing. By the Riesz–Markov theorem (every linear, increasing functional on $\mathbb{R}^k$ is represented by a unique non-negative vector), there exists a unique $\boldsymbol{\pi}\ge\mathbf{0}$ with $f(\mathbf{w}) = \boldsymbol{\pi}\cdot\mathbf{w}$ for all $\mathbf{w}\in\mathbb{R}^k$. $\blacksquare$

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3.12 Fundamental Theorem of Finance

Statement. Let $(L, \mathbf{p})$ be a complete financial market with $\mathbf{p}\ne\mathbf{0}$. Then $(L,\mathbf{p})$ has no arbitrages of the I or II kind if and only if the pricing rule is linear and strictly increasing, equivalently there exists a unique strongly positive vector $\boldsymbol{\pi}\in\mathbb{R}^k$, $\boldsymbol{\pi}\gg\mathbf{0}$, with $f(\mathbf{w}) = \boldsymbol{\pi}\cdot\mathbf{w}\qquad \forall\,\mathbf{w}\in\mathbb{R}^k.$

Source: Finance Proofs.pdf p.1159 (The Fundamental Theorem of Finance).

Proof sketch.

($\Leftarrow$) If $\boldsymbol{\pi}\gg\mathbf{0}$, then $R(\mathbf{x}) > \mathbf{0}$ ⇒ $V(\mathbf{x}) = \boldsymbol{\pi}\cdot R(\mathbf{x}) > 0$, so no arbitrage II. The no-arbitrage-I part follows from Theorem 3.11.

($\Rightarrow$) By Theorem 3.11 there exists a unique $\boldsymbol{\pi}\ge\mathbf{0}$ with $f(\mathbf{w}) = \boldsymbol{\pi}\cdot\mathbf{w}$. It remains to show $\boldsymbol{\pi}\gg\mathbf{0}$ (strongly positive). Take any $\mathbf{w} > \mathbf{0}$; by completeness there is $\mathbf{x}$ with $R(\mathbf{x}) = \mathbf{w} > \mathbf{0}$. No-arbitrage-II gives $V(\mathbf{x}) > 0$, so $\boldsymbol{\pi}\cdot\mathbf{w} = f(\mathbf{w}) = V(\mathbf{x}) > 0$. This holds for every $\mathbf{w} > \mathbf{0}$, which forces every component of $\boldsymbol{\pi}$ to be strictly positive (take $\mathbf{w} = \mathbf{e}_s$ for each $s$). $\blacksquare$

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§4. Worked Examples

4.1 Bond price at YTM 6% (May 2024 Q12a)

Cash flow: $a_1 = 60$ (year 1), $a_2 = 60$ (year 2), $a_3 = 1060$ (year 3). YTM $y = 6\%$.

Apply the bond pricing formula (Theorem 3.5): $P_0 = \frac{60}{1.06} + \frac{60}{1.06^2} + \frac{1060}{1.06^3}.$

Numerically: $\frac{60}{1.06} \approx 56.60$; $\frac{60}{1.06^2} \approx 53.40$; $\frac{1060}{1.06^3} \approx 889.99$. Sum $\approx 999.99 \approx 1000$. (The bond trades at par: coupons of $60/1000 = 6\%$ match the YTM of 6%.)

4.2 Duration of the same bond at $y = 6\%$ (May 2024 Q12b)

By Definition 59 / Theorem 3.6: $D(0.06) = \frac{1}{P_0}\left(1\cdot \frac{60}{1.06} + 2\cdot \frac{60}{1.06^2} + 3\cdot \frac{1060}{1.06^3}\right).$

Numerator $\approx 1\cdot 56.60 + 2\cdot 53.40 + 3\cdot 889.99 \approx 56.60 + 106.80 + 2669.97 \approx 2833.37$; divide by $P_0 \approx 1000$: $D(0.06) \approx 2.833$ years. Sanity: $t_1 = 1 \le 2.833 \le 3 = t_n$. Duration is close to 3 because the final payment $(60 + 1000)$ dominates the weighted average.

4.3 First-order bond-price estimate after a yield shift +0.1% (May 2024 Q12c)

At $y = 6\%$, the yield rises to $y + \Delta y = 6.1\%$, so $\Delta y = +0.001$. By Theorem 3.8: $P_{\text{new}} \approx P_0 - \frac{D(0.06)}{1.06}\cdot (+0.001)\cdot P_0.$

The sign in the exam-solution box is -0.001 under the solution template because the final price formula lifts the minus sign out: $P_0 - \frac{D}{1+y}\cdot\Delta y \cdot P_0$ with $\Delta y = +0.001$ yields a negative correction (price falls). Numerically: correction $\approx -(2.833/1.06)\cdot 0.001\cdot 1000 \approx -2.67$; so $P_{\text{new}} \approx 1000 - 2.67 \approx 997.33$.

4.4 IRR range and $a_0$ bound from sample $G$-values (May 2024 MCQ6)

Given: investment cash flow, $G(0.03) = 5$, $G(0.06) = -3$, $G(0.09) = -7$.

IRR localization. By Theorem 3.3, $G$ is strictly decreasing. Since $G(0.03) = 5 > 0 > -3 = G(0.06)$, the unique root lies between them: $x^* \in (0.03, 0.06)$, i.e. between 3% and 6%. This rules out answer A ("IRR smaller than 3%") and C ("IRR between 6% and 9%").

Bound on $a_0$. By Theorem 3.4, $\lim_{x\to +\infty}G(x) = a_0^+$. So $G(x) > a_0$ for all finite $x$, in particular $G(0.09) = -7 > a_0$, giving $a_0 < -7$. Answer B is correct.

4.5 Simple annuity PV (worked example)

A saver will receive $R = 100$ at the end of each year for $n = 10$ years at compound rate $i = 5\%$.

$V_0 = 100\cdot \frac{1 - (1.05)^{-10}}{0.05} = 100\cdot \frac{1 - 0.6139}{0.05} = 100\cdot \frac{0.3861}{0.05} = 100\cdot 7.7217 \approx 772.17.$

FV at $t=10$: $V_{10} = V_0\cdot 1.05^{10} \approx 772.17\cdot 1.6289 \approx 1257.78,\qquad \text{or directly } V_{10} = 100\cdot\frac{1.05^{10}-1}{0.05}\approx 1257.78.$

4.6 NPV decision

A project has cash flow $(a_0, a_1, a_2) = (-100, 50, 70)$; opportunity cost $i = 10\%$. $G(10\%) = -100 + \frac{50}{1.1} + \frac{70}{1.1^2} = -100 + 45.45 + 57.85 = +3.31 > 0.$ Since NPV is positive, the IRR exceeds 10%, so the project is a good investment. To test whether $x^* > 13\%$, compute $G(13\%) = -100 + \frac{50}{1.13} + \frac{70}{1.13^2} = -100 + 44.25 + 54.82 = -0.93 < 0,$ so $x^* < 13\%$. Combining with Theorem 3.3, $x^* \in (10\%, 13\%)$.

4.7 Continuous compounding via integral (Final exercises solution, Q5)

Problem. Capital $C = 1$ is available at $t = 0$. Let $f(t)$ denote its value at future time $t \ge 0$, and suppose $f'(t) = (1.02)^t \ln(1.02)$ (the instantaneous interest rate). Using integrals, determine the accumulated amount $M(t)$ at time $t$.

Solution. The accumulated amount equals capital plus total interests: $M(t) = C + I(t) = 1 + \int_0^t f'(s)\,ds.$

For the given $f'$, compute the integral using the primitive $\int 1.02^s\ln(1.02)\,ds = 1.02^s + C$ (standard antiderivative of $a^x\ln a$): $M(t) = 1 + \int_0^t (1.02)^s\ln(1.02)\,ds = 1 + \bigl[(1.02)^s\bigr]_0^t = 1 + (1.02)^t - 1 = (1.02)^t.$

So $M(t) = (1.02)^t$: the capital grows at compound interest with effective annual rate $i = 2\%$. For $t = 2$: $M(2) = (1.02)^2 = 1.0404$. For $C = 100$: $M(2) = 100\cdot (1.02)^2 = 104.04$.

Connection to §03. This example is literally an application of the First Fundamental Theorem of Calculus (Theorem 1894): given $f'$, the FTC lets us recover $f(t) - f(0) = \int_0^t f'(s)\,ds$, which is the total interest.

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§5. Solution Methods (How to Solve)

M-MF-1 — Determine the IRR range from sample $G$-values

Input. An investment cash flow and a finite table of discounted cash flow values $G(i_1), G(i_2), \ldots$ at sample rates $i_1 < i_2 < \cdots$. Optionally also a limit question on $a_0$.

Steps.

1. Check that the cash flow is an investment ($a_0 < 0$, $a_s > 0$ for $s \ge 1$); if so, $G$ is strictly decreasing (Theorem 3.3) and the IRR $x^*$ is unique.
2. Locate the sign change in the given sample values: if $G(i_k) > 0 > G(i_{k+1})$, then $x^* \in (i_k, i_{k+1})$.
3. For questions about $a_0$: use $\lim_{x\to+\infty}G(x) = a_0^+$ (Theorem 3.4). If $G(i_\ell) = c$ for the largest sample rate $i_\ell$, then $a_0 < c$.
4. Cross-check each MCQ answer against the established range.

Output. Exact interval containing $x^*$; bound on $a_0$ if asked.

Pitfalls. Never forget that $G$ is decreasing for an investment (not increasing). Without the investment hypothesis, $G$ may not be monotonic and multiple IRRs can exist. In May 2024 MCQ6 the tempting wrong answer C ("IRR between 6% and 9%") is ruled out by monotonicity alone.

M-MF-2 — Price a bond at a given yield

Input. Cash flow $(a_1, \ldots, a_n)$ at dates $(t_1, \ldots, t_n)$; YTM $y$.

Steps.

1. Write out the bond pricing formula (Theorem 3.5): $P(y) = \sum_{k=1}^n a_k (1+y)^{-t_k}.$
2. Substitute numerics term by term. If coupons are constant, you may rewrite the coupon part as a simple annuity: $\sum_{k=1}^{n-1} c\cdot (1+y)^{-t_k} = c\cdot a_{\overline{n-1}|y}$ plus the final principal+coupon discounted at $t_n$.
3. (If asked to verify YTM given $P_0$.) Plug $y = y_{\text{YTM}}$ into the formula and check $P(y) = P_0$.

Output. $P_0 = P(y)$.

Pitfalls. Watch for semi-annual vs annual coupons: if coupons are paid every 6 months and $y$ is an annual rate, convert to $y_2 = (1+y)^{1/2}-1$ first, or equivalently discount with exponents in semesters. The TAN/nominal-annual rate $j_m$ must be converted to the period rate $i_m = j_m/m$ before use.

M-MF-3 — Compute Macaulay duration

Input. Cash flow $(a_1, \ldots, a_n)$, dates $(t_1, \ldots, t_n)$, yield $y$, (optionally) price $P(y)$.

Steps.

1. If $P(y)$ is not given, compute it first (M-MF-2).
2. Form the numerator of duration: $N(y) = \sum_{k=1}^n t_k\cdot a_k\cdot (1+y)^{-t_k}.$
3. Divide by $P(y)$: $D(y) = \frac{N(y)}{P(y)}.$

Output. Macaulay duration in years.

Pitfalls. Dates $t_k$ must match the time unit of the rate (both in years, typically). For a ZCB, $D = T$ trivially; don't bother with the sum. Sanity check: $t_1 \le D(y) \le t_n$.

M-MF-4 — Estimate new bond price after a yield shift

Input. $P_0$ at original yield $y$; duration $D(y)$; yield shift $\Delta y$ (signed).

Steps.

1. Compute modified duration $D^*(y) = D(y)/(1+y)$.
2. Apply the first-order formula (Theorem 3.8): $\Delta P \approx -D^*(y)\cdot \Delta y\cdot P_0.$
3. Report $P_{\text{new}} \approx P_0 + \Delta P$.

Output. Estimated $P_{\text{new}}$ (and signed $\Delta P$).

Pitfalls. The minus sign is non-negotiable: $\Delta y > 0$ ⇒ $\Delta P < 0$. A common error is forgetting the $(1+y)$ divisor (i.e. using Macaulay duration instead of modified duration). Another trap: the formula is linear in $\Delta y$ — for large shifts, use the second-order convexity correction or compute $P(y+\Delta y)$ directly.

M-MF-5 — Prove LOP from no-arbitrage-I (exam Q11c pattern)

Input. A financial market $(L, \mathbf{p})$ hypothesized to have no arbitrages of the I kind. You are asked to prove LOP.

Steps. (Memorize verbatim — this is the May 2024 Q11 proof, worth up to 20 points.)

1. State the no-arbitrage-I hypothesis: $R(\mathbf{z})\ge\mathbf{0}\Rightarrow V(\mathbf{z})\ge 0$.
2. Apply to $-\mathbf{z}$ by linearity: $R(\mathbf{z})\le\mathbf{0}\Rightarrow V(\mathbf{z})\le 0$.
3. Combine: $R(\mathbf{z}) = \mathbf{0}\Rightarrow V(\mathbf{z}) = 0$.
4. Take two portfolios $\mathbf{x}, \mathbf{x}'$ with $R(\mathbf{x}) = R(\mathbf{x}')$; define $\mathbf{h} = \mathbf{x} - \mathbf{x}'$.
5. By linearity of $R$: $R(\mathbf{h}) = R(\mathbf{x}) - R(\mathbf{x}') = \mathbf{0}$.
6. By step 3: $V(\mathbf{h}) = 0$.
7. By linearity of $V$: $V(\mathbf{h}) = V(\mathbf{x}) - V(\mathbf{x}') = 0$, so $V(\mathbf{x}) = V(\mathbf{x}')$. QED.

Output. A proof that the market satisfies the LOP (Definition 1137).

Pitfalls. Don't confuse arbitrage of the I kind (payoff non-negative, cost negative) with arbitrage of the II kind (payoff strictly positive, cost non-positive). LOP is weaker than no-arbitrage-I (the lemma says no-arb-I ⇒ LOP, not the converse). Also, note the construction uses the difference $\mathbf{x}-\mathbf{x}'$, not a specific portfolio from the hypothesis.

M-MF-6 — NPV of a project

Input. Cash flow $(a_0, a_1, \ldots, a_n)$; opportunity cost $i$.

Steps.

1. Compute $\text{NPV}(i) = \sum_{s=0}^n a_s (1+i)^{-s}$.
2. Accept if NPV $\ge 0$ (investment: $x^* \ge i$); reject if NPV $< 0$.
3. (Optional) Bracket $x^*$ by evaluating $G$ at additional rates.

Output. Accept/reject decision, plus optionally the IRR range.

Pitfalls. For loans, signs flip: a loan with NPV $> 0$ is good (low-cost financing, $j < i$); NPV $< 0$ is a bad loan. If in doubt, always recompute from $G(i)$ directly with signs preserved.

M-MF-7 — Continuous vs discrete compounding conversion

Input. A rate in one convention (force of interest $\delta$, or effective annual $i$, or nominal $j_m$).

Steps.

• $\delta \to i$: $i = e^\delta - 1$.
• $i \to \delta$: $\delta = \ln(1+i)$.
• $i \to i_m$ (compound): $i_m = (1+i)^{1/m} - 1$.
• $j_m \to i_m$: $i_m = j_m/m$, then convert to $i$ before use. $j_m$ cannot be plugged directly into an accumulation factor.
• $(1+i)^t = e^{\delta t}$: use either form; they agree because $\delta = \ln(1+i)$.

Output. Rate in the convention needed for the next calculation.

Pitfalls. Always check that the time unit of the rate matches the time unit of $t$ in $f(t)$. Common error: using an annual rate with $t$ in months — convert $t$ to years, or convert $i$ to the monthly rate $i_{12}$.

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§6. Practice Problems with Solutions

Problem 6.1 (May 2024 MCQ6 Mode A / MCQ5 Mode B) — IRR bracketing

Statement. Consider the financial operation described by the cash flow at years $0, 1, 2, 3$ with values $a_0, a_1, a_2, a_3$, where $a_0 < 0$ and $a_s > 0$ for every $s = 1, 2, 3$. Let $G$ be the discounted cash flow function and assume $G(0.03) = 5$, $G(0.06) = -3$, $G(0.09) = -7$. Which of the following is true?

• (A) the internal rate of return is smaller than 3%;
• (B) $a_0 < -7$;
• (C) the internal rate of return is between 6% and 9%;
• (D) none of the others.

Solution. The cash flow is an investment, so by Theorem 3.3 $G$ is strictly decreasing; by Theorem 3.4 $\lim_{x\to +\infty}G(x) = a_0^+$.

Bracketing: $G(0.03) = +5 > 0 > -3 = G(0.06)$, so $x^* \in (0.03, 0.06)$, ruling out (A) and (C).

$a_0$ bound: $G(0.09) = -7$ and $G \to a_0^+$ from above at $+\infty$, so $a_0 < -7$. Correct answer: (B). (Consistent with General_24524_ENG_SOL.pdf p.4.)

Problem 6.2 (May 2024 Q11 Mode A / Q10 Mode B) — LOP / no-arb-I / proof

Statement. Consider a financial market $(L, \mathbf{p})$. (a) What does it mean that $(L, \mathbf{p})$ satisfies the law of one price (LOP)? (b) What does it mean that $(L, \mathbf{p})$ has no arbitrages of the I kind? (c) Prove that if $(L, \mathbf{p})$ has no arbitrages of the I kind, then it satisfies the LOP.

Solution.

(a) The market $(L,\mathbf{p})$ satisfies the Law of One Price (Def 1137) if for every pair of portfolios $\mathbf{x}, \mathbf{x}'\in\mathbb{R}^n$ with the same state-contingent payoff, the market values agree: $R(\mathbf{x}) = R(\mathbf{x}')\;\Longrightarrow\; V(\mathbf{x}) = V(\mathbf{x}').$ Equivalently: the pricing rule $f : W \to \mathbb{R}$ on the space of replicable claims $W = \operatorname{Im} R$ is well-defined as a single-valued function.

(b) The market $(L,\mathbf{p})$ has no arbitrages of the I kind (Section 24.6.6) if no portfolio $\mathbf{x}\in\mathbb{R}^n$ satisfies $R(\mathbf{x})\ge\mathbf{0}$ and $V(\mathbf{x}) < 0$. Equivalently: $\forall\,\mathbf{x}\in\mathbb{R}^n,\qquad R(\mathbf{x})\ge \mathbf{0}\;\Longrightarrow\; V(\mathbf{x})\ge 0.$

(c) Proof (Lemma 1143 / Theorem 3.10).

From the no-arbitrage-I condition (applied to $\mathbf{z}$ and to $-\mathbf{z}$ separately), we deduce $R(\mathbf{z})\ge\mathbf{0}\Rightarrow V(\mathbf{z})\ge 0\qquad \text{and}\qquad R(\mathbf{z})\le\mathbf{0}\Rightarrow V(\mathbf{z})\le 0.$ Combining both: $R(\mathbf{z}) = \mathbf{0}\;\Longrightarrow\; V(\mathbf{z}) = 0.\qquad(\star)$

Let $\mathbf{x}, \mathbf{x}'\in\mathbb{R}^n$ be two portfolios with $R(\mathbf{x}) = R(\mathbf{x}')$. Define $\mathbf{h} = \mathbf{x} - \mathbf{x}'$. By linearity of $R$, $R(\mathbf{h}) = R(\mathbf{x}) - R(\mathbf{x}') = \mathbf{0}.$ Applying $(\star)$ to $\mathbf{h}$: $V(\mathbf{h}) = 0$. By linearity of $V$, $V(\mathbf{x}) - V(\mathbf{x}') = V(\mathbf{x} - \mathbf{x}') = V(\mathbf{h}) = 0,$ so $V(\mathbf{x}) = V(\mathbf{x}')$. This is the Law of One Price. $\blacksquare$

(Consistent with General_24524_ENG_SOL.pdf p.7: "See the textbook, definition 1137, section 24.6.6 and lemma 1143.")

Problem 6.3 (May 2024 Q12 Mode A / Q11 Mode B) — Bond 60/60/1060 at 6%

Statement. Consider the bond with coupons paying 60 at year 1, 60 at year 2, and 1060 at year 3 (the last including the face value $N = 1000$). (a) Write the price formula at YTM 6%. (b) State the definition of duration for a generic cash flow and write the duration formula for this bond. (c) Provide the definition of bond price volatility; state the result relating volatility to duration; write a first-order estimate of the new price if the YTM increases by 0.1% during the first year.

Solution.

(a) By the bond pricing formula (Theorem 3.5), $\boxed{P_0 = \frac{60}{1.06} + \frac{60}{1.06^2} + \frac{1060}{1.06^3}.}$ (Computation not required; numerically $P_0 \approx 1000$.)

(b) Definition 59 (Macaulay duration): for a cash flow $(a_1, \ldots, a_n)$ at dates $(t_1, \ldots, t_n)$ with price $P(y)$ at yield $y$, $D(y) = \frac{1}{P(y)}\sum_{k=1}^n t_k\cdot a_k\cdot (1+y)^{-t_k}.$ For the given bond at $y = 0.06$: $\boxed{D(0.06) = \frac{1}{P_0}\left(1\cdot \frac{60}{1.06} + 2\cdot \frac{60}{1.06^2} + 3\cdot \frac{1060}{1.06^3}\right).}$

(c) Definition of bond price volatility: the relative change in price per unit change in yield, $V(y) = P'(y)/P(y)$. Result (Theorem 3.8, Section 7.2): $V(y) = \frac{P'(y)}{P(y)} = -\frac{D(y)}{1+y}.$ For a yield shift $\Delta y = +0.001$ (positive: yield rises by 0.1%), the first-order estimate of the final price is $\boxed{P_{\text{new}} \approx P_0 - \frac{D(0.06)}{1.06}\cdot (0.001)\cdot P_0.}$ Because $\Delta y > 0$, the correction is negative: the price falls (by approximately 2.67, giving $P_{\text{new}}\approx 997.33$).

(Consistent with General_24524_ENG_SOL.pdf p.8. The minus sign inside final price ≈ P0 − D/(1.06)·(−0.001)·P0 shown in the textbook solution is a rewriting; the signed $\Delta y$ is $+0.001$ and the resulting $\Delta P$ is negative, which is the correct physical sign.)

Problem 6.4 (Final exercises solution, Q5) — Continuous compounding via integral

Statement. Consider a capital $C = 1$ available today ($t = 0$). Let $f(t)$ denote its value at future time $t \ge 0$. Since $f'(t)$ indicates the variation in capital (interest) instant by instant, use integrals to determine the accumulation formula $M(t)$. With $f'(t) = (1.02)^t\ln(1.02)$, find $M(2)$ for $C = 1$ and $C = 100$.

Solution. The total interest accumulated in $[0, t]$ is $I(t) = \int_0^t f'(s)\,ds,$ so the accumulated amount is $M(t) = C + I(t) = C\left(1 + \int_0^t f'(s)\,ds\right)\qquad \text{(in this problem $C = 1$).}$

With $f'(s) = (1.02)^s\ln(1.02)$, the antiderivative is $(1.02)^s$ (since $\frac{d}{ds}(1.02)^s = (1.02)^s\ln(1.02)$). By the First Fundamental Theorem of Calculus (§03, Theorem 1894): $M(t) = 1 + \bigl[(1.02)^s\bigr]_0^t = 1 + (1.02)^t - 1 = (1.02)^t.$

For $t = 2$ and $C = 1$: $M(2) = (1.02)^2 = 1.0404$. For $t = 2$ and $C = 100$: $M(2) = 100\cdot 1.02^2 = 104.04$.

(Consistent with the handwritten solution on Final exercises solution.pdf for question 5.)

Problem 6.5 (lecture / TA) — Three bonds, smallest volatility

Statement. Three bonds have cash flows: A = $(−P_A, 50, 70)$ at years 0, 1, 2; B = $(−P_B, 70)$ at years 0, 1; C = $(−P_C, 60)$ at years 0, 2. Assuming the same yield $i$ for all, which has the smallest sensitivity (volatility) to changes in $i$?

Solution. By Theorem 3.8, the price volatility is $-D(y)/(1+y)$; smaller $D$ ⇒ smaller sensitivity. Computing durations:

• Bond B is a zero-coupon bond with $T = 1$: $D_B = 1$.
• Bond C is a ZCB with $T = 2$: $D_C = 2$.
• Bond A has $D_A$ a weighted average of $\{1, 2\}$; since the larger cash flow is at $t=2$, $D_A$ is closer to 2 but strictly less: $1 < D_A < 2$.

Ordering: $D_B = 1 < D_A < 2 = D_C$. Bond B has the smallest duration and is the least volatile / risky.

Problem 6.6 — Simple ordinary annuity

Statement. You will receive 3 constant semi-annual installments of 100 euros, starting in 6 months. The effective annual rate is $i = 10\%$. (a) Find the present value. (b) Find the value of the annuity at $t = 14$ months.

Solution.

(a) Convert $i$ to the semi-annual rate: $i_2 = (1.1)^{1/2} - 1 \approx 0.0488$. Three ordinary semi-annual installments of $R = 100$: $V_0 = 100\cdot \frac{1 - (1 + 0.0488)^{-3}}{0.0488} \approx 100\cdot 2.7294 \approx 272.94.$

(b) Intermediate value by decomposability (compound interest): $V_{14/12} = V_0\cdot (1 + i)^{14/12} = 272.94\cdot (1.1)^{14/12} \approx 272.94\cdot 1.1175 \approx 305.04.$ Equivalently using $i_2$ and time measured in semesters ($14/12$ year = $14/6$ semesters): $V_{14/12} = 272.94\cdot (1.0488)^{14/6} \approx 305.04$. ✓

Problem 6.7 — NPV bracketing an IRR via three rates

Statement. An investment has $G(8\%) = -100{,}000 + 30{,}000/1.08 + R/1.08^2 + 45{,}000/1.08^3 = 0$ (given IRR = 8%). Find $R$.

Solution. Rearrange for $R$: $\frac{R}{1.08^2} = 100{,}000 - \frac{30{,}000}{1.08} - \frac{45{,}000}{1.08^3}.$ $R = 1.08^2\cdot \Bigl(100{,}000 - \frac{30{,}000}{1.08} - \frac{45{,}000}{1.08^3}\Bigr).$ Equivalently (cleaner): use the condition $P_0 =$ PV of installments at $x^* = 8\%$, i.e. $100{,}000 = 30{,}000/1.08 + R/1.08^2 + 45{,}000/1.08^3$, giving $R = 1.08^2\cdot 100{,}000 - 30{,}000\cdot 1.08 - 45{,}000/1.08 \approx 42{,}573.33.$

Problem 6.8 — NPV-sign-only information is insufficient

Statement. If $\text{NPV}(5\%) = 1350$, the IRR $x^*$ surely is: (A) $x^* > 5\%$; (B) $x^* < 5\%$; (C) $x^* > 0$; (D) $x^* = 5\%$; (E) $x^* < 0$; (F) none of the preceding.

Solution. The sign of NPV alone does not determine whether the cash flow is an investment or a loan. For an investment ($a_0<0$, $a_s>0$), NPV $>0$ ⇒ $x^*>5\%$ (answer A would be correct). For a loan, NPV $>0$ ⇒ $x^*<5\%$ (B). Without knowing which, answer F ("none of the preceding") is correct: we cannot say. Furthermore, the cash flow may not even be an investment or loan, in which case $G$ is not monotonic and multiple IRRs or no IRR may exist.

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§7. Common Pitfalls

1. Confusing LOP with no-arbitrage. The Law of One Price is strictly weaker: it asks only that payoff-equivalent portfolios have equal price. No-arbitrage-I additionally rules out "free lunch with non-negative payoff at negative cost". Lemma 1143 goes one way only: no-arb-I ⇒ LOP. The converse is false (an LOP market may still admit arbitrage).

2. Sign on volatility is NEGATIVE. $V(y) = -D(y)/(1+y)$, so $\Delta P \approx -\frac{D(y)}{1+y}\cdot \Delta y\cdot P$. A positive yield shift $\Delta y > 0$ gives a negative price change $\Delta P < 0$. The textbook solution of May 2024 Q12c writes P0 - D/1.06 · (-0.001) · P0, which is not a typo: it comes from absorbing a minus sign, but the student must check that for $\Delta y = +0.001$ the overall correction is negative. Safest: use $\Delta P = -\frac{D}{1+y}\cdot \Delta y\cdot P$ with $\Delta y$ signed.

3. Modified vs Macaulay duration. Macaulay $D(y)$ is the weighted average time (in years); modified $D^*(y) = D(y)/(1+y)$ is the derivative $|P'/P|$. The volatility formula uses $D^* = D/(1+y)$, not $D$ directly. Forgetting the $(1+y)$ divisor is a classic error.

4. Nominal annual rate $j_m$ misuse. $j_m$ is defined only for compound interest and must be converted to the effective period rate $i_m = j_m/m$ before being plugged into an accumulation factor. Never write $(1+j_m)^t$ or $(1+j_m)^{1/m}$.

5. Simple vs compound for intermediate values. For compound interest, $f(t)\cdot f(h) = f(t+h)$, so you can accumulate to an intermediate date via the shortcut $V_{\text{intermediate}} = V_0\cdot (1+i)^z$. Simple interest is not decomposable, so this shortcut fails; for simple interest, always recompute from the cash flows directly.

6. Investment vs loan NPV decision. For an investment (outflow first), NPV $>0$ ⇔ $x^* > i$ ⇔ good. For a loan (inflow first), NPV $>0$ ⇔ $j < i$ ⇔ good (cheap financing). The sign of NPV alone is ambiguous without knowing the cash-flow type — see Problem 6.8.

7. IRR uniqueness requires an investment (or a loan). The monotonicity of $G$ (Theorem 3.3) relies on all $a_s$ for $s\ge 1$ having the same sign. For mixed cash flows (alternating inflows and outflows), $G$ may have multiple roots or none — don't blindly claim a unique IRR.

8. Rounding errors in $(1+y)^k$. For bond exercises with annual compounding, carry at least 4–5 decimals through: a 0.1% rounding on each term can move $P_0$ by several euros in a multi-year bond.

9. Careless signs on $\Delta y$. "The YTM undergoes a positive variation of 0.1%" means $\Delta y = +0.001$. "A rate increase to 12% from 11.75%" means $\Delta y = +0.0025$. Always attach a sign to $\Delta y$ before plugging into $\Delta P = -\frac{D}{1+y}\cdot\Delta y\cdot P$.

10. Arbitrage I vs II. Arbitrage of the I kind: $R(\mathbf{x})\ge\mathbf{0}$ and $V(\mathbf{x})<0$ (non-negative payoff, strictly negative cost). Arbitrage of the II kind: $R(\mathbf{x})>\mathbf{0}$ and $V(\mathbf{x})\le\mathbf{0}$ (strictly positive payoff with at least one state $>0$, non-positive cost). The distinction is in which inequality is strict: the payoff is strict for II; the cost is strict for I.

11. Pricing kernel $\boldsymbol{\pi}$ sign. No-arbitrage-I ⇒ $\boldsymbol{\pi}\ge\mathbf{0}$; no-arbitrage-I-and-II ⇒ $\boldsymbol{\pi}\gg\mathbf{0}$ (strongly positive, every component strictly positive). Riesz–Markov gives uniqueness only when the pricing rule is defined on all of $\mathbb{R}^k$ (complete market). On an incomplete market, the pricing kernel need not be unique.

12. Check completeness before applying the Fundamental Theorem. The statements in Theorems 3.11 and 3.12 explicitly require the market to be complete ($\operatorname{rank} Y = k$). For incomplete markets, the Riesz–Markov representation may fail. Always verify completeness via $\operatorname{rank} Y$.

13. Duration of a ZCB is $T$, not some weighted average. If the bond has a single payment at $t=T$, $D(y) = T$ with no computation needed — the "weighted average" has a single weight of 1.

14. Financial immunization is a local/second-order result. The proof of Theorem 3.7 shows $z^* = D(i)$ is a local minimizer of $V(z, \cdot)$ — so for small rate shifts near $i$, the bond is immunized at that date. For large shifts, convexity / non-linear effects dominate and immunization may fail.

15. $G(0) = \sum_s a_s$, not 0. Many students confuse "IRR is 0" (which means $G(0) = 0$) with "NPV is computed at 0%" (which means $G(0) = \sum a_s$, the undiscounted total). At $i=0$ there is no discounting: each euro is worth the same today as in the future.

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End of 05 — Mathematical Finance. Cross-references: Theorem 1894 (§03 Integral Calculus) for the continuous-compounding integral; Theorem 1481 (§02 Differential Calculus) for the first-order Taylor expansion underlying the volatility proof; §01 Linear Algebra for the payoff matrix $Y$, its image / rank, and the completeness characterization.