Equity & Fixed Income

Question 1

(a) (4 marks) $P(y) = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n}$ For this bond: $P(y) = \dfrac{80}{1+y} + \dfrac{80}{(1+y)^2} + \dfrac{1080}{(1+y)^3}$. (2 marks) YTM is the value $y=y^\*$ that solves $P(y^\*) = P_{market}$; equivalently it is the root of $g(y)=P(y)-P_{market}=0$ — the single internal rate of return equating discounted cash flows to price. (2 marks)

(b) (6 marks) Let $v = 1+y$. Each term is $c_t v^{-t}$ with $c_t>0$.

• $P'(y) = \sum_t c_t \cdot (-t) v^{-t-1} = -\sum_t t\,c_t v^{-t-1}$. Since $c_t>0$, $t>0$, $v>0$ (as $y>-1$), every term is negative $\Rightarrow P'(y)<0$. Strictly decreasing. (3 marks)
• $P''(y) = \sum_t c_t \cdot t(t+1) v^{-t-2} > 0$ (all factors positive) $\Rightarrow$ strictly convex. (3 marks) This gives the inverse price–yield relation and positive convexity.

(c) (6 marks) Newton: $y_1 = y_0 - \dfrac{P(y_0)-950.25}{P'(y_0)}$. At $y_0=0.08$, $v=1.08$: $P(0.08)=80/1.08+80/1.08^2+1080/1.08^3 = 74.074+68.587+857.338=1000.00$. (2 marks) $P'(0.08) = -\left[\frac{80}{1.08^2}+\frac{2\cdot80}{1.08^3}+\frac{3\cdot1080}{1.08^4}\right] = -[68.587+126.943+2381.49] = -2577.02$. (2 marks) $y_1 = 0.08 - \dfrac{1000.00-950.25}{-2577.02} = 0.08 - \dfrac{49.75}{-2577.02} = 0.08 + 0.01931 = 0.09931$. (2 marks) (Converges toward $y^\*\approx0.10$.)

(d) (4 marks)

def ytm(price, face, coupon, n, tol=1e-8):
    def P(y):
        return sum(coupon/(1+y)**t for t in range(1, n+1)) + face/(1+y)**n
    lo, hi = -0.9, 1.0          # bracket
    for _ in range(200):
        mid = (lo+hi)/2
        if (P(mid)-price)*(P(lo)-price) <= 0:
            hi = mid
        else:
            lo = mid
    return (lo+hi)/2

Bracketing: $P$ is continuous and strictly decreasing (Q1b), $P\to\infty$ as $y\to-1^+$ and $P\to0$ as $y\to\infty$; so for any positive price there is exactly one root. $[-0.9,1.0]$ brackets it since $P(-0.9)$ is huge $>950.25$ and $P(1.0)$ is small $<950.25$. (2 marks method + 2 marks uniqueness/existence)

(e) (4 marks) At $y^\*=0.10$, $v=1.1$: PVs $= 72.727,\,66.116,\,811.42$; sum $=950.26$ ✓. Macaulay $D = \dfrac{\sum_t t\,PV_t}{P} = \dfrac{1(72.727)+2(66.116)+3(811.42)}{950.26} = \dfrac{72.727+132.231+2434.26}{950.26}=\dfrac{2639.22}{950.26}=2.7774$ yrs. (2) $D_{mod} = D/(1+y) = 2.7774/1.1 = 2.5249$. (1) $\Delta P \approx -D_{mod}\cdot P \cdot \Delta y = -2.5249\times950.26\times0.005 = -12.00$. Price falls ≈ $12.00. (1)

Question 2

(a) (5 marks) $P = 1000/(1.06)^3 = 1000/1.191016 = 839.62$. (3) Proof duration = maturity: A zero has a single cash flow at $t=n$, so $D=\frac{n\cdot PV_n}{PV_n}=n$. (2)

(b) (5 marks) Spot rates rising with maturity ⇒ upward-sloping (normal) yield curve (2), interpreted as expectations of higher future short rates / positive term (liquidity) premium / economic expansion. (1) Forward rate: $(1+z_3)^3=(1+z_2)^2(1+f_{2,3})$: $f_{2,3}=\dfrac{(1.06)^3}{(1.055)^2}-1 = \dfrac{1.191016}{1.113025}-1 = 1.070071-1 = 0.070071 \approx 7.01\%$. (2)

(c) (5 marks) $P = \dfrac{60}{1.05}+\dfrac{60}{1.055^2}+\dfrac{1060}{1.06^3}$ $=57.143+53.909+889.99 = 1001.04$. (each term 1½ mk, total 5; accept 1000.99–1001.10)

(d) (5 marks) A BBB corporate carries default (credit) risk — positive probability of missed coupons/principal — whereas the government bond is (near) risk-free. Investors demand extra compensation, so they pay less (lower price) for the same promised cash flows, raising the yield. (3) The gap is the credit spread: $\text{spread}=y_{corp}-y_{govt}$; wider for lower ratings. (2)

Question 3

(a) (4 marks) Conversion ratio = number of shares received per bond = 20. (1) Conversion value = ratio × share price. (1) $=20\times46=920$. (2)

(b) (6 marks) No-arbitrage:

• If $V_{conv} <$ straight value: hold-to-maturity cash flows dominate a cheaper investment ⇒ buy convertible, its debt cash flows alone exceed cost ⇒ arbitrage. So $V_{conv}\ge$ straight value. (3)
• If $V_{conv} <$ conversion value: buy bond, immediately convert to 20 shares worth conversion value, sell shares for riskless profit $=$ conversion value $- V_{conv}>0$. So $V_{conv}\ge$ conversion value. (2)
• Hence $V_{conv}\ge\max(\text{straight},\text{conversion})$; the optionality adds value so typically strictly greater. (1)

(c) (3 marks) Set $20 S^\* = 940 \Rightarrow S^\* = 47.00$. (2) Above $47 conversion dominates the straight value (equity-like); below, the bond floor dominates (debt-like). (1)

(d) (3 marks) A debenture is an unsecured bond backed only by the issuer's creditworthiness/general assets, not by specific collateral; a secured bond is backed by pledged assets. (2) Convertibles pay lower coupons because the embedded conversion option has value to the holder, who accepts less current income in exchange for potential equity upside. (1)

[
  {"claim":"Q1 bond price at y=0.08 equals 1000", "code":"P=80/1.08+80/1.08**2+1080/1.08**3; result = abs(P-1000)<1e-2"},
  {"claim":"Q1 Newton one step gives y1 approx 0.09931", "code":"P0=1000; Pp=-(80/1.08**2+2*80/1.08**3+3*1080/1.08**4); y1=0.08-(P0-950.25)/Pp; result = abs(y1-0.09931)<1e-4"},
  {"claim":"Q1 Macaulay duration at y=0.10 approx 2.7774", "code":"y=0.10; pv=[80/(1+y),80/(1+y)**2,1080/(1+y)**3]; P=sum(pv); D=(1*pv[0]+2*pv[1]+3*pv[2])/P; result = abs(D-2.7774)<1e-3"},
  {"claim":"Q1 duration price change approx -12.00", "code":"y=0.10; pv=[80/(1+y),80/(1+y)**2,1080/(1+y)**3]; P=sum(pv); D=(1*pv[0]+2*pv[1]+3*pv[2])/P; Dmod=D/1.1; dP=-Dmod*P*0.005; result = abs(dP+12.0)<0.1"},
  {"claim":"Q2 forward rate f23 approx 0.070071", "code":"f=(1.06**3)/(1.055**2)-1; result = abs(f-0.070071)<1e-4"},
  {"claim":"Q2c coupon bond spot-priced approx 1001.04", "code":"P=60/1.05+60/1.055**2+1060/1.06**3; result = abs(P-1001.04)<0.2"},
  {"claim":"Q3 conversion value 920 and breakeven price 47", "code":"cv=20*46; Sstar=940/20; result = (cv==920) and (Sstar==47.0)"}
]