Equity & Fixed Income

Question 1 (12)

(a) Bond price = PV of all future cash flows discounted at yield $y$:

$P = \sum_{t=1}^{n}\frac{C}{(1+y)^t} + \frac{F}{(1+y)^n}$

• Each coupon $C$ paid at time $t$ discounted by $(1+y)^t$ (money later is worth less). (2)
• Final term is redemption of face value $F$ at maturity $n$. (1)
• Sum = fair present value / market price. (1)

(b) $C = 0.08\times1000 = 80$, $y=0.10$, $n=3$.

• $t=1$: $80/1.10 = 72.7273$
• $t=2$: $80/1.21 = 66.1157$
• $t=3$: $80/1.331 = 60.1052$
• Face: $1000/1.331 = 751.3148$

$P = 72.7273+66.1157+60.1052+751.3148 = \mathbf{950.26}$ (5) (1 per CF, 1 for total)

(c) Trades at a discount ($950.26 < 1000$). Because coupon rate ($8\%$) < yield ($10\%$): the bond pays less than the market demands, so its price must fall below par to compensate the buyer. (3)

Question 2 (10)

(a) YTM is the single discount rate that equates the present value of all a bond's future cash flows to its current market price. It is the IRR because it is exactly the rate $y$ that makes NPV(price − PV of cashflows) = 0 — the return earned if held to maturity with coupons reinvested at $y$. (3)

(b) $y=0.06$:

• $80/1.06 = 75.4717$
• $80/1.1236 = 71.1997$
• $80/1.191016 = 67.1695$
• $1000/1.191016 = 839.6193$

$P = 75.4717+71.1997+67.1695+839.6193 = \mathbf{1053.46}$ (4)

(c) As yield fell from $10\%$ to $6\%$, price rose from $950.26$ to $1053.46$ — inverse relationship: lower discount rate → higher PV of fixed cash flows. The curve is convex (curved, not linear): price rises more for a yield drop than it falls for an equal yield rise. (3)

Question 3 (10)

(a) Set $C=0$ in $P=\sum C/(1+y)^t + F/(1+y)^n$: all coupon terms vanish, leaving

$P = \frac{F}{(1+y)^n}$ (3)

(b) $680 = 1000/(1+y)^5$

$(1+y)^5 = 1000/680 = 1.470588$ $1+y = 1.470588^{1/5} = 1.08024$ $y = \mathbf{0.0802}$ (≈8.02%) (4)

(c) A zero has a single cash flow at maturity, so its full value depends on $(1+y)^n$ with the longest possible exponent — no intermediate coupons return capital early. Its duration equals its maturity (the maximum), giving the greatest price sensitivity to yield changes. (3)

Question 4 (12)

(a) $D_{Mac} = \frac{\sum_{t} t\cdot \frac{CF_t}{(1+y)^t}}{P}$ It is the PV-weighted average time to receive the bond's cash flows (in years). (3)

(b) Coupon $=60$, $y=6\%$, par price $P=1000$.

$D_{Mac} = 1943.3963/1000 = \mathbf{1.9434}$ years (6)

(c) $D_{mod} = D_{Mac}/(1+y) = 1.9434/1.06 = 1.8334$. $\%\Delta P \approx -D_{mod}\times\Delta y = -1.8334\times0.005 = -0.00917 = \mathbf{-0.92\%}$. (3)

Question 5 (8)

def bond_price(face, coupon_rate, ytm, years, freq=1):
    n = years * freq          # total periods
    c = face * coupon_rate / freq   # coupon per period
    r = ytm / freq            # periodic yield
    price = 0.0
    for t in range(1, n + 1):
        price += c / (1 + r) ** t          # discounted coupon
    price += face / (1 + r) ** n           # discounted face value
    return price
# zero-coupon: coupon_rate=0 -> loop adds 0, returns face/(1+r)**n  ✓
 
# Q1 call:
bond_price(1000, 0.08, 0.10, 3)   # -> 950.26

Marks: loop/discounting (3), face value term (2), freq handling (2), correct Q1 call (1). (8)

Question 6 (8)

(a) An inverted yield curve means short-term yields exceed long-term yields. It often signals expected economic slowdown or recession, as markets anticipate future rate cuts. (2)

(b) A debenture is an unsecured bond backed only by the issuer's creditworthiness (no specific collateral); a secured bond is backed by pledged assets and ranks higher in default. A convertible bond is typically a debenture that also carries an option to convert into equity shares. (3)

(c) After a downgrade AAA→BBB, perceived default risk rises, so investors demand a higher yield; since yield and price move inversely, the bond's price falls and its yield rises. (3)

[
  {"claim":"Q1 bond price at 10% yield is 950.26","code":"C=80; F=1000; y=Rational(1,10); P=sum(C/(1+y)**t for t in range(1,4))+F/(1+y)**3; result = round(float(P),2)==950.26"},
  {"claim":"Q2 bond price at 6% yield is 1053.46","code":"C=80; F=1000; y=Rational(6,100); P=sum(C/(1+y)**t for t in range(1,4))+F/(1+y)**3; result = round(float(P),2)==1053.46"},
  {"claim":"Q3 zero-coupon YTM approx 0.0802","code":"y=(1000/680)**(Rational(1,5))-1; result = round(float(y),4)==0.0802"},
  {"claim":"Q4 Macaulay duration is 1.9434 years","code":"y=Rational(6,100); pv1=60/(1+y); pv2=1060/(1+y)**2; P=pv1+pv2; D=(1*pv1+2*pv2)/P; result = round(float(D),4)==1.9434"},
  {"claim":"Q4 percent price change for +50bp is about -0.92%","code":"y=Rational(6,100); pv1=60/(1+y); pv2=1060/(1+y)**2; P=pv1+pv2; Dmac=(1*pv1+2*pv2)/P; Dmod=Dmac/(1+y); chg=-Dmod*Rational(5,1000); result = round(float(chg)*100,2)==-0.92"}
]