Level 4 — ApplicationEquity & Fixed Income

Equity & Fixed Income

⏱ 60 minutes50 marksprintable — key stays hidden on paper

Level: 4 (Application — novel problems, no hints) Time Limit: 60 minutes Total Marks: 50

Question 1. (10 marks)

A corporate bond has a face value of $\$ 1{,}000 $, an annual coupon rate of$ 8% $paid annually, and$ 3 $years remaining to maturity. The market's required yield (YTM) for bonds of this risk is currently$ 10%$.

(a) Compute the fair price of the bond today. (5 marks)

(b) The company is downgraded from A to BBB, and the required yield jumps to $12\%$ . Recompute the price and state, with a one-sentence justification grounded in the price–yield relationship, why the price moved in the direction it did. (3 marks)

(c) An investor argues "a downgrade only matters if the company actually defaults, so the price shouldn't change." Refute this claim in 2–3 sentences using the concept of default risk premium. (2 marks)

Question 2. (12 marks)

A zero-coupon government bond with face value $\$ 1{,}000 $matures in$ 5 $years and is currently priced at$ $783.53$.

(a) Determine the annually-compounded yield-to-maturity of this bond. (4 marks)

(b) Two years later, with $3$ years remaining, the prevailing yield for equivalent bonds is $6\%$ . Compute the new price and the total percentage gain an investor who bought at $\$ 783.53$ would realise if they sold now. (5 marks)

(c) Explain why zero-coupon bonds have no reinvestment risk whereas a coupon-paying bond of the same maturity does. (3 marks)

Question 3. (12 marks)

You are handed the following yields on newly-issued government bonds:

Maturity	Yield
1 year	5.2%
2 years	4.8%
5 years	4.1%
10 years	3.6%

(a) Name the shape of this yield curve and sketch/describe it. (2 marks)

(b) Give one plausible macroeconomic interpretation that market participants might draw from this shape. (3 marks)

(c) A colleague holds two bonds of equal price: Bond X (10-year, low coupon) and Bond Y (2-year, high coupon). The central bank unexpectedly signals a series of rate hikes. Which bond loses more value and why? Reference the concept of duration explicitly. (4 marks)

(d) Define modified duration in words and state what a modified duration of $7.5$ predicts for a $0.5\%$ (50 bps) rise in yields. (3 marks)

Question 4. (10 marks)

A company issues a convertible bond: face value $\$ 1{,}000 $, convertible into$ 40 $shares of the company at the holder's option. The current share price is$ $22$.

(a) Compute the conversion value and state whether it is currently rational to convert if the bond trades at $\$ 950$. (3 marks)

(b) The share price later rises to $\$ 28$. Compute the new conversion value and explain why the convertible bond's market price will now tend to track the share price more closely than a plain bond would. (4 marks)

(c) A convertible bond typically carries a lower coupon than an otherwise identical straight bond from the same issuer. Explain why an investor accepts this lower coupon. (3 marks)

Question 5. (6 marks)

Classify each of the following statements as TRUE or FALSE and justify each in one sentence.

(a) A debenture is always secured against specific physical assets of the company. (2 marks)

(b) All else equal, a bond with a higher credit rating (e.g. AAA vs BB) will offer a lower yield. (2 marks)

Answer keyMark scheme & solutions

Question 1 (10 marks)

(a) Price = PV of coupons + PV of face value. Coupon = $0.08 \times 1000 = \$ 80$.

$P = \frac{80}{1.10} + \frac{80}{1.10^2} + \frac{1080}{1.10^3}$ $= 72.727 + 66.116 + 811.420 = \boxed{\$950.26}$

Correct coupon $\$ 80$ and setup — 2 marks
Correct discounting of each cash flow — 2 marks
Final answer $\approx\$ 950.26$ — 1 mark

(b) At YTM = 12%: $P = \frac{80}{1.12} + \frac{80}{1.12^2} + \frac{1080}{1.12^3} = 71.43 + 63.78 + 768.72 = \boxed{\$903.93}$

Correct recomputation — 2 marks
Justification: Price and yield are inversely related — future cash flows are discounted at a higher rate, lowering present value. — 1 mark

(c) Refutation: Even without actual default, the market prices in a higher probability/severity of default via a larger risk premium. Investors demand higher yield to compensate, and since yield rose the price falls today regardless of whether default ever occurs. — 2 marks (1 for premium concept, 1 for "price adjusts now").

Question 2 (12 marks)

(a) $783.53 = \dfrac{1000}{(1+y)^5}$

$(1+y)^5 = \frac{1000}{783.53} = 1.27627 \Rightarrow 1+y = 1.27627^{1/5} = 1.05000$ $y = \boxed{5.0\%}$

Correct zero-coupon pricing equation — 2 marks
Correct root/solution $y=5\%$ — 2 marks

(b) New price with 3 years left at 6%: $P = \frac{1000}{1.06^3} = \frac{1000}{1.191016} = \boxed{\$839.62}$

Gain = $\dfrac{839.62 - 783.53}{783.53} = \dfrac{56.09}{783.53} = \boxed{7.16\%}$

New price correct — 2 marks
Percentage gain formula & value — 3 marks

(c) A zero-coupon bond pays no interim coupons, so there are no intermediate cash flows to reinvest — the holder's return is locked in at purchase (assuming hold to maturity). A coupon bond forces the investor to reinvest each coupon at unknown future rates, creating reinvestment risk. — 3 marks (2 for "no interim cash flow," 1 for coupon-bond contrast).

Question 3 (12 marks)

(a) Yields fall as maturity rises → inverted (downward-sloping) yield curve. — 2 marks

(b) Any one valid interpretation — 3 marks:

Markets expect future interest-rate cuts / falling rates.
Anticipation of economic slowdown or recession.
Flight to safety pushing down long-term yields.

(c) Bond X (10-year, low coupon) loses more. Longer maturity and lower coupon both increase duration; higher duration means greater price sensitivity to a given change in yield. When rates rise, the higher-duration bond's price falls more. — 4 marks (2 for correct bond, 2 for duration reasoning).

(d) Modified duration measures the approximate percentage change in a bond's price for a 1% (100 bps) change in yield. — 2 marks. For MD $=7.5$ and $+0.5\%$ : price falls by $7.5 \times 0.5\% = \boxed{3.75\%}$ . — 1 mark.

Question 4 (10 marks)

(a) Conversion value $= 40 \times 22 = \boxed{\$ 880} $. Bond trades at$ $950 > $880$, so converting now would give less value than selling the bond — not rational to convert. — 3 marks (2 value, 1 decision).

(b) New conversion value $= 40 \times 28 = \boxed{\$ 1120} $. Since conversion value ($ $1120$) now exceeds face/bond value, the option is "in the money"; the bond behaves like equity because its worth is driven by the underlying shares, so its price tracks the stock. — 4 marks (2 value, 2 explanation).

(c) The investor accepts a lower coupon because the conversion option has value — the potential upside of converting to appreciating equity compensates for the reduced interest income. — 3 marks.

Question 5 (6 marks)

(a) FALSE — a debenture (esp. in many markets) is typically unsecured, backed only by the issuer's creditworthiness, not specific physical assets. — 2 marks.

(b) TRUE — higher rating implies lower default risk, so investors accept a lower yield. — 2 marks.

(c) FALSE — Bondholders rank ahead of equity holders; equity holders have the residual (last) claim in liquidation. — 2 marks.

[
  {"claim":"Q1a bond price at 10% YTM approx 950.26","code":"P=80/1.10+80/1.10**2+1080/1.10**3; result = abs(float(P)-950.26)<0.05"},
  {"claim":"Q1b bond price at 12% YTM approx 903.93","code":"P=80/1.12+80/1.12**2+1080/1.12**3; result = abs(float(P)-903.93)<0.05"},
  {"claim":"Q2a zero-coupon YTM is 5%","code":"y=(1000/783.53)**(Rational(1,5))-1; result = abs(float(y)-0.05)<0.001"},
  {"claim":"Q2b new price 839.62 and gain approx 7.16%","code":"P=1000/1.06**3; g=(P-783.53)/783.53; result = abs(float(P)-839.62)<0.05 and abs(float(g)-0.0716)<0.001"},
  {"claim":"Q4 conversion values 880 and 1120","code":"result = (40*22==880) and (40*28==1120)"}
]