Introduction to Basic Fixed Income Securities :Introduction to Financial Engineering and Risk Management (Introduction to Financial Engineering and Risk Management) Answers 2025

Question 1 – Lottery payments (Present Value)

• Payment = $0.5 million per year

• Number of payments = 20

• Interest rate = 10% annually

• First payment = immediate → annuity due

Formula (Annuity Due):

PV=PMT×1−(1+r)−nr×(1+r)PV = PMT \times \frac{1 – (1+r)^{-n}}{r} \times (1+r)PV=PMT×r1−(1+r)−n​×(1+r) PV=0.5×1−(1.1)−200.10×1.1PV = 0.5 \times \frac{1 – (1.1)^{-20}}{0.10} \times 1.1PV=0.5×0.101−(1.1)−20​×1.1 $$PV\approx 0.5\times 8.5136\times 1.1=4.68$$

✅ Answer: 4.68 million

Question 2 – Sunk Costs

• Old apartment rent = $1000/month

• New apartment rent = $900/month

• Duration = 6 months

• Deposits are fully refundable → sunk cost ignored

Monthly savings:

$$1000-900=100$$

Total savings over 6 months:

$$100\times 6=600$$

Interest rate does not change the decision.

✅ Answer: Switch

Question 3 – Discount rate $d(0,2)$

Spot rate for 2 years:

s2=6.9%=0.069s_2 = 6.9\% = 0.069s2​=6.9%=0.069

Discount factor:

d(0,2)=1(1+s2)2=1(1.069)2d(0,2) = \frac{1}{(1+s_2)^2} = \frac{1}{(1.069)^2}d(0,2)=(1+s2​)21​=(1.069)21​ $$d(0,2)\approx 0.875$$

✅ Answer: 0.875

Question 4 – Forward rate f1,2f_{1,2}f1,2​

(1+s2)2=(1+s1)(1+f1,2)(1+s_2)^2 = (1+s_1)(1+f_{1,2})(1+s2​)2=(1+s1​)(1+f1,2​) (1.069)2=(1.063)(1+f)(1.069)^2 = (1.063)(1+f)(1.069)2=(1.063)(1+f) 1+f=1.14281.063≈1.07471+f = \frac{1.1428}{1.063} \approx 1.07471+f=1.0631.1428​≈1.0747 $$f\approx 7.47\%$$

✅ Answer: 7.5%

Question 5 – Forward contract on stock

• Stock price = $400

• Interest rate = 8% compounded quarterly

• Time = 9 months = 0.75 years

• Quarters = 3

F=S0(1+r/4)3=400(1.02)3F = S_0 (1 + r/4)^3 = 400(1.02)^3F=S0​(1+r/4)3=400(1.02)3 $$F\approx 400\times 1.0612=424.49$$

✅ Answer: 424.49

Question 6 – Bounds on perpetuity

• Payment = $10,000

• Lending rate = 8%

• Borrowing rate = 10%

Upper bound:

100000.08=125000\frac{10000}{0.08} = 1250000.0810000​=125000

Lower bound:

100000.10=100000\frac{10000}{0.10} = 1000000.1010000​=100000

Difference:

$$125000-100000=25000$$

✅ Answer: 25000

Question 7 – Value of forward contract (intermediate time)

• Original maturity = 1 year

• Time passed = 6 months → remaining = 0.5 years

• Semiannual compounding → 1 period

• $r=10\%\Rightarrow r/2=5\%$

Forward price at initiation:

F0=100(1.05)2=110.25F_0 = 100(1.05)^2 = 110.25F0​=100(1.05)2=110.25

Value today:

V=St−F01.05V = S_t – \frac{F_0}{1.05}V=St​−1.05F0​​ $$V=125-105=20$$

✅ Answer: 20

🧾 Summary Table (Final & Correct)