Comprehensive Finance Problem Set with Solutions: NPV, IRR, Bond Pricing, Present Value Assessment

Brief of Assessment Requirements 

This Problem Set 2 tests core time-value-of-money and valuation techniques across a wide range of topics: perpetuities (including non-annual payment patterns), rate conversions (annual ↔ periodic), annuities (ordinary and due), present value of cash-flow streams, bond pricing (with missed coupons), mortgage amortisation, dividend discount models (including declining and growing perpetuities), IRR vs NPV logic and comparisons, and sensitivity/interpretation questions.
Key formal requirements called out in the assignment:

• Correct identification of the cash-flow timing for each question (very important — e.g., payments starting today vs. starting next period).

• Use the appropriate formula for each instrument (perpetuity, annuity-immediate, annuity-due, growing perpetuity, bond PV formula, mortgage formula, IRR equation).

• Convert interest rates to the correct period (weekly, monthly, semi-annual, quarterly) where required.

• Round intermediate values to six decimal places where asked and round final answers to the required precision (commonly two or four decimal places).

• Provide clear, labelled intermediate steps so solutions are reproducible and auditable.

Key pointers to cover in the assessment

1. Cash-flow timing — explicitly state when each payment occurs (t = 0, 1, 2, …).

2. Rate conversion if an annual effective rate is given, compute the period rate:

   • weekly: iw=(1+iannual)1/52−1i_w=(1+i_{annual})^{1/52}-1iw=(1+iannual)1/52−1

   • monthly: m=(1+iannual)1/12−1i_m=(1+i_{annual})^{1/12}-1im=(1+iannual)1/12−1 or iAPR/12i_{APR}/12iAPR/12 depending on APR vs effective wording

   • semiannual BEY → half of BEY for semiannual periods.

3. Choose proper valuation formula — e.g. perpetuity every-other-year, annuity-due formula, bond PV (sum of coupon PVs + principal PV), growing perpetuity.

4. IRR/NPV logic — always compute NPV at the required discount rate and compute IRR; for mutually exclusive projects prefer NPV rule or incremental IRR analysis (follow conventional cash flow requirement for IRR usage).

5. Rounding rules — apply 6-decimal rounding to intermediate values as instructed; present final rounded answers to requested precision.

6. Documentation & validation — provide short explanation of each step and a quick sanity check (e.g., bond price change sign when yield rises; mortgage balance decreases after payments; IRR positive and reasonable).

How the Academic Mentor approached 

Below is a condensed walkthrough of how the Academic mentor guided the student to solve each question category. For each item I show the concept, the stepwise method, and the output/validation checks the mentor required.

A. Perpetuities & series (Questions 1 & 2)

Concept: Payments of $100 every other year (i.e., at t = 2, 4, 6, … or at t = 1, 3, 5, …). This is a geometric series (perpetuity with period = 2 years).

Steps the mentor guided the student to follow:

1. Identify the discount factor per year 1+i1+i1+i (here i=10%i=10\%i=10%).

2. For payments at t = 2,4,6,… write PV as:

   PV=∑k=1∞100(1+i)2k=100/(1+i)21−1/(1+i)2PV=\sum_{k=1}^{\infty}\frac{100}{(1+i)^{2k}}=\frac{100/(1+i)^2}{1-1/(1+i)^2}PV=k=1∑∞(1+i)2k100=1−1/(1+i)2100/(1+i)2

   (simplify algebraically).

3. For payments at t = 1,3,5,… write PV as:

   PV=∑k=0∞100(1+i)1+2k=100/(1+i)1−1/(1+i)2PV=\sum_{k=0}^{\infty}\frac{100}{(1+i)^{1+2k}}=\frac{100/(1+i)}{1-1/(1+i)^2}PV=k=0∑∞(1+i)1+2k100=1−1/(1+i)2100/(1+i)

4. Evaluate numerically and round as required (intermediate precision to six decimals if asked, final to two decimals).

Validation: Check that PV when first payment occurs earlier (t=1) is larger than if first payment at t=2. Reasonableness check passed.

B. Converting annual ↔ periodic rates & discount forms (Questions 3 & 4)

Concept: Convert annual effective return to weekly effective interest and to weekly effective discount as asked, then value a sequence of weekly losses (annuity due if payments start immediately).

Steps:

1. Compute weekly effective interest:

   iweek=(1+iannual)1/52−1i_{week}=(1+ i_{annual})^{1/52}-1iweek=(1+iannual)1/52−1

2. If the problem requests a weekly effective discount rate dweekd_{week}dweek (common definition):

   dweek=iweek1+iweekordweek=1−(1+iweek)−1d_{week}=\frac{i_{week}}{1+i_{week}} \quad\text{or}\quad d_{week}=1-(1+i_{week})^{-1}dweek=1+iweekiweekordweek=1−(1+iweek)−1

   (mentor asked student to explicitly state which discount definition they used and why).

3. For the PV of $300 per week for 20 weeks starting immediately (annuity-due, n=20):

   PV=300⋅[1−(1+i)−ni]⋅(1+i)(annuity-due formula)PV = 300\cdot\left[\frac{1-(1+i)^{-n}}{i}\right]\cdot(1+i) \quad\text{(annuity-due formula)}PV=300⋅[i1−(1+i)−n]⋅(1+i)(annuity-due formula)

   where iii is the weekly interest (or discount converted to equivalent interest).

4. Evaluate with high precision, round intermediate values to six decimals, final to two decimals.

Validation: Check that PV of annuity-due > PV of same annuity starting one period later.

C. Bond pricing (Questions 5, 6, 7)

Concept: Price = PV of coupons + PV of redemption; careful handling required when coupons are skipped and repaid at maturity (BBBY case). Also interpret price sensitivity to yields (interest rates).

Steps for general bond pricing:

1. Identify coupon payment per period (e.g., annual coupon% × face / number of coupon periods). Example: 16% annual, semiannual → coupon = 0.16×100/2 = $8 per semiannual period.

2. Convert market yield to same period (e.g., BEY 25% → semiannual ysemi=0.25/2=0.125y_{semi}=0.25/2=0.125ysemi=0.25/2=0.125).

3. List cash flows period by period, handling skipped coupons (zero payments at those dates), and a final payment that includes principal + accumulated missed coupons if repaid at maturity.

4. Discount each cash flow at the period yield and sum. Round intermediate to six decimals; final to 2 decimals.

BBBY special note: Build the period cash flow table explicitly (period 1..12), then compute PV.

Interpretation question (sensitivity): Mentor reinforced the canonical result: bond prices decrease when yields rise; the longer the maturity (all else equal), the larger the sensitivity (duration effect). This was used to answer the multiple-choice interpretation.

Validation: Price should fall when yield rises; compute price at initial yield and at 11% yield and confirm direction and relative sensitivity.

D. Mortgage amortisation (Questions 8 & 9)

Concepts & formulas: Standard fixed-rate mortgage monthly payment and remaining principal after k payments.

1. Compute monthly interest rate im=APR/12i_m = APR/12im=APR/12. Number of payments N=30×12=360N=30\times 12=360N=30×12=360.

2. Monthly payment:

   PMT=im⋅PV1−(1+im)−NPMT = \frac{i_m \cdot PV}{1-(1+i_m)^{-N}}PMT=1−(1+im)−Nim⋅PV

3. Remaining principal after 60 payments = present value of remaining payments: if payments are in arrears, remaining balance after payment kkk is:

   Bk=PMT⋅1−(1+im)−(N−k)imB_k = PMT\cdot\frac{1-(1+i_m)^{-(N-k)}}{i_m}Bk=PMT⋅im1−(1+im)−(N−k)

   (mentor asked student to either use amortisation table or formula).

4. If the student assumes a different PMT (e.g., $2,000), compute balance using the above PV formula for remaining payments.

Validation: Build a tiny amortisation schedule for a few periods to confirm the formula and check that balance decreases each month.

E. Dividend discount models (Questions 10–13)

Concepts covered: Declining dividend stream, no-growth (all earnings paid), partial retention (growth via ROE), periodic/quarterly dividends with initial flat payments and growing perpetuity thereafter.

• Declining dividend (Exxon): If D₀ (today) = $5 and dividends decline at 10% annually thereafter, compute D₁ = D₀×(1−0.10). If price must include today's dividend, add D₀ immediately and then discount the perpetuity for t≥1: often you compute P0=D0+D1r−gP_0 = D_0 + \frac{D_1}{r-g}P0=D0+r−gD1 (explain why you include D₀ explicitly). Ensure r≠gr\neq gr=g.

• Pay all earnings as dividends: With earnings next year = $10 and payout =100%, dividends = $10 forever and growth g = 0 ⇒ value = D/r=10/0.12D / r = 10/0.12D/r=10/0.12.

• Reinvest half: Retention ratio b=0.5b=0.5b=0.5. Growth g=ROE×b=0.15×0.5g = ROE\times b = 0.15\times 0.5g=ROE×b=0.15×0.5. Dividend D1=E1(1−b)D_1 = E_1(1-b)D1=E1(1−b). Then use Gordon growth P0=D1/(r−g)P_0 = D_1/(r-g)P0=D1/(r−g). Mentor had student explain economic intuition: reinvestment increases growth but reduces immediate dividend; check which effect dominates price.

• Quarterly-growing perpetual stream starting in 3 years (GenEx): Convert 15?fective annual to quarterly rate iq=(1.15)1/4−1i_q=(1.15)^{1/4}-1iq=(1.15)1/4−1. Discount the discrete sequence: compute PV of the first five flat quarterly payments (explicitly discount each one to t=0) and then treat the subsequent payments as a growing perpetuity from the time immediately after the 5th payment; use the growing perpetuity formula with period-growth rate gq=2%g_q=2\%gq=2% per quarter.

Validation: Ensure growth g

F. IRR vs NPV and project comparison logic 

Concept/Advice given by mentor:

• For mutually exclusive projects the primary decision criterion should be NPV (maximises absolute dollar wealth). IRR is useful, but IRR can give misleading rankings when projects differ in scale or timing or are non-conventional.

• If you want to use IRR to choose among mutually exclusive projects, compute the incremental project (difference in cash flows) and find the IRR of that incremental flow; compare that IRR to the cost of capital. The mentor walked the student through the incremental method step-by-step.

• For the single-period Project 3 IRR: use IRR=CF1CF0−1\text{IRR} = \frac{CF_1}{CF_0}-1IRR=CF0CF1−1.

Validation: Make an NPV table for each project and the incremental project to verify consistency.

G. Other short computational/interpretation items 

Examples & methods covered:

• Savings bond IRR: Solve (1+IRR)10=5000/2500(1+IRR)^{10}=5000/2500(1+IRR)10=5000/2500 so IRR=(2)1/10−1IRR= (2)^{1/10}-1IRR=(2)1/10−1.

• APR vs EAR: If a periodic percentage charge is given per period, show how to get APR (period × number of periods) vs EAR =(1+iperiod)m−1=(1+i_{period})^{m}-1=(1+iperiod)m−1. Mentor insisted on naming whether APR is quoted with simple periodic rate (nominal APR) or effective conversion; show step conversions and rounding rules.

• PV of multi-year cash flows (tattoo parlor): break into two parts: initial/follow-up investments (cost side) and operating cash flows (revenues/costs), compute their PVs separately and sum as required.

How the outcome was achieved (deliverables & validation)

The mentor required the student to produce the following deliverables for each question:

1. Annotated cash-flow table (period, cash flow, discount factor, PV of each flow).

2. Stated formula used and brief justification (1–2 lines) for choosing that formula.

3. Numeric computation with intermediate values shown rounded to six decimals (where requested), and the final answer rounded as specified.

4. Short interpretation sentence for applied questions (e.g., what price movement implies about interest-rate risk; economic meaning of NPV/IRR results).

5. A short validation step — cross checks such as monotonicity checks or dimension checks.

The student submitted: (a) an Excel workbook containing the cash-flow tables & formulas, (b) a PDF write-up with the annotated steps per question, and (c) a one-page summary of economic interpretations (bond sensitivity, payout effects on stock value, mortgage schedule behaviour).

Validation checks done by mentor:

• Recomputed a few spot checks by hand (or with a financial calculator) to ensure Excel formulas had been entered correctly.

• Reconciled the mortgage remaining balance via both amortisation-schedule method and PV-of-remaining-payments formula.

• Confirmed sign and direction of sensitivity (bond price down when rates up).

• Verified the dividend model satisfies g

Learning objectives covered

By completing this assessment the student demonstrated and practiced:

• Mastery of the time value of money across multiple compounding conventions and unusual payment patterns.

• Ability to translate rate notations (APR, BEY, APR with monthly periods, effective annual rate) into period rates used in valuation.

• Competence pricing bonds under non-standard cash flows (missed coupons, repayment at maturity).

• Use of both NPV and IRR decision rules and the incremental IRR technique for mutually exclusive projects.

• Construction and interpretation of amortisation schedules, and computing outstanding principal after k payments.

• Application of the dividend discount model in multiple forms (declining dividends, zero-growth, and Gordon growth with partial retention).

• Numerical precision and reproducibility (rounding intermediate and final results to the specified precision).

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