Matching two known cash payments

The case and its answers

Every question of this case, with its answer and why each other option is wrong.

An institution must pay ₹30 lakh exactly two years from today and ₹40 lakh exactly four years from today. Its manager can buy two hypothetical zero-coupon bonds that mature on those exact dates and repay their face amounts. Both are priced using a 5% annual yield compounded annually. Assume no default, taxes, transaction costs or early withdrawals. Fractional quantities are available. The manager buys face amounts matching the two payments and intends to hold the bonds to maturity. Use present value = future payment/(1+yield)^years. For these zero-coupon bonds, Macaulay duration equals maturity, so the combined duration is the present-value-weighted average of two and four years.

Default, transaction costs, taxes and early withdrawal needs are excluded solely for this teaching example.

1. Q1How much must be invested today in the two-year zero-coupon bond?

   • A

     Option A. ₹30 lakh

     Why it is wrong: This ignores the two-year discounting.

   • B

     Option B. ₹28.5714 lakh

     Why it is wrong: This discounts for only one year.

   • C

     Option C. About ₹27.2109 lakh — Correct answer

   • D

     Option D. ₹33.0750 lakh

     Why it is wrong: This compounds the liability instead of discounting it.

   Answer: C. ₹30 lakh/1.05² = ₹27.210884 lakh. The bond’s face amount remains ₹30 lakh. Remember: Separate a bond’s current price from its maturity payment.

   Hint, shown before you answer in practice mode: Discount the ₹30 lakh payment for two years at 5%.

   XXIB-S3-C02-Q1easynumerical

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2. Q2What is the total amount invested today to match both payments?

   • A

     Option A. ₹57.5892 lakh

     Why it is wrong: This discounts both payments as if they were due in four years.

   • B

     Option B. About ₹60.1190 lakh — Correct answer

   • C

     Option C. ₹70 lakh

     Why it is wrong: This is the undiscounted total of the payments.

   • D

     Option D. ₹63.4921 lakh

     Why it is wrong: This discounts both payments as if they were due in two years.

   Answer: B. ₹30/1.05² + ₹40/1.05⁴ = ₹27.210884 + ₹32.908099 = ₹60.118983 lakh. Remember: A cash-flow ladder values each liability on its own due date.

   Hint, shown before you answer in practice mode: Value each payment at its own date and add the results.

   XXIB-S3-C02-Q2mediumnumerical

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3. Q3Which expression correctly gives the combined Macaulay duration?

   • A

     Option A. 2 + 4 = 6 years

     Why it is wrong: Portfolio duration is a weighted average, not a sum.

   • B

     Option B. Exactly 3 years, the simple average of the maturities

     Why it is wrong: The invested present values are not equal.

   • C

     Option C. [2×(30/1.05²) + 4×(40/1.05⁴)] ÷ [30/1.05² + 40/1.05⁴] — Correct answer

   • D

     Option D. (2×30 + 4×40) ÷ 70

     Why it is wrong: This weights by future face amounts instead of current values.

   Answer: C. Each zero-coupon bond’s Macaulay duration is its maturity. Portfolio duration therefore weights two and four years by their respective present values. Remember: Use market-value weights for portfolio duration.

   Hint, shown before you answer in practice mode: Use today’s value of each bond as its weight.

   XXIB-S3-C02-Q3hardapplication

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4. Q4If market yields immediately rise to 6%, what happens to the market value of the matching bonds?

   • A

     Option A. It falls to about ₹58.3836 lakh — Correct answer

   • B

     Option B. It becomes ₹70 lakh immediately

     Why it is wrong: ₹70 lakh is the undiscounted total at two different future dates.

   • C

     Option C. It stays at ₹60.1190 lakh

     Why it is wrong: Holding to maturity does not freeze today’s market price.

   • D

     Option D. It rises because future payments are unchanged

     Why it is wrong: A higher discount rate lowers the present value of fixed payments.

   Answer: A. New value = ₹30/1.06² + ₹40/1.06⁴ = about ₹58.3836 lakh. The stated maturity payments do not change. Remember: Cash-flow matching does not prevent interim price changes.

   Hint, shown before you answer in practice mode: Revalue the same two future payments at the new yield.

   XXIB-S3-C02-Q4mediumnumerical

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5. Q5Under the stated hold-to-maturity and no-default assumptions, does the yield rise prevent the scheduled liabilities being paid?

   • A

     Option A. No. All bond risks have been eliminated in every real situation

     Why it is wrong: The exercise excludes risks such as default and unexpected liquidity needs rather than proving they cannot occur.

   • B

     Option B. Yes. Lower market prices reduce the contractual face payments

     Why it is wrong: A market price change does not change the assumed face amounts.

   • C

     Option C. Yes. The manager must sell both bonds immediately

     Why it is wrong: The institution has no early withdrawal requirement.

   • D

     Option D. No. The bonds still pay the matching face amounts on the liability dates — Correct answer

   Answer: D. The bonds still deliver ₹30 lakh in year two and ₹40 lakh in year four. Interim valuation changes do not create a funding shortfall under these explicit assumptions. Remember: Evaluate a matching strategy against the liability cash flows and its assumptions.

   Hint, shown before you answer in practice mode: Compare contractual cash-flow dates with the required payment dates.

   XXIB-S3-C02-Q5mediumapplication

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