Consider a $1,000,000 cash flow to be paid in 5 years. How would you compute the present value? Assume the five-year USD swap rate is 3.5%.

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BlackTom AI · Accepted Answer

Question restatement: You are asked to compute the present value of a $1,000,000 cash flow due in 5 years, given a five-year USD swap rate of 3.5%. No answer options are provided in the prompt.

Step 1 — Clarify what information is needed: To discount a single future cash flow to present value, you need a discount factor for 5 years. That discount factor comes from the zero-coupon (risk-free) yield for 5 years or from the appropriate discount curve.

Step 2 — Identify what the provided 5-year swap rate can tell you: A swap rate for 5 years is a par rate. It is the fixed rate that sets the present value of a 5-year fixed-for-floating swap to zero, given an underlying set of floating payments and the corresponding discount factors. It is not the same thing as the 5-year zero-coupon yield unless you have the full yield/discount curve or you assume certain conditions.

Step 3 — Why you cannot directly use the 5-year swap rate as a zero rate: If you only have the 5-year swap rate, you do not automatically know the individual year-by-year zero-coupon discount factors for years 1 through 5. Discounting a single payment 5 years out requires the 5-year zero rate or the full discount curve to accurately reflect the time value of money, especially if the yield curve is not flat.

Step 4 — How one typically proceeds (bootstrapping): In practice, to obtain the 5-year zero-coupon discount factor from swap data you would bootstrap the zero curve using a set of instruments: overnight/short-term rates, 1-, 2-, 3-, 4-, and 5-year par swap rates (or corresponding government/IBOR swap data), solving for the annual or semi-annual zero-coupon rates so that the present value of those instruments matches their market prices. This requires multiple maturities and their par rates, not just the 5-year swap rate.

Step 5 — A practical (approximate) approach if you must proceed with limited data: If you are allowed to make a simplifying assumption, you could treat the 5-year swap rate as a proxy for the 5-year zero-coupon yield and use a standard discounting formula. With annual compounding, the present value would be calculated as 1,000,000 divided by (1 + 0.035) raised to the 5th power. With continuous compounding, you would use exp(-0.035 × 5). Both approaches yield a present value that is less than the face amount, reflecting the time value of money, but note that this is an approximation and depends on the chosen compounding convention.

Step 6 — Highlight key caveats and potential sources of error: The core issue is that swap par rates reflect the entire par curve and the sum of discounted fixed payments equals the swap rate times the annuity factor, given the assumed discount factors. Using a single 5-year par rate to back out a 5-year zero requires additional information or assumptions about the short-end of the curve. If the curve is not flat or if there are basis risks, the simple approximate approach can misstate the true present value.

Step 7 — Summary of the conceptual path you would take: To compute the precise present value, you would ideally obtain the full 5-year zero-coupon discount factor from the current yield/discount curve via bootstrapping using a set of instruments across maturities. If you must proceed with limited data, you can use a simplifying assumption to approximate the 5-year discount factor from the 5-year swap rate, recognizing that this introduces approximation error and depends on the matching of compounding conventions (annual vs. semi-annual vs. continuous) and curve shape.

Note on formatting and interpretation: All of the above is intended to explain the correct conceptual approach and the practical steps you would take, rather than providing a single definitive numeric result, given the data constraints in the prompt.

IFEPIA7022_001_2025_3 - Economics of Finance EOF in-class quiz 9-23-25

Consider a $1,000,000 cash flow to be paid in 5 years. How would you compute the present value? Assume the five-year USD swap rate is 3.5%.

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