5Y5Y Forward Rate Calculation

This question considers a fixed income setting where an investor compares investing over a long horizon at a single quoted spot rate with splitting that horizon into two segments using spot and forward rates. The concrete setup is a medium-term and a longer-term maturity, and the candidate is asked to infer the implied rate for the later sub-period that makes the two investment strategies equivalent. This style of problem is typical in entry-level trading, structuring, and fixed income quant interviews, where understanding of term structures and no-arbitrage relationships is essential.

The problem leans on basic time value of money, compounding, and the no-arbitrage relation between spot and forward rates. It requires manipulating compounded growth factors and rearranging them to isolate the unknown forward rate. An interviewer is watching for clean algebra, comfort moving between different maturities, and correct use of annual compounding assumptions. They also look for a clear explanation of why these relationships must hold in an arbitrage-free market, not just mechanical formula use.

What it tests

Forward rates are derived from spot rates by equating the compounded returns over different intervals, ensuring that investing for the full period at the long-term spot rate yields the same result as sequentially investing at the short-term spot rate and then at the forward rate. This arises because the absence of arbitrage requires that the total return from locking in a rate today for the entire period must match the return from rolling over shorter-term investments at prevailing rates. The key is that the product of the growth factors for each sub-period must equal the growth factor for the total period. This structure holds for any time intervals and rates, not just the specific years or rates in this problem. The underlying logic is that money invested for $n$ years at the $n$-year spot rate must grow to the same amount as money invested for $m$ years at the $m$-year spot rate and then rolled over at the forward rate for the remaining $n-m$ years.