Complications in Bank Valuations  751 model. The rates are all derived from the current yield curve. To illustrate, the expected three-year interest rate in 2021 follows from the current three- and six-year yields: r Y Y 2021 2024 2024 6 2021 3 1 3 6 1 1 1 1 2 82 1 1 6 − = + + −    = + + ( ) ( ) ( . %) ( . 6 1 4 0 3 1 3 %) . % −    = where r2021–2024 is the expected three-year interest rate as of 2021, Y2021 is the current three-year interest rate, and Y2024 is the current six-year interest rate. In practice, forward rate curves derived from the yield curve will rarely follow the smooth patterns of Exhibit 38.11. Small irregularities in the cur- rent yield curve can lead to large spikes and dents in the forward rate curves, which would produce large fluctuations in net interest income fore- casts. As a practical solution, use the following procedure. First, obtain the forward one-year interest rates from the current yield curve. Then smooth these forward one-year rates to even out the spikes and dents arising from irregularities in the yield curve. Finally, derive the two-year and longer- maturity forward rates from the smoothed forward one-year interest rates. As the exhibit shows, all interest rates should converge toward the current yield curve in the long term. As a result, the bank’s income contribution from any maturity difference in deposits and loans disappears in the long term as well. EXHIBIT 38.11  Yield Curve and Future Interest Rates Interest rate, % 2020 2024 2028 2032 2036 2040 2044 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Current yield curve Forward 5-year rates Forward 3-year rates Forward 1-year rates Forward 10-year rates 752  Banks Loan Loss Provisions For our ABC Bank valuation, we did not model any losses from defaults on loans outstanding to customers. In real life, your analysis and valuation have to include loan loss forecasts, because loan losses are among the most impor- tant factors determining the value of retail and wholesale banking activities. For estimating expected loan losses from defaults across different loan catego- ries, a useful first indicator would be a bank’s historical additions to loan loss provisions or sector-wide estimates of loan losses (see Exhibit 38.12). As the exhibit shows, these losses increased sharply during the 2008 credit crisis but recovered to pre-crisis levels by 2013. Credit cards typically have the highest losses, and mortgages the lowest, with business loans somewhere in between. All default losses are strongly correlated with overall economic growth, so use through-the-economic-cycle estimates of additions to arrive at future annual loan loss rates to apply to your forecasts of equity cash flows. To project the future interest income from a bank’s loans, deduct the es- timated future loan loss rates from the future interest rates on loans for each year. You should also review the quality of the bank’s current loan portfolio to assess whether it is under- or overprovisioned for loan losses. Any required increase in the loan loss provision translates into less equity value. EXHIBIT 38.12  Annual Losses for U.S. Banks by Loan Category Write-off charges as % of loans outstanding 0.0 1985Q1 1986Q1 1987Q1 1988Q1 1989Q1 1990Q1 1991Q1 1992Q1 1993Q1 1994Q1 1995Q1 1996Q1 1997Q1 1998Q1 1999Q1 2000Q1 2001Q1 2002Q1 2003Q1 2004Q1 2005Q1 2006Q1 2007Q1 2008Q1 2009Q1 2010Q1 2011Q1 2012Q1 2013Q1 2014Q1 2015Q1 2016Q1 2017Q1 2018Q1 2019Q1 2.0 4.0 6.0 8.0 10.0 12.0 Credit cards  Consumer loans Mortgages Business loans Source: Federal Reserve, “Charge-Off and Delinquency Rates on Loans and Leases at Commercial Banks,” www.federalreserve.gov.