Your suggestion that interest rates are expected to increase is the pure expectations theory of the yield curve; that’s not the only theory out there.
Nevertheless, if the yield curve slopes upward, the rate at which you can reinvest the coupons is less than the YTM: you’re reinvesting the coupons for a shorter period than the original time to maturity. Lower reinvestment means that the realized yield will be less than the YTM.
Am I getting lower reinvestment because the rate at which I can reinvest the coupons is less than the YTM (why) or because of shorter period (also why)?
Because the yield curve is upward sloping, that means the shorter time periods have a smaller yield, so the more time passes by, the less yield you get with each reciept re-invested at the time remaining to maturity.
If the yield curve was downward sloping, you would re-invest all your reciepts at the shortest tenor, because it gives the highest possible yield, and the effect would be reversed.
Because the yield curve is upward sloping, that means the shorter time periods have a smaller yield, so the more time passes by, the less yield you get with each reciept re-invested at the time remaining to maturity.
If the yield curve was downward sloping, you would re-invest all your reciepts at the shortest tenor, because it gives the highest possible yield, and the effect would be reversed.
For a flat yield curve, reinvestment yield = YTM.
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With a downward sloping yield curve, we are still reinvesting for a shorter period of time. So why is the reinvestment yield > YTM?
you are earning a higher interest for that time period - so you reinvestment return is higher than the flat YTM over the entire life of the investment.