Term Structures of Interest Rates and Simple Stochastic Asset Return Models
Term structure basics
This domain tests spot rates, forward rates, discount factors and simple models for asset returns. A spot rate y_t is the annual effective rate for a zero-coupon cashflow at time t, with discount factor P(0,t)=(1+y_t)^-t. The yield curve plots spot rates against term. A forward rate f_{t,t+1} is the one-period rate agreed now for lending between t and t+1: (1+y_{t+1})^{t+1}=(1+y_t)^t(1+f_{t,t+1}). The expectations hypothesis says forward rates reflect expected future spot rates plus possible risk premia; liquidity preference suggests longer terms may carry premia.
Stochastic returns
A simple model treats log returns as independent and identically distributed normal variables. If log(S_t/S_0) has mean μt and variance σ^2 t, then S_t is lognormal, with E[S_t]=S_0 exp(μt+0.5σ^2 t). In continuous time, geometric Brownian motion is dS=μS dt+σS dW. The variance grows with horizon, and the median is S_0 exp(μt), below the mean when σ>0.
Worked example
Suppose the 1-year spot rate is 3% and the 2-year spot rate is 4%. The forward rate for year 2 is (1.04^2/1.03)-1=0.050097, or 5.0097%. If a 2-year zero-coupon bond pays 100, its present value is 100/1.04^2=92.456. Under lognormal returns with μ=0.06 and σ=0.20, the expected value after one year of a 100 asset is 100 exp(0.06+0.5(0.20^2))=108.33, while the median is 106.18.
Traps
Using forward rates as certain future spot rates; confusing effective annual spot rates with continuously compounded rates; applying the lognormal mean formula without the 0.5σ^2 term; and assuming independent returns implies constant prices rather than a random walk in log price.