Stochastic Interest Rate models Flashcards
Formula for St/S0
St/S0 = E((1+i)^t)
Formula for S0/St
E(1/(1+i)^t)
Formula for s10 - certain accumulated value of ten cash flows invested annually in arrears
((1+i)^10-1)/i
Formula for a10 - annuity certain present value of an annuity certain paid annually in arrears
(1-(1+i)^-10)/i
If returns are varied every year, but known in advance how could we maximise the accumulated value of s10 ( or any accumulation of cash flows) by placing certain returns on each year?
To maximise the impact of the largest returns larger returns should be applied in the final years implying the lower returns should arise in the first few years.
If returns are varied every year, but known in advance how could we maximise the present value of a10 (or any annuity certain) by placing certain returns on each year?
Annuity value increases as further cash flows are included. To maximise the impact of the largest negative returns with the largest discount factor, these should be applied in the first few years, this implies the higher return should arise in the final years.
Describe the relationship between annuity and accumulated cash flows
The accumulation of regular cash flows is the annuity multiplied by the accumulation of a bullet cash flows as the annual cash flows in arrears are the same in both cases, all that is changing is the accumulation date.
How would you test if it is possible to find a model with independent annual returns that replicates the expectations of a model which has dependence
Test by expressing the expected product as the product of expectations for growth factors.
How to check if something is a martingale or not
Mt-1= Et-1 (Mt) . This is the one step ahead martingale proof.
Name some risks that might be important for investment decisions that are not taken into account in a model
Statistical noise in parameter estimates
Human blunders or deliberate deceit
Data selection and treatment of outliers
Modelling endogenous quantities as exogenous. Assuming market returns have a statistical distribution which does not allow for decisions made using the model to feedback into market prices for example ignoring the possibility that a large investors sell decision could move the market price and change future return distributions
Optimising model errors
Unquantifiable risks.
Explain steps that might have been used to calibrate the parameters for a stationary time series model for log equity market returns.
Collecting data of historical equity returns
Not returns on a single share - need to aggregate individual share returns to form an index
Calculating the log returns
Considering issues of relevance ex: discarding or underweighting earlier periods
Calculating autocorrelation of historic returns.
What is annualized volatility
Annualised volatility is the SD of log returns divided by the square root of the time period.
If Xt had been an independent identically distributed sequence, then the annualised volatility (for all k) is the standard deviation of Xt. For processes with dependent
returns, the annualised volatility in general depends on the time period.
How would one calculate the variance of log returns across 3 years for assets that are correlated with each other. Matrix answer
Variance of 3 year log returns = (SD of each year^Transpose) * correlation matrix *SD of each year
What kind of model would be associated with weakly efficient markets?
Random walks where returns are independent from one period to the next. Often this doesn’t apply as prices are positively autocorrelated. This suggests new information is incorporated only slowly, hence weak-form efficiency.
