Mr. Darcy is a single man of good fortune, so good, in fact, that it allows him to sustain a lavish lifestyle simply on the interest he earns on his savings without ever having to work. Like many of the British upper class of his time, his considerable wealth is invested in consols, short for consolidated annuities, a type of government bond that never matures and continues to pay interest forever. By purchasing these bonds he effectively extends the government a loan for which the principal is never repaid, allowing him and his descendants to earn annual coupon payments in perpetuity. Indeed, this form of investment was so popular that people often expressed their wealth in terms of this annual income, rather as a net worth as we do today.
(a) Mr. Darcy has “ten thousand a year” as any prospective young bride in his social circle would have known well. That is, his entire net worth is held in consols which entitle him to annual coupon payments totalling C = 10,000. If the nominal interest rate is i = 0.04, what is Mr. Darcy’s total wealth today?
(b) One facet of modern economies that wasn’t nearly as relevant in Mr. Darcy’s time is inflation. While £10,000 a year would represent a fortune in the 19th century, today it is hardly enough to sustain a decent standard of living. Let the expected annual inflation rate be π = 0.02 and assume there is no unexpected inflation. What is the real interest rate r?
(c) Suppose Mr. Darcy instead wishes to hold his wealth in inflation indexed consols whose coupon payments grow annually so that their purchasing power remains constant. Assuming his initial wealth remains unchanged and he wishes to maintain the same real annual income forever, what is his nominal income in each of the first three years?
(d) What is Mr. Darcy’s nominal wealth at the end of each the next three years? What is his real wealth in current pounds in each of these years?
(e) How many years would it take before income from the inflation indexed consols would exceed that from the non-indexed consols?