Silicon Valley Bank and Burton Malkiel’s Mathematics of Bond Prices

Silicon Valley Bank could easily have avoided its mark-to-market write-down and bankruptcy.  Its managers, their outside consultants, and the steward of the banking system – the Federal Reserve Bank – should all have been required to read, understand, confirm and apply the constraints immanent in Burton Malkiel’s five theorems, especially Theorem 2, in his 1962 essay:

Malkiel, Burton, Expectations, Bond Prices, and the Term Structure of Interest Rates, Quarterly Journal of Economics Vol. LXXVI (May 1962) pp. 197-218 (Cambridge, Mass.: Harvard University Press) [Malkiel, 1962]:

In the essay’s Section II, THE MATHEMATICS OF BOND PRICES, Malkiel explains that the market price, P, of a bond, is determined by four factors: 1) the face value, F, or the principal amount to be paid at maturity, 2) the bond’s coupon or interest paid periodically, C, 3) the effective interest rate per period, i, and 4) the number of years to maturity, N.   Malkiel:

P = C/(1+i) + C/(1+i)2 + …+ C/(1+i)N + F/(1+i)N

 Summing the geometric progression and simplifying, we obtain

P = C/i[1 – 1/(1+i)N] + F/(1+i)N 


 P = C/i  +  (F – c/i)/(1+i)N

Malkiel’s five theorems treat the prices, absolute changes in prices, and the percentage changes in prices due to the factors of yield i, change in yield δi, and relevant term to maturity N.  Using the calculus of differentiating, Malkiel proves each of the five theorems immediately after he states them.  We will simply state each theorem and leave it to the serious student to study, understand, and confirm the differentiations constituting Malkiel’s proofs.

Theorem 1: Bond Prices move inversely to bond yields.

Theorem 2: For a given change in yield from the nominal yield, changes in bond prices are greater, the longer is the term to maturity.

Theorem 3: The percentage price changes described in Theorem 2 increase at a diminishing rate as N increases.

 Theorem 4: Price movements resulting from equal absolute (or, what is the same, from equal proportionate) increases and decreases in yield are asymmetric; i.e. a decrease in yields raises bond prices more than the same increase in yields lowers prices.

Theorem 5: The higher is the coupon carried by the bond, the smaller will be the percentage price fluctuation for a given change in yield except for one-year securities and consols.

Also, relevant to recent mistakes or oversights by bank management and the Central Bank, we quote Schumpeter re the responsibility of the banking system:

Banks are not there to “force their money upon people,”4 nor “do they congratulate themselves if they are loaned up.”5  A banking committee is not “an automaton” but understanding and attentive to purpose and situation, “ judging chances of success of each purpose and, as means to this end, the kind of man the borrower is, watching him as he proceeds …”6  “It should be observed how important it is for the system of which we are trying to construct a model, that the banker should know, and be able to judge, what his credit is for and that he should be an independent agent.  To realize this is to understand what banking means.”7  “the banker’s function is essentially a critical, checking, admonitory one.  Alike in this respect to economists, bankers are worth their salt only if they make themselves thoroughly unpopular with governments, politicians and the public.  This does not matter in times of intact capitalism.  In the times of decadent capitalism, this piece of machinery is likely to be put out of gear by legislation.”McShane, Philip (quoting Joseph Schumpeter’s Business Cycles I and II) Implementing Lonergan’s Economics, in The Lonergan Review, Culture Science and Economics, Vol. III, No 1, Spring 2011, Seton Hall University, pp. 196-204


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