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What is the time value of a stream of investments?

Part of: Mathematical economics Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Ragnar Norberg  and
Mogens Steffensen
Ragnar Norberg*
Affiliation:
London School of Economics
Mogens Steffensen*
Affiliation:
University of Copenhagen
*
∗Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email address: [email protected]
∗∗Postal address: Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark. Email address: [email protected]
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Abstract

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The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

Keywords

Cash flowcompounding assetdiscountingstochastic exponential

MSC classification

Secondary: 91B30: Risk theory, insurance 65C30: Stochastic differential and integral equations
Type
Short Communications
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 861 - 866
Copyright
© Applied Probability Trust 2005 

References

Aase, K. K. (2002). Equilibrium pricing in the presence of cumulative dividends following a diffusion. Math. Finance 12, 173–198.Google Scholar
Holm Nielsen, P. (2002). On optimal control of bonus in life insurance. , Laboratory of Actuarial Mathematics, University of Copenhagen.Google Scholar
Kalashnikov, V. and Norberg, R. (2002). Power tailed ruin probabilities in the presence of risky investments. Stoch. Process. Appl. 98, 211–228.Google Scholar
Paulsen, J. (1993). Risk theory in a stochastic economic environment. Stoch. Process. Appl. 46, 327–361.Google Scholar
Paulsen, J. (1996). Stochastic calculus with application to risk theory. Lecture notes, Department of Lecture notes.Google Scholar
Protter, P. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
Steffensen, M. (2001). On valuation and control in life and pension insurance. , Laboratory of Actuarial Mathematics, University of Copenhagen.Google Scholar