MELON PROTOCOL: A BLOCKCHAIN PROTOCOL FOR DIGITAL ASSET MANAGEMENT DRAFT 6 Appendix B. Portfolio Formally we expand the portfolio as follows. Assuming there are n ∈ N digital assets available. This constitutes the following vector set a of assets: a1 .  (5) a =  .  . an where ak is the k-th available asset, for k ∈ N. By convention the first asset a1 represents Ether. ti Then the portfolio h , is defined as the vector set of asset holdings of a portfolio m at time ti. m   ti h a1 t  .  n (6) h i =  .  ∈ R≥0 m . hti a n ti where h is the amount, in token units of a , a portfolio m holds at time t , for k ∈ N. ak k i Appendix C. Gross Asset Value t t Let p i be the vector set of asset prices, at time ti. Then p i is:  ti  p a1 ti  .  n (7) p = . ∈R≥0 . t p i a n ti where p is the price per token unit of asset ak in Ether, at time ti, for k ∈ N. ak ti Note, since by convention a1 represents Ether, and the prices are given in Ether the first price p is always equal to a1 one. ti ti The Gross Asset Value or GAV vˆ in Ether of portfolio h at time ti is: hm m n ti t t Xt t (8) vˆ =hpi,h ii = p i h i h m a a m k k k=1 with the standard scalar product on Rn. The GAV can be seen as the gross value of the portfolio. Appendix D. Net Asset Value ti ti The Net Asset Value or NAV v in Ether of portfolio h at time ti is: h m m t t t t (9) v i =vˆ i −Management Fees i −Performance Fees i h h m m ti ti Management Fees resp. Performance Fees is the management resp. performance fees given to the Portfolio Man- ager for timestep ti. Appendix E. Delta TodefinetheDelta ∆ of a portfolio m, within the time (t ,t ], where t < t , we first define the Delta of ∆ , (t ,t ] i j i j (t ,t ] i j i i+1 i.e. the Delta of a single time step. Let be: t0 = time of contract creation tl = time of first investment = min {vtk 6= 0} t ∈[t ,t ] hm k 0 i t I i = Sumof all investments within (ti−1,ti] t Wi = Sumofall withdrawals within (ti−1,ti] t t Where both I i and W i is a value in Ether. Then the Delta of a single time step is: t v i+1 −Iti+1 +Wti+1 (10) ∆ = hm (ti,ti+1] vti h m By design, the Delta of a portfolio, is independent of funds invested or withdrawn within the time (ti,ti+1]. By factoring together these Deltas of single time steps we get the general definition of the Delta ∆ of a portfolio m as: (t ,t ] i j   1 if tj ≤ t  l   j−1 Q  tk  ∆ if t