Deep Dive

The Marian Mathematical Model

This page collects every quantitative claim in the docs into one derivation. All symbols are consistent with the module pages; a symbol table is at the bottom.

1. The identity

A user's total gross yield on a locked, collateralized basket position, expressed as an APY on their basket NAV, is:

Y  =  Y_div  +  Y_emit  +  Y_fee  +  Y_reinvest

Y_div      =  Σ_i w_i · d_i        (basket dividend yield)
Y_emit     =  b_i · E_g · P_MRN / V_i    (boosted emissions in $)
Y_fee      =  φ_L · R_proto · ve_i / ve_total / V_i
Y_reinvest =  LTV · ( Y_alt − r_borrow )   (redeployed mUSD alpha)

Symbols:
   V_i        = value of user's basket position (USD)
   w_i, d_i   = weight and div yield of constituent i
   b_i        = boost multiplier (Emissions page)
   E_g        = annual $MRN emitted to gauge g
   P_MRN     = $MRN market price
   φ_L        = locker fee share = 0.70
   R_proto    = annual protocol revenue (USD)
   LTV        = user's chosen loan-to-value
   Y_alt      = external redeploy APY of borrowed mUSD
   r_borrow   = current mUSD borrow APR

Y is the annualized dollar return divided by NAV of the basket position.

2. Baseline scenario

Parameters used for the reference number quoted throughout the docs:

Parameter	Symbol	Value
Basket NAV of position	V_i	$10,000
Basket div yield	Y_div	1.65%
Lock duration	L	24 months
Basket CF	CF_i	72%
Chosen LTV	LTV	60%
Borrow APR	r_borrow	4.85%
Redeploy APY (stable pool)	Y_alt	6.20%
Protocol revenue	R_proto	$14.05M / yr
User veMRN share	ve_i / ve_total	0.000625
Boost multiplier	b_i	1.75×
Emissions to user's gauge	E_g	$412 MRN / yr / $10k

Substituting baseline into the identity

Y_div       = 0.0165                             =  1.650%
Y_emit      = 1.75 · 412 · 0.85 / 10,000         =  6.128%
Y_fee       = 0.70 · 14,050,000 · 0.000625 / 10,000  =  61.47%*
Y_reinvest  = 0.60 · (0.0620 − 0.0485)           =  0.810%

* Y_fee only applies to the MRN Alice locks, not the basket.
  To make units comparable we express it relative to basket NAV:
  Alice locked 50,000 MRN = $42,500 (P_MRN=0.85).
  Fee $ received = 0.70 · 14,050,000 · 0.000625 = $6,147 / yr
  Fee yield on basket NAV alone (excluding lock capital) is
  moved out of Y — see "unit hygiene" below.

Basket-only yield (Y excluding Y_fee)
   =  1.65% + 6.13% + 0.81%
   =  8.59%  APY

Position-level yield (basket + lock capital):
   = ( div + emit + fee + redeploy ) / ( V_i + P_MRN·amount_lock )
   = ( 165 + 612.8 + 6,147 + 81 ) / ( 10,000 + 42,500 )
   =   7,005.8 / 52,500
   =  13.34%  APY blended

Unit hygiene

Yields on the basket and yields on the lock are on different capital bases. The 8.59% APY headline uses basket NAV as the denominator (what a user gets for their equity exposure); the 13.34% blended APY includes the veMRN capital they also committed. Both are true; they answer different questions.

3. Sensitivity — what moves the number

Δ Variable	New value	New basket yield	Δ vs baseline
Baseline	—	8.59%	—
Higher div basket (bDIV25)	Y_div = 3.22%	10.16%	+1.57%
Longer lock (48 mo)	b_i = 2.50, CF = 77%	10.85%	+2.26%
Higher $MRN price	P_MRN = $1.50	12.42%	+3.83%
Utilization to 90%	r_borrow = 32.5%	8.59% (locker share ↑ elsewhere)	≈0
Bear case: −40% $MRN, 6-mo lock	b_i = 1.10, P_MRN = $0.50	3.87%	−4.72%

4. Protocol-level revenue as a function of TVL

R_proto(TVL, U)  =  f_mgmt · TVL
                 +  r_borrow(U) · U · CF_avg · TVL
                 +  f_redeem · Turnover · TVL
                 +  E[penalty] · TVL

With defaults:
   f_mgmt      = 0.0018     (blended basket mgmt fee)
   CF_avg      = 0.65       (average collateral factor)
   f_redeem    = 0.0005
   Turnover    = 0.35       (annual redemption turnover)
   E[penalty]  = 0.0012     (empirical liquidation contribution)

At TVL = $500M, U = 0.50:
   R_proto  =  900,000  +  0.05·0.50·0.65·500M
                        +  0.0005·0.35·500M  +  600,000
            =  900,000  +  8,125,000  +  87,500  +  600,000
            =  9,712,500  USD / year

Note the difference between this figure ($9.7M) and the $14.05M used in the locker-yield example. The locker page assumes CF_avg of 100% of borrowed collateral (users borrow to their max) and higher penalty realization in a volatility spike. Both are illustrative bands.

5. Solvency invariant

Σ_i D_i  ≤  Σ_i Σ_j  c_i,j · L_j · P_j    −   ε · TVL

ε is the insurance buffer, currently 3% of TVL.
When the inequality is violated, the RiskEngine halts new borrows across
all markets until the insurance module (backed by 8% of $MRN supply)
recapitalizes to restore the buffer.

For any oracle price vector P, total debt cannot exceed liquidation-thresholded collateral.

6. Symbol table

Symbol	Meaning	Unit
V_i	Basket position value	USD
w_i, d_i	Constituent weight, dividend yield	%, APY
ve_i	User's veMRN balance	veMRN
b_i	Emissions boost multiplier	1.0–2.5×
CF_i	Collateral factor	%
HF_i	Health factor	unitless
LTV	Loan-to-value chosen by user	%
U	Market utilization = borrowed/supplied	%
r_borrow(U)	Kinked rate curve	APR
φ_L, φ_I, φ_T	Fee splits (0.70 / 0.20 / 0.10)	—
R_proto	Annual protocol revenue	USD

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