Analyst Dojo

CFA Level I Alternative Investments formulas

Fund fees and waterfalls, private equity and private credit returns, real estate and commodity pricing, and leverage in alternative funds.

52 formulas from the Alternative Investments lessons (6–9% of the exam), 7 to memorize. Each says how deeply to learn it and opens to a quick worked example with the keystrokes for the TI BA II Plus and the HP 12C.

How deeply to learn each part
Know the idea
Understand it in general. It frames other topics, but you will not calculate with it.
Use it
Understand it well enough to solve a question with it. Recognize when it applies.
Memorize
Learn it by heart. The exam will not give it to you.

Module 1 Alternative Investment Features, Methods and Structures

Fees and waterfalls: hurdle, catch-up, high-water mark, clawback

Management fee Use it
F=f×BF = f \times B
Quick example

A hedge fund with USD 800 million of assets under management charges 1.5%. Fee = 800 × 0.015 = USD 12 million a year. A private equity fund would use committed capital as the base instead.

On your calculator
  1. 800×0.015= 12
  1. 800ENTER0.015× 12

ENTER enter (on the HP, puts the number on the stack)

GP return with a hard hurdle Memorize
rGP=max⁡[0,  p (r−rh)]r_{GP} = \max\left[0,\; p\,(r - r_h)\right]
Quick example

Fund return 22%, hard hurdle 10%, performance fee 25%. 0.25 × (22 − 10) = 3.0% to the GP, leaving 19.0% to LPs.

On your calculator
  1. 22−10=×0.25= 3
  1. 22ENTER10−0.25× 3

ENTER enter (on the HP, puts the number on the stack)

Catch-up return Memorize
rcu=p×rh1−pr_{cu} = \dfrac{p \times r_h}{1 - p}
Quick example

Hurdle 10% and performance fee 20%: r_cu = 0.20 × 10 ÷ 0.80 = 2.5%. The GP takes 100% of the next 2.5% so that its share reaches 20% of the total return so far.

On your calculator
  1. 0.2×10÷0.8= 2.5
  1. 0.2ENTER10×0.8÷ 2.5

ENTER enter (on the HP, puts the number on the stack)

GP return with a full catch-up Memorize
rGP=max⁡[0,  rcu+p (r−rh−rcu)]r_{GP} = \max\left[0,\; r_{cu} + p\,(r - r_h - r_{cu})\right]
Quick example

Return 20%, hurdle 10%, fee 20%, so r_cu = 2.5%. r_GP = 2.5 + 0.20 × (20 − 10 − 2.5) = 4.0%, which is 20% of the whole 20%.

On your calculator
  1. 0.2×10÷0.8=STO1 r_cu = 2.5
  2. 20−10−RCL1=×0.2+RCL1= 4

STO store · RCL recall a stored number

  1. 0.2ENTER10×0.8÷STO1 r_cu = 2.5
  2. 20ENTER10−RCL1−0.2×RCL1+ 4

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

High-water mark performance fee Use it
fee=p×max⁡[0,  V−H]\text{fee} = p \times \max\left[0,\; V - H\right]
Quick example

High-water mark 128, value before the fee 140, performance fee 20%. Fee = 0.20 × (140 − 128) = 2.4. A fall below 128 earns nothing.

On your calculator
  1. 140−128=×0.2= 2.4
  1. 140ENTER128−0.2× 2.4

ENTER enter (on the HP, puts the number on the stack)

Clawback Use it
CB=max⁡[0,  Cpaid−p×P]CB = \max\left[0,\; C_{paid} - p \times P\right]
Quick example

Carry paid so far is 9. The fund turns out to have earned a total profit of only 25, so carry at 20% should be 5. Clawback = 9 − 5 = 4.

On your calculator
  1. 25×0.2=STO1 carry really earned
  2. 9−RCL1= 4

STO store · RCL recall a stored number

  1. 9ENTER25ENTER0.2×− 4

ENTER enter (on the HP, puts the number on the stack)

Module 2 Alternative Investment Performance and Returns

IRR and multiple of invested capital (MOIC)

Internal rate of return Use it
∑t=0nCFt(1+IRR)t=0\sum_{t=0}^{n} \frac{CF_t}{(1 + IRR)^t} = 0
Quick example

Cash flows of −100, +30, +40 and +90 at years 0 to 3. The rate that sets the present value to zero is 22.56%.

On your calculator
  1. CF2NDCLR WORK
  2. 100+|-ENTER↓
  3. 30ENTER↓↓
  4. 40ENTER↓↓
  5. 90ENTER
  6. IRRCPT 22.56

CF cash flow worksheet · 2ND second function (the shifted label above a key) · CLR WORK clear the worksheet · +|- change sign · ENTER enter (on the HP, puts the number on the stack) · IRR internal rate of return · CPT compute

  1. fCLEAR REG
  2. 100CHSgCF0
  3. 30gCFj
  4. 40gCFj
  5. 90gCFj
  6. fIRR 22.56

f gold shift key · CLEAR REG clear all registers · CHS change sign · g blue shift key · CF0 cash flow at time zero · CFj cash flow j · IRR internal rate of return

Multiple of invested capital Memorize
MOIC=Realized value+Unrealized valueInvested capital\text{MOIC} = \dfrac{\text{Realized value} + \text{Unrealized value}}{\text{Invested capital}}
Quick example

Invested capital 36, realized value 20, remaining holdings valued at 61. (20 + 61) ÷ 36 = 2.25×.

On your calculator
  1. 20+61=÷36= 2.25
  1. 20ENTER61+36÷ 2.25

ENTER enter (on the HP, puts the number on the stack)

Invested capital Memorize
Invested capital=Paid-in capital−Fees and expenses\text{Invested capital} = \text{Paid-in capital} - \text{Fees and expenses}
Quick example

Investors have paid in 320 and fees and expenses took 24. Invested capital is 320 − 24 = 296.

On your calculator
  1. 320−24= 296
  1. 320ENTER24− 296

ENTER enter (on the HP, puts the number on the stack)

A multiple as a yearly rate Use it
r=MOIC1/T−1r = \text{MOIC}^{1/T} - 1
Quick example

A single investment returns 3.0× its cost after 8 years. 3^(1/8) − 1 = 14.72% a year.

On your calculator
  1. 3y^x(1÷8)=−1= 0.1472

y^x power

  1. 3ENTER81/xy^x1− 0.1472

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Leverage, redemptions, lockups and gates

Leveraged return Use it
rL=r (Vc+Vb)−Vb rbVcr_L = \dfrac{r\,(V_c + V_b) - V_b\,r_b}{V_c}
Quick example

Own capital 80, borrowed 40 at 3%, portfolio return 7% on 120. (7 × 120 − 40 × 3) ÷ 80 = 720 ÷ 80 = 9.0%.

On your calculator
  1. (7×120)−(40×3)=÷80= 9
  1. 7ENTER120×40ENTER3×−80÷ 9

ENTER enter (on the HP, puts the number on the stack)

Leveraged return, rearranged Use it
rL=r+VbVc (r−rb)r_L = r + \dfrac{V_b}{V_c}\,(r - r_b)
Quick example

The same fund: 7% + (40 ÷ 80) × (7% − 3%) = 7 + 0.5 × 4 = 9.0%. Borrowing adds 2.0 points.

On your calculator
  1. 40÷80=×(7−3)+7= 9
  1. 40ENTER80÷7ENTER3−×7+ 9

ENTER enter (on the HP, puts the number on the stack)

Breakeven borrowing rate Use it
rL=r  ⟺  rb=rr_L = r \iff r_b = r
Quick example

The portfolio earns 6%. Borrowing at 6% leaves the fund exactly where it started: (V_b ÷ V_c)(6% − 6%) = 0.

Calculating returns after fees: management, incentive, hurdle and high-water mark

Management fee Use it
F=m×AUMF = m \times \text{AUM}
Quick example

A fund with USD 400 million of assets charges 1.5% on the AUM stated in its terms. Fee = 400 × 0.015 = USD 6.0 million. Use start or end of period AUM as the terms say.

On your calculator
  1. 400×0.015= 6
  1. 400ENTER0.015× 6

ENTER enter (on the HP, puts the number on the stack)

Total fees, calculated independently Use it
RGP=P1 m+max⁡[0,  (P1−P0) p]R_{GP} = P_1\,m + \max\left[0,\; (P_1 - P_0)\,p\right]
Quick example

Start 100, end 112 before fees, management fee 2% of year-end value, incentive fee 20% of the gain. 112 × 0.02 + 0.20 × 12 = 2.24 + 2.40 = 4.64.

On your calculator
  1. 112×0.02=STO1 2.24
  2. 112−100=×0.2=STO2 2.4
  3. RCL1+RCL2= 4.64

STO store · RCL recall a stored number

  1. 112ENTER0.02×STO1 2.24
  2. 112ENTER100−0.2×STO2 2.4
  3. RCL1RCL2+ 4.64

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Investor net return Use it
ri=P1−P0−RGPP0r_i = \dfrac{P_1 - P_0 - R_{GP}}{P_0}
Quick example

Gain 12 on a start of 100, total fees 4.64. (12 − 4.64) ÷ 100 = 7.36%. Divide by the starting capital, not the ending value.

On your calculator
  1. 112−100−4.64=÷100= 0.0736
  1. 112ENTER100−4.64−100÷ 0.0736

ENTER enter (on the HP, puts the number on the stack)

Total fees, incentive fee net of the management fee Use it
RGP=P1 m+max⁡{0,  [P1(1−m)−P0]p}R_{GP} = P_1\,m + \max\left\{0,\; \left[P_1(1 - m) - P_0\right] p\right\}
Quick example

Start 500, end 560, management fee 1% of year-end value, incentive fee 20% net of it. Incentive = 0.20 × (560 × 0.99 − 500) = 0.20 × 54.4 = 10.88. Fees = 5.60 + 10.88 = 16.48, so the investor earns (60 − 16.48) ÷ 500 = 8.70%.

On your calculator
  1. 560×0.01=STO1 5.6
  2. 560×0.99−500=×0.2=STO2 10.88
  3. 560−500−RCL1−RCL2=÷500= 0.087

STO store · RCL recall a stored number

  1. 560ENTER0.01×STO1 5.6
  2. 560ENTER0.99×500−0.2×STO2 10.88
  3. 560ENTER500−RCL1−RCL2−500÷ 0.087

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Total fees with a hard hurdle Use it
RGP=P1 m+max⁡{0,  [P1(1−m)−P0(1+h)]p}R_{GP} = P_1\,m + \max\left\{0,\; \left[P_1(1 - m) - P_0(1 + h)\right] p\right\}
Quick example

Start 120, end 146, management fee 1.5% of year-end value, incentive fee 20% net of it, hurdle 4%. Incentive = 0.20 × (146 × 0.985 − 120 × 1.04) = 0.20 × 19.01 = 3.802. Fees = 2.19 + 3.802 = 5.992, so the investor earns (26 − 5.992) ÷ 120 = 16.67%.

On your calculator
  1. 146×0.015=STO1 2.19
  2. 120×1.04=STO3 124.8, the value needed to clear the hurdle
  3. 146×0.985−RCL3=×0.2=STO2 3.802
  4. 146−120−RCL1−RCL2=÷120= 0.1667

STO store · RCL recall a stored number

  1. 146ENTER0.015×STO1 2.19
  2. 120ENTER1.04×STO3 124.8, the value needed to clear the hurdle
  3. 146ENTER0.985×RCL3−0.2×STO2 3.802
  4. 146ENTER120−RCL1−RCL2−120÷ 0.1667

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Total fees with a high-water mark Use it
RGP=Pt m+max⁡{0,  [Pt(1−m)−H]p}R_{GP} = P_t\,m + \max\left\{0,\; \left[P_t(1 - m) - H\right] p\right\}
Quick example

High-water mark 300, value before fees 310, management fee 1% of year-end value, incentive fee 20% net of it. Incentive = 0.20 × (310 × 0.99 − 300) = 0.20 × 6.9 = 1.38. Below 300 the incentive fee would be zero.

On your calculator
  1. 310×0.99−300=×0.2= 1.38
  1. 310ENTER0.99×300−0.2× 1.38

ENTER enter (on the HP, puts the number on the stack)

Fund of funds fees, survivorship bias and backfill bias

Return after the hedge fund layer Use it
r1=r−m1−p1 rr_1 = r - m_1 - p_1\,r
Quick example

Hedge funds earn 12% gross and charge a 1% management fee on beginning capital plus 20% of the return, calculated independently. 12 − 1 − 0.20 × 12 = 8.6%.

On your calculator
  1. 12×0.8−1= 8.6
  1. 12ENTER0.8×1− 8.6

ENTER enter (on the HP, puts the number on the stack)

Investor return through a fund of funds Use it
ri=r1−m2−p2 r1r_i = r_1 - m_2 - p_2\,r_1
Quick example

The fund of funds receives 8.6% and charges 0.5% plus 5% of that return, calculated independently. 8.6 − 0.5 − 0.05 × 8.6 = 7.67%. Charge the second incentive fee on 8.6, not on the gross 12.

On your calculator
  1. 8.6×0.95−0.5= 7.67
  1. 8.6ENTER0.95×0.5− 7.67

ENTER enter (on the HP, puts the number on the stack)

Fund of funds fees on the combined value Use it
RFoF=V1 m2+max⁡[0,  (V1−V0) p2]R_{FoF} = V_1\,m_2 + \max\left[0,\; (V_1 - V_0)\,p_2\right]
Quick example

Invest 70 in total; the funds are worth 82.6 after their own fees. Fund of funds charges 1% of year-end value and 10% of the gain. Fees = 0.826 + 1.26 = 2.086, so the investor earns (12.6 − 2.086) ÷ 70 = 15.02%.

On your calculator
  1. 82.6×0.01=STO1 0.826
  2. 82.6−70=×0.1=STO2 1.26
  3. 82.6−70−RCL1−RCL2=÷70= 0.1502

STO store · RCL recall a stored number

  1. 82.6ENTER0.01×STO1 0.826
  2. 82.6ENTER70−0.1×STO2 1.26
  3. 82.6ENTER70−RCL1−RCL2−70÷ 0.1502

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Module 3 Private Capital: Private Equity and Private Debt

Private equity strategies: buyouts, venture capital and growth equity

Equity multiple of a buyout Use it
M=PT−DTP0 (1−w)M = \dfrac{P_T - D_T}{P_0\,(1 - w)}
Quick example

A fund buys a company for 320 with 65% debt and sells it for 450 with 150 of debt still outstanding. Equity in: 320 × 0.35 = 112. Equity out: 450 − 150 = 300. M = 300 ÷ 112 = 2.68×.

On your calculator
  1. 320×0.35=STO1 equity invested, 112
  2. 450−150=÷RCL1= 2.6786

STO store · RCL recall a stored number

  1. 450ENTER150−320ENTER0.35×÷ 2.6786

ENTER enter (on the HP, puts the number on the stack)

Yearly return on buyout equity Use it
r=M1/T−1r = M^{1/T} - 1
Quick example

An equity multiple of 2.68× earned over 4 years with no interim cash flows. 2.6786^(1/4) − 1 = 27.93% a year.

On your calculator
  1. 2.6786y^x(1÷4)=−1= 0.2793

y^x power

  1. 2.6786ENTER41/xy^x1− 0.2793

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Private debt: direct lending, mezzanine, venture and distressed debt

Floating-rate coupon Use it
Coupon=Reference rate+Spread\text{Coupon} = \text{Reference rate} + \text{Spread}
Quick example

A private loan pays SOFR + 525 bps. SOFR is 3.20%. The coupon is 3.20% + 5.25% = 8.45%.

On your calculator
  1. 3.2+5.25= 8.45
  1. 3.2ENTER5.25+ 8.45

ENTER enter (on the HP, puts the number on the stack)

Blended unitranche spread (illustration) Know the idea
u=w s+(1−w) ju = w\,s + (1 - w)\,j
Quick example

A senior secured spread of 3.5% at weight 60% and an unsecured spread of 7.5% at weight 40% blend to 0.6 × 3.5 + 0.4 × 7.5 = 5.1% over SOFR, between the two.

On your calculator
  1. (0.6×3.5)+(0.4×7.5)= 5.1
  1. 0.6ENTER3.5×0.4ENTER7.5×+ 5.1

ENTER enter (on the HP, puts the number on the stack)

Loan to value Use it
LTV=LV\text{LTV} = \dfrac{L}{V}
Quick example

A loan of 38 is secured on property valued at 50. LTV = 38 ÷ 50 = 76%.

On your calculator
  1. 38÷50= 0.76
  1. 38ENTER50÷ 0.76

ENTER enter (on the HP, puts the number on the stack)

Module 4 Real Estate and Infrastructure

Real estate: features and ways to invest (direct, REITs, funds)

Funds from operations (FFO) Use it
FFO=Net income+Depreciation−Gains on property sales\text{FFO} = \text{Net income} + \text{Depreciation} - \text{Gains on property sales}
Quick example

A REIT reports net income of 18.0, depreciation of 9.5 and a gain on a property sale of 2.0 (EUR m). FFO = 18.0 + 9.5 − 2.0 = 25.5.

On your calculator
  1. 18+9.5−2= 25.5
  1. 18ENTER9.5+2− 25.5

ENTER enter (on the HP, puts the number on the stack)

Real estate returns: core to opportunistic, leverage and diversification

Total return on a property (unlevered) Use it
R=NOIP0+P1−P0P0R = \dfrac{NOI}{P_0} + \dfrac{P_1 - P_0}{P_0}
Quick example

A building bought at 60 earns net operating income of 3.6 and is valued at 63 a year later. Income 3.6 ÷ 60 = 6.0% plus appreciation 3 ÷ 60 = 5.0% gives 11.0%.

On your calculator
  1. 3.6+63−60=÷60= 0.11
  1. 3.6ENTER63+60−60÷ 0.11

ENTER enter (on the HP, puts the number on the stack)

Levered return on equity Use it
RE=NOI−i D+ΔVP0−DR_E = \dfrac{NOI - i\,D + \Delta V}{P_0 - D}
Quick example

Price 60, NOI 3.6, value up 5% (3.0), debt 30 at 4% (interest 1.2), equity 30. (3.6 − 1.2 + 3.0) ÷ 30 = 18.0%, against 11.0% unlevered.

On your calculator
  1. 3.6−(30×0.04)+(60×0.05)=÷30= 0.18
  1. 3.6ENTER30ENTER0.04×−60ENTER0.05×+30÷ 0.18

ENTER enter (on the HP, puts the number on the stack)

Loan to value Use it
LTV=LV\text{LTV} = \dfrac{L}{V}
Quick example

A loan of 33 is secured on property now valued at 44. LTV = 33 ÷ 44 = 75%.

On your calculator
  1. 33÷44= 0.75
  1. 33ENTER44÷ 0.75

ENTER enter (on the HP, puts the number on the stack)

Loan repayment that restores the covenant Use it
Paydown=L−LTVmax V\text{Paydown} = L - \text{LTV}_{max}\,V
Quick example

Loan 33, value 40, covenant cap 75%. Maximum loan = 0.75 × 40 = 30, so repay 33 − 30 = 3.0.

On your calculator
  1. 33−(0.75×40)= 3
  1. 33ENTER0.75ENTER40×− 3

ENTER enter (on the HP, puts the number on the stack)

Extra collateral that restores the covenant Use it
Extra=LLTVmax−V\text{Extra} = \dfrac{L}{\text{LTV}_{max}} - V
Quick example

Loan 33, value 40, cap 75%. The collateral must be worth 33 ÷ 0.75 = 44, so add 44 − 40 = 4.0.

On your calculator
  1. 33÷0.75−40= 4
  1. 33ENTER0.75÷40− 4

ENTER enter (on the HP, puts the number on the stack)

Infrastructure: economic vs social, greenfield vs brownfield

Take-or-pay payment Use it
Payment=max⁡(Qtaken, Qmin)×P\text{Payment} = \max(Q_{taken},\,Q_{min}) \times P
Quick example

A buyer must pay for at least 6.0 million units at 1.50 each. It takes only 4.5 million, so it still pays for 6.0: 6.0 × 1.50 = 9.0 million. Had it taken 7.5 million, it would pay 11.25 million.

On your calculator
  1. 6×1.5= 9
  1. 6ENTER1.5× 9

ENTER enter (on the HP, puts the number on the stack)

Usage-based revenue Know the idea
Revenue=Volume per day×Price×365\text{Revenue} = \text{Volume per day} \times \text{Price} \times 365
Quick example

A tunnel with 20,000 vehicles a day at a toll of 3.00 earns 20,000 × 3.00 × 365 = 21.9 million a year.

On your calculator
  1. 20000×3×365= 21,900,000
  1. 20000ENTER3×365× 21,900,000

ENTER enter (on the HP, puts the number on the stack)

Infrastructure risk, return and diversification

Fund target return (weighted average) Know the idea
Rfund=∑kwk RkR_{fund} = \sum_k w_k\,R_k
Quick example

A fund holds 50% in low-risk assets targeting 7%, 30% in medium-risk assets targeting 11% and 20% in high-risk assets targeting 15%. 0.5 × 7 + 0.3 × 11 + 0.2 × 15 = 9.8%.

On your calculator
  1. (0.5×7)+(0.3×11)+(0.2×15)= 9.8
  1. 0.5ENTER7×0.3ENTER11×+0.2ENTER15×+ 9.8

ENTER enter (on the HP, puts the number on the stack)

Purchasing power of a fixed payment Know the idea
V=N(1+π)TV = \dfrac{N}{(1 + \pi)^{T}}
Quick example

A fixed fee of 50 with no inflation link, 2% inflation, 15 years out. 50 ÷ 1.02^15 = 37.15 in today's money, so the fee has lost about a quarter of its value.

On your calculator
  1. 1.02y^x15=1/x×50= 37.15

y^x power · 1/x reciprocal

  1. 50ENTER1.02ENTER15y^x÷ 37.15

ENTER enter (on the HP, puts the number on the stack) · y^x power

Module 5 Natural Resources: Land, Timber, Farmland and Commodities

Raw land, timberland and farmland as investments

Present value of waiting to harvest Use it
PVwait=Q (1+g) P11+kPV_{wait} = \dfrac{Q\,(1 + g)\,P_1}{1 + k}
Quick example

Standing timber of 6,000 units, volume growth 4%, price now 300 and 340 expected next year, opportunity cost 8%. 6,000 × 1.04 × 340 ÷ 1.08 = 1,964,444, against 1,800,000 from cutting now, so waiting pays.

On your calculator
  1. 6000×1.04×340÷1.08= 1964444.44
  1. 6000ENTER1.04×340×1.08÷ 1964444.44

ENTER enter (on the HP, puts the number on the stack)

Break-even future price for waiting Know the idea
P1∗=P0 (1+k)1+gP_1^{*} = \dfrac{P_0\,(1 + k)}{1 + g}
Quick example

Price now 300, growth 4%, opportunity cost 8%. 300 × 1.08 ÷ 1.04 = 311.54. Waiting pays only if next year’s price beats this.

On your calculator
  1. 300×1.08÷1.04= 311.5385
  1. 300ENTER1.08×1.04÷ 311.5385

ENTER enter (on the HP, puts the number on the stack)

Return from waiting to harvest Know the idea
Rwait=(1+g) P1P0−1R_{wait} = \dfrac{(1 + g)\,P_1}{P_0} - 1
Quick example

Growth 4%, price 300 now and 340 next year. 1.04 × 340 ÷ 300 − 1 = 17.87%. Waiting beats any opportunity cost below this.

On your calculator
  1. 1.04×340÷300−1= 0.1787
  1. 1.04ENTER340×300÷1− 0.1787

ENTER enter (on the HP, puts the number on the stack)

Commodities: sectors, ways to invest, contango and backwardation

Commodity forward price with convenience yield Memorize
F0(T)=S0 e(r+c−i)TF_0(T) = S_0\,e^{(r + c - i)T}
Quick example

Spot 1,600, risk-free rate 3%, carry cost 1%, convenience yield 0.5%, two years. 1,600 e(0.03+0.01−0.005)×21{,}600\,e^{(0.03 + 0.01 - 0.005)\times 2} = 1,600 × e^0.07 = 1,716.01.

On your calculator
  1. 0.03+0.01−0.005=×2=2NDe^x×1600= 1716.01

2ND second function (the shifted label above a key) · e^x e to the power x

  1. 0.03ENTER0.01+0.005−2×ge^x1600× 1716.01

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Implied convenience yield Use it
i=r+c−1Tln⁡ ⁣(F0S0)i = r + c - \dfrac{1}{T}\ln\!\left(\dfrac{F_0}{S_0}\right)
Quick example

Spot 1,600, one-year futures 1,630, r = 3%, c = 1%. ln(1,630 ÷ 1,600) = 1.858%, so i = 4% − 1.86% = 2.14%. Futures above spot, so contango, because i is below r + c = 4%.

On your calculator
  1. 1630÷1600=LN+|-+0.03+0.01= 0.0214

LN natural logarithm · +|- change sign

  1. 0.03ENTER0.01+1630ENTER1600÷gLN− 0.0214

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · LN natural logarithm

Curve shape rule Memorize
i<r+c⇒contango;i>r+c⇒backwardationi < r + c \Rightarrow \text{contango};\quad i > r + c \Rightarrow \text{backwardation}
Quick example

r = 3.5%, c = 1%, so the cut-off is 4.5%. A convenience yield of 3% gives contango; 6% gives backwardation; exactly 4.5% gives a flat curve.

Natural resources: risk, return, inflation hedging and diversification

Two-asset portfolio standard deviation Use it
σp=w12σ12+w22σ22+2 w1w2 ρ σ1σ2\sigma_p = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2\,w_1 w_2\,\rho\,\sigma_1\sigma_2}
Quick example

Weights 75% and 25%, standard deviations 18% and 28%, correlation 0.30. Variance = 0.5625 × 324 + 0.0625 × 784 + 2 × 0.75 × 0.25 × 0.3 × 18 × 28 = 182.25 + 49 + 56.7 = 287.95, so σ = 16.97%.

On your calculator
  1. 0.75x²×18x²+(0.25x²×28x²)+(2×0.75×0.25×0.3×18×28)=√x 16.97

x² square · √x square root

  1. 0.75ENTER×18ENTER××0.25ENTER×28ENTER××+2ENTER0.75×0.25×0.3×18×28×+g√x 16.97

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Module 6 Hedge Funds

What a hedge fund is, and how it differs from other funds

Incentive fee with a high-water mark Use it
F=p×max⁡[0,  V1−H]F = p \times \max\left[0,\; V_1 - H\right]
Quick example

The fund last paid a fee when its value was 80, so H=80H = 80. It now stands at 88 and the incentive fee is 20%. F=0.20×(88−80)=F = 0.20 \times (88 - 80) = 1.6, even if the year began at 70.

On your calculator
  1. 88−80=×0.2= 1.6
  1. 88ENTER80−0.2× 1.6

ENTER enter (on the HP, puts the number on the stack)

Hedge fund strategies: equity, event-driven, relative value, opportunistic

Merger arbitrage spread (cash deal) Use it
S=O−PPS = \dfrac{O - P}{P}
Quick example

A target trades at 90 after a cash offer of 96. S=(96−90)÷90=S = (96 - 90) \div 90 = 6.67% if the deal closes.

On your calculator
  1. 96−90=÷90= 0.0667
  1. 96ENTER90−90÷ 0.0667

ENTER enter (on the HP, puts the number on the stack)

Annualized spread Use it
A=(1+S)12/T−1A = (1 + S)^{12/T} - 1
Quick example

The 6.67% spread is earned in 4 months (T=4T = 4). A=1.06673−1=A = 1.0667^{3} - 1 = 21.36%.

On your calculator
  1. 96÷90=y^x3=−1= 0.2136

y^x power

  1. 96ENTER90÷3y^x1− 0.2136

ENTER enter (on the HP, puts the number on the stack) · y^x power

Stock-deal spread Use it
D=e PA−PTD = e\,P_A - P_T
Quick example

Each target share is exchanged for 0.8 acquirer shares (e=0.8e = 0.8). The acquirer trades at 50 and the target at 37. D=0.8×50−37=D = 0.8 \times 50 - 37 = 3.00, or 8.1% of 37. Buy one target share and short 0.8 acquirer shares.

On your calculator
  1. 0.8×50−37= 3
  1. 0.8ENTER50×37− 3

ENTER enter (on the HP, puts the number on the stack)

Hedge fund returns: market beta, strategy beta, alpha and diversification

Hedge fund alpha Use it
α=R−βRM−S\alpha = R - \beta R_M - S
Quick example

A fund returns 15%. Its beta to equities is 0.4, equities returned 12%, and strategy beta explains 4%. α=15−0.4×12−4=\alpha = 15 - 0.4 \times 12 - 4 = 6.2%.

On your calculator
  1. 0.4×12=STO1 4.8
  2. 15−RCL1−4= 6.2

STO store · RCL recall a stored number

  1. 15ENTER0.4ENTER12×−4− 6.2

ENTER enter (on the HP, puts the number on the stack)

Alpha after fees Use it
αnet=α−m−p R\alpha_{net} = \alpha - m - p\,R
Quick example

Alpha before fees is 7%, the fund returned 13%, and fees are a 1% management fee plus 20% of the return, calculated independently. αnet=7−1−0.20×13=\alpha_{net} = 7 - 1 - 0.20 \times 13 = 3.4%.

On your calculator
  1. 13×0.2=STO1 2.6
  2. 7−1−RCL1= 3.4

STO store · RCL recall a stored number

  1. 13ENTER0.2×STO1 2.6
  2. 7ENTER1−RCL1− 3.4

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Module 7 Digital Assets

Ways to invest in digital assets: direct and indirect

Coin trust premium or discount Use it
Premium or discount=P−NAVNAV\text{Premium or discount} = \frac{P - NAV}{NAV}
Quick example

A trust holds crypto worth 18.50 per share and its shares trade at 20.35. (20.35 − 18.50) ÷ 18.50 = +10.0%, a premium. A negative result is a discount.

On your calculator
  1. 20.35−18.5=÷18.5= 0.1, a 10% premium
  1. 20.35ENTER18.5−18.5÷ 0.1, a 10% premium

ENTER enter (on the HP, puts the number on the stack)

Digital assets: risk, return and diversification

Coefficient of variation Use it
CV=sRˉCV = \frac{s}{\bar{R}}
Quick example

A coin has a mean monthly return of 3.2% and a standard deviation of 24%. CV = 24 ÷ 3.2 = 7.5 units of risk per unit of return.

On your calculator
  1. 24÷3.2= 7.5
  1. 24ENTER3.2÷ 7.5

ENTER enter (on the HP, puts the number on the stack)

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