Reading 77
MODULE 77.1: PERFORMANCE APPRAISAL AND RETURN CALCULATIONS
Describe the performance appraisal of alternative investments.
Alternative investments are typically exposed to greater risks than unleveraged long-only traditional investments. These additional risks arise from:
- Timing of cash flows over an investment's life cycle
- Use of leverage by fund managers
- Valuation of investments that may or may not have observable market prices
- Complexity of fees, taxes, and accounting
Ideally, returns should be adjusted for these risks, although that may be difficult in practice. Evaluating alternative investment returns (or expected returns) without considering these additional risks would be naive and possibly misleading.
另類投資相對於無槓桿、僅做多的傳統投資承擔更高風險，主要來自四個層面：
- 投資生命週期中現金流的時間分布不均
- 基金經理人使用槓桿放大部位
- 投資標的可能缺乏可觀察市價，估值依賴假設
- 費用、稅負與會計結構複雜
理想上應對這些風險作調整後再評估報酬，但實務上不易執行。若忽略這些風險直接看報酬數字，將會低估風險並產生誤導。
Timing of Cash Flows
Alternative investments often have a life cycle exhibiting three phases:
- Phase 1:Capital commitment phase. Managers identify investments and make capital calls from limited partners. LPs commit a stated amount but do not deliver the entire amount immediately. Because of cash outflows and the long-term nature of investments, returns tend to be negative.
- Phase 2:Capital deployment phase. Managers fund and often involve themselves directly in the firms or projects. Returns typically remain negative, especially if invested in start-ups or troubled firms being turned around.
- Phase 3:Capital distribution phase. If investments succeed and generate income, returns turn positive and accelerate.
A J-curve effect reflects the norm of negative returns in the commitment phase, increasing returns during deployment, and maximum returns during distribution. Returns may plateau toward the end of a fund's life as managers exit remaining investments.
Given the variability of cash flows and the importance of management timing decisions, an IRR (money-weighted rate of return) over the life of a fund is the most appropriate measure of after-tax investment performance — appropriate when a manager controls the timing of cash inflows and outflows. A drawback is that IRR rests on assumptions about the cost of capital and reinvestment rate.
A simpler measure is the multiple of invested capital (money multiple) — the ratio of total capital returned plus the value of any remaining assets, to the total capital paid in. Because it does not consider timing, it is somewhat naive.
另類投資生命週期常呈現三階段：
- 資本承諾期（Capital commitment）：經理人尋找投資標的，依需要向 LP 進行 capital call。由於現金流出且尚未產生收益，報酬通常為負。
- 資本部署期（Capital deployment）：經理人投入資金並直接參與被投資企業。對新創或翻轉中的公司，報酬仍多為負。
- 資本分配期（Capital distribution）：投資若成功，現金流回流，報酬轉正並加速成長。
三階段組合形成J 曲線（J-curve）：先負後正，最後在退場前報酬趨平。
由於現金流時點由經理人主導，最適合衡量基金績效的指標是IRR（貨幣加權報酬率）。缺點是 IRR 對資金成本與再投資率有隱含假設。
較簡單的指標是已投入資本倍數（money multiple）＝（已分配金額＋剩餘資產價值）／累計投入資本。但此指標不考慮時間因素，相對粗略。
Use of Leverage
Some alternative investments, particularly hedge funds, use borrowing to magnify gains (at the risk of magnifying losses). Hedge funds may arrange margin financing with prime brokers or employ leverage via derivatives.
Consider a fund that can invest the amount $V_0$ without leverage and earn return $r$. The unleveraged dollar return is simply $r \times V_0$. Now suppose the fund borrows $V_B$ at interest rate $r_B$ and earns $r$ on the proceeds. The leveraged dollar return, after subtracting interest cost, is:
$$r \times (V_0 + V_B) - (r_B \times V_B)$$
Stated as a rate of return on the initial portfolio value $V_0$, the leveraged rate of return is:
$$\text{leveraged return} = \frac{r(V_0 + V_B) - r_B V_B}{V_0}$$
Funds use leverage because some strategies attempt to exploit relatively small pricing anomalies that would not produce meaningful results without leverage. Risks include:
- Margin calls — a lender may issue margin calls if equity falls below a level, forcing the fund to realize losses by closing positions or liquidating at unfavorable prices. Selling a large position may further depress its price.
- Reduced access to borrowing — lenders may limit additional borrowing capacity.
部分另類投資（尤其避險基金）使用借款放大獲利，但同時放大損失。避險基金可向 prime broker 取得保證金融資，或透過衍生品變相加槓桿。
設無槓桿可投資金額為 $V_0$、報酬率為 $r$；若再借入 $V_B$（利率 $r_B$）並按 $r$ 投資。槓桿後的金額報酬：
$$r(V_0+V_B)-r_B V_B$$
以初始本金 $V_0$ 計算的槓桿報酬率：
$$\text{槓桿報酬率}=\frac{r(V_0+V_B)-r_B V_B}{V_0}$$
使用槓桿的原因：許多策略利用的價差很小，不加槓桿難以獲取顯著報酬。風險：①保證金追繳（margin call）可能被迫於不利價位平倉，大部位拋售更會壓低價格；②貸方可能限制可動用的額外借款。
Valuation of Investments
Many alternative investments involve illiquid assets that do not trade frequently in transparent markets. While funds must recognize investments at fair value for accounting purposes, fair value may rest on assumptions. A fair value hierarchy groups these into three levels:
| Level | Description | Examples |
|---|---|---|
| Level 1 | Assets trade in active markets with quoted prices readily available. | Exchange-traded securities |
| Level 2 | No quoted prices, but valued based on directly or indirectly observable inputs. | Many derivatives priced via models |
| Level 3 | Require unobservable inputs to establish fair value. | Real estate, private equity (few/no transactions) |
Particularly for Level 3 investments, the absence of market activity can result in valuations remaining near initial cost for long periods. As a result, those values might not reflect the actual exit costs. Importantly, this lack of change can make reported returns appear higher, less risky, and less correlated with traditional investments than they really are.
許多另類投資為缺乏透明市場的非流動資產。雖然會計上仍以公允價值（fair value）列報，但其背後的估值假設應被理解。公允價值依輸入可觀察程度分三級：
- Level 1：活躍市場有報價（如交易所掛牌證券）。
- Level 2：無直接報價，但可用可觀察輸入（直接或間接）作模型估值（如多數衍生品）。
- Level 3：須使用不可觀察輸入，例如不動產與私募股權（市場交易稀少）。
Level 3 資產長時間維持接近成本價，未必反映實際退場價格。此「估值黏滯」現象會讓另類投資的報酬看起來較高、波動較小、與傳統資產相關性偏低，但這只是統計假象。
Fee Structures
Fee specifications such as high-water marks, hard or soft hurdle rates, and waterfall structures all affect investor returns. Fee structures are subject to negotiation (e.g., a limited partner might agree to a higher management fee in exchange for fewer redemption restrictions) and may differ depending on how early in a fund's life cycle an investor commits. Different investors in the same fund might therefore realize significantly different returns.
費用結構如高水位線（high-water mark）、硬／軟門檻報酬率（hurdle rate）、瀑布分配（waterfall）等都會影響投資人實得報酬。費用條款可協商（例如以更高管理費換取更寬鬆的贖回條件），且越早投入的投資人通常條件越優；因此同一支基金的不同投資人實得報酬可能差距很大。
Calculate and interpret alternative investment returns both before and after fees.
Just as margin calls may force a leveraged fund to exit at unfavorable times, similar risks arise from investor redemptions. The more negative a fund's returns, the more likely investors are to redeem. To restrict early redemptions, alternative investment funds (particularly hedge funds) typically take measures such as:
- Lockup period — time after initial investment over which LPs cannot request redemptions or incur significant fees for redemptions.
- Notice period (typically 30 to 90 days) — the amount of time a fund has to fulfill a redemption request, allowing managers to reduce positions in an orderly manner.
- Redemption fees — offset transaction costs incurred when shares are redeemed.
- Gate — manager discretion to restrict redemptions for a temporary period.
Other investors may receive terms that differ from those stated in the partnership agreement. Customized fee structures are contained in side letters with individual investors detailing how their terms differ from the standard offering documents.
Although "2 and 20" and "1 and 10" (for fund-of-funds) were once standard, fees remain under competitive pressure. Larger commitments can negotiate lower fees. There can also be a tradeoff between liquidity provisions and fees — investors can negotiate for lower fees or shorter lockup/notice periods. Hurdle rates, hard versus soft hurdles, and catch-up provisions may also be negotiable.
Early investors may receive lower fees or better liquidity as an incentive to invest at the fund's inception. These investment interests are called founders class shares.
Annual investor fees can also be either-or fees — the maximum of the management fee or the incentive fee. Under such a structure, with a 1% management fee and a 30% incentive fee, fees each year would simply be the management fee unless the calculated incentive fee is higher. Such a structure may also stipulate that the 1% management fee be subtracted from the incentive fee in a subsequent year.
類似槓桿基金被迫平倉，投資人贖回（redemption）也是壓力來源；表現越差，贖回壓力越大。常見的贖回限制工具：
- 鎖定期（lockup period）：投資後一段期間不得贖回，或贖回需付高額費用。
- 通知期（notice period，30–90 天）：基金有此期間以有序方式減倉、滿足贖回請求。
- 贖回費（redemption fee）：抵銷因贖回產生的交易成本。
- 閘門（gate）：經理人可暫時限制贖回比例。
不同投資人可透過側函（side letter）取得異於標準募集文件的條件。早期進場的投資人常以較低費率或更佳流動性作為誘因，稱為創始份額（founders class shares）。
「2 and 20」（基金）與「1 and 10」（FoF）已不再是固定範本，承諾額大者可議價，亦可在費率與流動性之間取捨。
Either-or 費用結構：每年取管理費與績效費中較高者；後續年度可能還需從績效費中扣回前期收取的管理費。
Biases in Alternative Investment Returns
Numerous alternative investment indexes exist, but they may not provide much meaningful information because each fund's structure is unique and funds can be in widely different phases of their life cycles. One way around the latter issue is to compare funds that originated in the same vintage year.
Survivorship bias is greater for hedge fund databases — by some estimates more than 25% of hedge funds fail in the first three years. Indexes that exclude failed funds will overstate returns and understate risk. These effects may be magnified by backfill bias, which occurs when managers only select their successful funds for inclusion in indexes.
另類投資指數雖多，但每檔基金結構獨特、處於不同生命週期階段，彙總指數的可比性不高。針對後者，可以僅比較相同創立年（vintage year）的基金。
避險基金資料庫的存活者偏誤（survivorship bias）特別嚴重——估計超過 25% 的避險基金在頭三年內收場。剔除失敗基金的指數會高估報酬、低估風險。回填偏誤（backfill bias）則進一步放大這個效應：經理人只把表現好的基金回補加入指數。
Return Calculations for Alternative Investments
Before-fee returns are calculated the same way as any other investment (holding period returns and periodic rates of return, with or without interim cash distributions).
For after-fee returns, adjust cash flows or values for management and performance fees. For a simple case in which management fees are a fixed percentage of end-of-period assets and performance fees are a fixed percentage of total return with no hurdle rate, total fees in money terms are:
$$\text{total fees} = m V_1 + \max[0,\; p(V_1 - V_0)]$$
where $m$ = management fee, $p$ = performance fee, $V_0$ = beginning-of-period assets, $V_1$ = end-of-period assets.
The investor's after-fee rate of return is:
$$r = \frac{V_1 - V_0 - \text{total fees}}{V_0}$$
Hurdle rates and high-water marks make calculation more complex. Other provisions specify whether the performance fee is net of the management fee, and whether the management fee is based on beginning-of-period or end-of-period assets.
稅前／費前報酬的計算與一般投資相同（持有期報酬、期間報酬，含或不含期中分配）。
費後報酬則需把管理費與績效費從現金流或期末值中扣除。當管理費按期末資產比例計、績效費按報酬比例計（無門檻），總費用：
$$\text{總費用}=mV_1+\max[0,\,p(V_1-V_0)]$$
投資人費後報酬率：
$$r=\frac{V_1-V_0-\text{總費用}}{V_0}$$
若加入門檻（hurdle）與高水位線（HWM）規則，計算會更複雜。其他條款還包括：績效費是否從管理費後的淨報酬計、管理費基底為期初或期末。
教授提醒：另類投資費用條款並無業界統一標準，凡是要計算的考題都會明示所有條件。題目沒寫的條款，不要自行假設。
BJI Funds is a hedge fund with a value of $110 million at initiation. BJI charges a 2% management fee based on assets under management at the beginning of the year and a 20% performance fee with a 5% soft hurdle rate, and uses a high-water mark. Performance fees are calculated on gains net of management fees. Year-end values before fees:
- Year 1: $100.2 million
- Year 2: $119.0 million
Calculate the total fees and the investor's after-fee return for both years.
Management fee: $110.0\text{M} \times 2\% = \$2.2\text{M}$
Return net of management fee: $\dfrac{100.2-2.2}{110.0}-1 = -10.9\%$
No performance fee — return is below the 5% soft hurdle.
Total fees: $2.2 million. Ending value net of fees: $100.2 - 2.2 = \$98.0\text{M}$. Year 1 after-fee return: −10.9%.
Management fee: $98.0\text{M} \times 2\% = \$1.96\text{M}$
Year-end value net of management fee: $119.0 - 1.96 = \$117.04\text{M}$
High-water mark: $110\text{M}$. Value above HWM: $117.04 - 110.0 = \$7.04\text{M}$.
Return net of management fee, above HWM: $\dfrac{7.04}{110.0}=6.4\%$ > 5% soft hurdle, so performance fee applies on the entire gain above HWM (with a hard hurdle the fee would only apply on gains above 5% of HWM).
Performance fee: $7.04 \times 0.20 = \$1.41\text{M}$
Total fees: $1.96 + 1.41 = \$3.37\text{M}$. Year-end value after fees: $119.0 - 3.37 = \$115.63\text{M}$.
Year 2 after-fee return: $\dfrac{115.63}{98.0}-1 = 18.0\%$
BJI Funds 起始規模 $110M$，2% 管理費（按年初 AUM）、20% 績效費（軟門檻 5%、設高水位線、績效費以扣管理費後淨額計）。年末費前淨值：Y1 $100.2M$、Y2 $119.0M$。
Y1：管理費 $110M\times2\%=\$2.2M$；扣管理費後報酬 $\frac{100.2-2.2}{110}-1=-10.9\%$，未達 5% 軟門檻 → 無績效費。總費用 $2.2M$，Y1 費後報酬 −10.9%，期末 $98M$。
Y2：管理費 $98M\times2\%=1.96M$；扣管理費後 $117.04M$。HWM $=110M$，超過 HWM 部分 $7.04M$。$\frac{7.04}{110}=6.4\%$ > 5%（軟門檻 → 全部超 HWM 的部分都收績效費；若為硬門檻，只對超過 5% 的部分收費）。
績效費 $7.04\times20\%=1.41M$；總費用 $1.96+1.41=3.37M$；期末費後 $115.63M$。Y2 費後報酬 = 115.63/98 − 1 = 18.0%。
An investor invests $60 million in a fund-of-funds with a "1 and 10" fee structure (management and performance fees calculated independently based on year-end values). $40M to Alpha fund and $20M to Beta fund. After one year, Alpha is worth $45M and Beta is worth $28M, both net of underlying fund fees. Calculate the investor's net-of-fees return for the year.
Year-end gross value: $45 + 28 = \$73\text{M}$.
FoF management fee: $1\% \times 73 = \$0.73\text{M}$.
Investor gain before FoF fees: $73 - 60 = \$13\text{M}$.
FoF performance fee: $10\% \times 13 = \$1.30\text{M}$.
Year-end value of FoF investment: $73 - 0.73 - 1.30 = \$70.97\text{M}$.
Investor one-year return after fees: $\dfrac{70.97}{60}-1 = 18.3\%$
Note: Direct investment in Alpha and Beta would have returned $\dfrac{73}{60}-1 = 21.7\%$.
投資人投入 $60M$ 至 FoF（1 and 10，管理費與績效費獨立計，皆按年末值）；其中 $40M$ 投 Alpha、$20M$ 投 Beta。一年後 Alpha 值 $45M$、Beta 值 $28M$（均為扣除底層基金費用後）。
FoF 期末總值 $73M$；FoF 管理費 $1\%\times73=0.73M$；FoF 績效費 $10\%\times(73-60)=1.30M$；期末扣費後 $70.97M$。投資人費後報酬 = 70.97/60 − 1 = 18.3%。若直接投 Alpha + Beta，報酬為 21.7%（差距即 FoF 二層費用拖累）。
A private equity fund invests $100M in a venture company sold for $130M. It also invests $100M in an LBO that goes poorly and is liquidated for $80M.
- If the GP's carried-interest performance fee is 20% with no clawback, calculate the investor's return after performance fees, assuming both outcomes are realized in the same year, under:
- (a) American-style (deal-by-deal) waterfall
- (b) European-style (whole-of-fund) waterfall
- How would the answers change if the venture deal was sold in Year 1 and the LBO was sold in Year 2?
- How would including a clawback provision affect the answers in Question 1?
(a) American-style: Performance fee on venture: $20\% \times (130-100) = \$6\text{M}$. No fee on LBO loss.
Investor return: $\dfrac{130 + 80 - 6}{200} - 1 = 2\%$
(b) European-style: Whole-of-fund gain $= 130+80-200 = \$10\text{M}$. Performance fee $= 20\% \times 10 = \$2\text{M}$.
Investor return: $\dfrac{130 + 80 - 2}{200} - 1 = 4\%$
The European-style waterfall would yield the same overall return as the American-style — performance fee of $6M paid on the venture in Year 1, and no fee on the LBO loss in Year 2.
With a clawback, the $6M paid on the venture is more than 20% of the total $10M net gain (it is 60%). The investor could "claw back" $4M, reducing the total performance fee to $2M (20% of $10M).
PE 基金投 $100M$ 入 venture，售得 $130M$；另投 $100M$ 入 LBO，僅清算回 $80M$。GP 績效費 20%。
Q1（同一年實現）：
- 美式瀑布（deal-by-deal）：venture 收 $20\%\times30=6M$，LBO 虧損不收。投資人報酬 $\frac{130+80-6}{200}-1=2\%$。
- 歐式瀑布（whole-of-fund）：整體獲利 $10M$，績效費 $20\%\times10=2M$。投資人報酬 $\frac{210-2}{200}-1=4\%$。
Q2（venture 在 Y1、LBO 在 Y2）：歐式瀑布結果與美式相同——Y1 對 venture 收 $6M$，Y2 對 LBO 損失不收。
Q3（加入 clawback）：美式情境下 GP 已先收 $6M$，但整體淨利只有 $10M$，績效費實佔 60%，遠超 20% 上限。投資人可追回 $4M$，使最終績效費回到 $2M$（即整體 $10M$ 的 20%）。
- A. distribution.
- B. deployment.
- C. commitment.
- A. 20.5%.
- B. 21.5%.
- C. 25.0%.
- A. hard hurdle rate.
- B. soft hurdle rate.
- C. catch-up provision.
Alternative investments are exposed to risks traditional investments are not: timing of cash flows, use of leverage, valuation assumptions for illiquid investments, and complex fee structures and taxation.
Returns typically exhibit a J-curve: negative during the capital commitment phase, less negative during deployment, positive during distribution.
Because managers control cash inflows and outflows, an IRR is appropriate for measuring returns. The multiple of invested capital is a simpler measure that does not depend on timing.
Leveraged return formula:
$$\text{leveraged return}=\frac{r(V_0+V_B)-r_B V_B}{V_0}$$
where $V_0$ = initial assets, $V_B$ = borrowed funds, $r$ = return on investments, $r_B$ = borrowing rate.
Fair value hierarchy: Level 1 (quoted prices), Level 2 (observable inputs), Level 3 (unobservable inputs). Level 3 valuations may be updated infrequently, biasing returns upward and risk measures downward.
Before-fee returns are calculated like any investment. After-fee returns adjust for management and performance fees, including hurdle rates (hard vs soft), high-water marks, and waterfall structures.
For a simple structure (management fee on end-of-period assets, no hurdle):
$$\text{total fees}=mV_1+\max[0,\,p(V_1-V_0)],\qquad r=\frac{V_1-V_0-\text{total fees}}{V_0}$$
Other negotiable items: founders class shares, either-or fees, side letters, lockup/notice periods, gates, redemption fees.
【LOS 77.a】另類投資特殊風險：①現金流時點；②槓桿；③非流動性資產估值假設；④複雜費用與稅負。報酬呈 J 曲線（承諾→部署→分配）。績效衡量首選 IRR；money multiple 簡單但忽略時間。槓桿報酬率 $=\frac{r(V_0+V_B)-r_B V_B}{V_0}$。公允價值三級：L1 報價、L2 可觀察輸入、L3 不可觀察輸入；L3 估值黏滯使報酬看似較高、風險較低、相關性較低。
【LOS 77.b】費前報酬同一般投資。費後報酬須扣管理費、績效費，並依規則套用 hurdle、HWM、瀑布等。簡單情境總費用 $=mV_1+\max[0,p(V_1-V_0)]$，費後報酬 $r=(V_1-V_0-\text{fees})/V_0$。其他可議項：founders class、either-or、side letter、鎖定期／通知期／閘門／贖回費等。