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Instruments and Markets 1. Asset Allocation a. How much of an wealth should be invested in each of the following broad areas i. Cash, equities, bonds, derivative securities ii. Private equity, hedge funds, commodities, and real estate 2. Investment strategies a. Passive management i. and a portfolio...

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Instruments and Markets 1. Asset Allocation a. How much of an investor’s wealth should be invested in each of the following broad areas i. Cash, equities, bonds, derivative securities ii. Private equity, hedge funds, commodities, and real estate 2. Investment strategies a. Passive management i. “buy and hold” a well-diversified portfolio of assets b. Active management i. Security selection attempts to identify securities that have been priced 1. Buy low/sell high ii. Market timing tilts the portfolio composition in favour of (away from) equities when the investor is “bullish” (“bearish”) about the stock market c. Portfolio insurance i. Use derivatives to “manage risk” 3. Financial instruments a. Financial security i. Legal contract ii. Confers the right to receive future benefits iii. Will focus on traded securities (not bank debt) that can be priced continuously iv. Usually traded in organised markets b. Classification i. Cash product vs. Derivative security ii. Debt vs. Equity c. Sub-classification of debt i. By issuer ii. By maturity 4. Money market securities a. Short-term (less than 1 year) debt b. Issued by Governments or Companies c. Examples include i. Treasury Bills ii. Repurchase Agreements (repos) iii. Certificates of Deposit iv. Commercial Paper 5. T-Bills a. Issued by the Government: no risk of default b. Short-term securities: little inflation risk c. Highly liquid (the transaction fee is low, easy to convert) d. Sold at a discount to face value: “zero-coupon” bonds 6. Repos(2017-1-a) a. Agreement to sell and re-purchase securities b. Collateralized debt, and provide cheap financing c. Sells the government securities to investors, usually on an overnight basis, and buy back the following day 7. Commercial Paper a. Issued by large companies as an alternative to borrowing directly from a bank b. Usually short-term ( 0 but a < 0 without having any skills: i. Buy index call options (more later; idea is calls give you exposure on the upside, protection on the downside; option cost leads to a < 0) ii. Or switch into equities when market goes up, back into bonds when it goes down 4. Multiple risk factors a. With different model for explaining returns (e.g. Fama French, or multi-asset CAPM) can readily generalise previous measures i. For example, Jensen’s alpha is simply the intercept on regressing the excess return on the portfolio on the excess return on the market ii. With multiple reference portfolios get:

b. Example: (Fama-French 3 factor model based Abnormal Returns) ARP = RP – {RF + b1[RM – RF]+b2 [SMB]+b3 [HML]} c. Use of inappropriate benchmark is dangerous i. Early academic work on mutual funds using S&P500 showed positive alpha, which was interpreted as showing manager skills ii. In fact, out-performance attributable to substantial exposure to small cap stocks that are under-represented in index and which out-performed d. Evidence of multiple priced factors (eg value v growth) i. If believe they represent sources of risk, should correct for them to get risk-adjusted return ii. If believe they simply represent higher returns, may still want to identify how much of performance is attributable to factor exposure and how much to security selection 5. Statistical significance a. Ex-post returns used to evaluate performance i. Do they reflect the true ability of Fund Manager to select stocks or to practise market timing? ii. Component of performance we cannot explain (“idiosyncratic risk”) has annual volatility of se iii. Over N years, average abnormal return has standard deviation of s e/N b. Suppose the manager has managed an average Jensen measure of J over the N years i. JN/se is a measure of the probability of the performance being due to luck ii. It is a t-statistic for the null hypothesis H0: performance due to luck iii. It is also the average appraisal ratio x N c. Implications i. Cannot normally expect to detect individual management skills purely from analysis of risk-adjusted performance ii. But can use in conjunction with declared philosophy to test consistency and plausibility iii. Can also use on populations of portfolios to ask questions about styles of investment, or persistence 6. Measurement and incentives a. Goodhart’s Law: i. “When a measure becomes a target, it ceases to be a good measure” (Strathern 1997) b. A measurement system that penalises risk will cause managers to take risks that are not measured i. Credit risk on bonds ii. Writing options iii. Illiquid or complex securities iv. Is this what is wanted?

Fixed income 1. Types of bond a. A bond is a security where the pay-out is pre-determined i. Defined face value or principal that is repaid at maturity ii. Defined stream of interest or coupon payments b. Issuer i. Generally sovereign or agency or corporate ii. May be guaranteed by parent or sponsor c. Interest generally fixed or floating (tied to some rate like LIBOR) d. Liquidity 2. Innovations in the Bond Market a. Growing importance of markets i. Many debt obligations are converted into traded bonds ii. Private risks are also being packaged into bonds b. Asset backed securities: i. Bank or building society lends money to company or individual ii. Bank then has promised stream of cash flows iii. Sells cash flows to a special vehicle that finances itself by issuing bonds iv. Most risks (default, pre-payment) borne by bond holders, though some retained (moral hazard) c. Other risk transfer i. Credit default swaps, catastrophe bonds, commodity bonds d. We will focus for the present on more traditional market, ignoring credit and liquidity issues for now 3. Bond definitions a. Bond: a security issued as promise to repay a loan i. The loan is paid off in a fixed number of payments. b. Face value: the amount of the loan to be repaid c. Coupon: a payment to the holder of the bond at a specified date. d. Coupon rate: the ratio of the coupon payment to the face value. e. Bond Indenture: the contract between the bondholder and the issuer. These are very complicated for corporate bonds. f. Zero-Coupon Bond: a bond that pays no coupons. g. Bond Valuation Formula:

C = Coupon Payment; F = Principal Payment; r = Discount Rate; n = maturity of the bond h. Price or Present Value of a Bond = PV of Coupons + PV of Final/Principal Payment. i. Because Coupons are regular (end of the period) payments, we can use the Present Value of Annuity equation to find the present value of Coupons

j.

Semi-Annual interest payments

i. When bonds pay semi-annual coupons

k. Bond premiums and discounts

l.

m.

n.

o.

p.

Bond quotes i. Price you pay is quoted price plus accrued interest – the share of the coupon you will receive at the next coupon date attributable to the period before you owned it ii. Why bother with clean price? 1. If the dirty price is what you have to pay, why bother with quoting a clean price? 2. Suppose interest rate is 8%, then bond would be worth $100 immediately after a coupon payment 3. It is worth $104 immediately before the coupon is paid 4. It is worth $104/(1.04D/180)D days before the coupon is paid 5. Quoting clean prices makes it easier to compare bonds with different coupons and coupon dates Interest yield i. Interest yield is computed by dividing interest due by clean price 1. With 8% bond, interest yield is 8/96.50 or 8.29% ii. But bond is trading below par (under 100) 1. So holder to maturity will receive capital gain of 3.50 over three years 2. Appreciation amounts to about 1.20%/yr (ie (3.50/96.50)/3) 3. Total return is about 9.49% (ie 8.29% + 1.20%) Yield to maturity(2017-4-b) i. The yield on a bond (redemption yield, yield to maturity) is the discount rate that makes the present value of the bond equal to its (dirty) price The trade based on YTM(2017-4-c) i. The appropriate yield to maturity, y*, for a bond depends on how risky it is, which as we shall see depends not only on the maturity of the bond, but also on the size of the coupons. 1. If y < y* sell because you believe the bond is overpriced. 2. If y > y* buy because you believe the bond is underpriced. Spot rates and zero-coupon bonds

i. A spot rate of interest is the YTM on a t-year zero... ii. Let yt be the spot rate for a t-year zero-coupon (“discount”) bond. iii. The price of a zero-coupon bond that pays out F at time t is equal to iv. If we know the bond price, we can figure out the spot rate and vice versa. v. The yield-to-maturity can be thought of as a weighted average of the spot rates for each coupon date. q. Strips i. Bonds are quite complex – the three year bond is a bundle of 6 cash flows ii. To aid liquidity, Government makes it possible to strip some bonds – unpackaging the individual elements and trading separately 1. Strips are zero coupon bonds 2. Price of bond equals sum of strips – or else arbitrage r. Analysis of term structure

s. Term structure of interest rate i. Even with single issuer, deep and liquid market, yields differ across bonds 1. Called the term structure of interest rates 2. Tax used to be an issue 3. Treatment of interest and capital gains differed across investors, led to clienteles with own term structure t. Interpreting the term structure i. The yield curve is a good predictor of the business cycle. 1. Long term rates tend to rise in anticipation of economic expansion. 2. Inverted yield curve may indicate that interest rates are expected to fall and signal a recession. u. Term structure i. Have inferred term structure from strip prices 1. No strips in many markets 2. Can readily infer from standard bond prices ii. In general, have M bonds 1. Bond m promises cash flow of Xm,t in year t 2. It costs Pm 3. If strips did exist, the price of a strip of maturity t would be St Then the following equations must hold:

Pm =Xm,1S1 + Xm,2S2 + … + Xm,TST iii. Have M equations with T unknowns 1. Can solve exactly if M=T 4. Forward rate

a. Difference between forward rates and (expected) future spot rates i. Forward Rate: 1 year rate between 2017 and 2018 fixed in 2016 ii. Expected future spot rates: 1 year spot rate between 2017 and 2018, but expected in 2016 and to be revealed in 2017 iii. Rough view: 1. Bond market is highly liquid 2. Many players (borrowers and lenders) who are not that fixed on a particular maturity 3. Little “inside” information; many smart analysts 4. So forward rate unlikely to be seriously out of line with market expectations of future spot rates 5. Clienteles a. There are preferences. If the Expectations Hypothesis holds, forward equals expected future spot, and investors will match their needs i. Pension funds will hold long dated ii. Investors with liquidity needs will hold short b. Supply is important too

i. Borrowers will match maturity to cash flow needs, and reflect risk management concerns ii. Government issuance integrated with monetary policy c. But if supply and demand don’t match prices will adjust i. If liquidity is important to investors and securing long term finance important for borrowers, there will be a liquidity premium ii. On average short rates will be lower than long rates iii. Forward rates will be higher than expected future spot rates iv. Fn = E(rn) + liquidity premium 6. Bond prices and interest rates a. Suppose you hold a 10-year 5% coupon bond in your portfolio i. Currently interest rates are 5%, and the bond is at par (100) b. Interest rates generally rise to 6% i. The value of your bond falls to ii. The cash flow remains the same iii. The expected return on your money has gone up c. Are you made better or worse off by the rate change? i. On a mark-to-market basis, worse off ii. If funding a long-term liability, the loss is offset by a fall in the value of the liability d. Interest Rate Sensitivity i. 1% rise in rates caused bond price to fall by 7.36% ii. Interest rate sensitivity strongly related to maturity, but also depends on coupon iii. Price P is a function of the yield y on the bond iv. Differentiating: v. But we don’t want £ change in price per 1% change in rates, but % change in price 7. Duration a. Dp/P = -Ddy/(1+y) where

b. D is called Duration i. If all cash flows are in year T , duration is T ii. If it is spread over the period 0…T it is a measure of the average life of the cash flows (with years with bigger cash flows having more weight) iii.

D=

PV ( X T )  PV ( X 2)  PV ( X 1 )  ∗2+…+ ∗1+ ∗T P P P

iv. To compute need to know cash flows and bond price v. Bond duration always calculated using bond’s own redemption yield c. Duration: (2017-4-a) i. Of a zero coupon bond equals its maturity ii. Is lower the higher the coupon iii. Is greater the longer the maturity iv. Goes to (1+y)/y for console (perpetual) bond v. Tends to decline over time

d. A very useful measure i. Duration is a measure of portfolio’s sensitivity to interest rates 1. Duration of portfolio is weighted average of durations of individual bond holdings 2. Many bond portfolios held to match liabilities; useful to check whether durations match 3. Many financial institutions mismatched (e.g. Banks); use duration as a measure of equity’s exposure to interest rates ii. Two cautions: 1. Only applies to small changes 2. When comparing across bonds implicitly assumes that they are subject to same yield change – i.e. That shifts in the yield curve are always parallel e. Duration, Modified Duration and Dollar Duration i. The formula is: ii. D/(1+y) is technically called modified duration 1.

dP  P

= -(Modified Duration)*dy

2. The 1+y comes in only because the yield is annually compounded 3. With yield componded n times per year, modified duration is D/(1 + y/n) iii. For brevity, will use term duration to mean modified duration hereafter iv. If you have $100m face value of bond, with market value of $97m and (modified) duration of 7 years, then a 1 bp rise in yields will cause value of holding to fall by 0.01%x7x$97m = $67.9k 1. Dollar duration = MV x D = $679m-yrs f. Floating rate notes i. Floating rate note pays coupon equal to current short-term interest rate 1. E.g. 10 year FRN paying LIBOR quarterly 2. If quarter begins today (“reset date”) and 3-month LIBOR rate is 4.4%, then coupon paid in three months’ time is £1.10/£100 nominal ii. Like a deposit that always pays going interest rate 1. So ignoring credit and liquidity issues, will trade at par (face value), at least at reset date 2. So bond will be worth £101.10 at next reset date whatever happens to interest rates 3. Price today is £101.10/(1+y/4)4t where t is time to next coupon and duration is t/(1+y/4) – which is close to 0 - even though maturity is ten years g. Immunisation i. Suppose portfolio contains many assets with values A1, A2 … and corresponding durations d1, d2 … 1. And liabilities have value L and duration dl 2. Then portfolio is immunised – i.e. Protected against small changes in interest rates – if dollar duration of assets and liabilities are the same A1d1 + A2d2 … = ldl ii. Immunisation works best using bonds that are similar to liabilities being hedged 1. Similar means that yield changes are similar 2. Actual change in value is -a1d1dy1 -a2d2dy2 …+ ldl dyl Where dyn is the change in the yield of bond n h. Using duration i. Pension fund has liability to pay £100m/year for 20 years ii. Intends to invest in 4% 10 year bonds and a Floating rate Security

iii. How do we immunize the pension fund’s interest rate risk using duration? 1. A1d1 + A2d2 = ldl 2. A1+ A2 = L 3. A1 =?; A2 = ?; d1 = ? D2 =? L =? Dl= ? iv.

dP  P

= -(Modified Duration)*dy

-(P(y+dy)-P(y)) P(y)* dy (P(y)-P(y+dy)) 2. Or, Modified Duration = P(y)* dy 1. Modified Duration =

8. Implications a. With short positions, can match any desired duration i. Will protect against any small and parallel shift in yield curve ii. But exposure to changes in slope or curvature of yield curve may be devastating b. Note that value of hedged position goes down for large move in either direction i. Position has negative convexity ii. True that you will tend to lose from large moves

9. Convexity a. The relationship between bond prices and yields is not linear. b. Duration rule is a good approximation for only small changes in bond yields. c. Bonds with greater convexity have more curvature in the price-yield relationship.

d. Duration is only a local measure of interest rate sensitivity

e. For greater precision: i. The convexity of a zero coupon bond that matures at time T is T(T+1)/(1+y)2 ii. The convexity of a portfolio is the weighted average of the convexity of components iii. If matching duration of assets and liabilities is like matching mean time to repay, then matching duration and convexity is like matching mean and standard deviation f. Why do investors like convexity i. Bonds with greater curvature gain more in price when yields fall than they lose when yields rise. ii. The more volatile interest rates, the more attractive this asymmetry. iii. Bonds with greater convexity tend to have higher prices and/or lower yields, all else equal.

Use of Derivatives 1. Overview a. Have considered the traditional building blocks of a portfolio – equities, bonds b. Each of these is a complex bet on management skills, economic variables, exchange rates, interest rates, credit spreads i. Problem if for example you like the management but don’t like the country, or you like the sector long term but think it will be hit by currency movements short term ii. Derivatives allow one to divest unwanted risks while keeping the others iii. Example: Buying Barrick Gold vs. Gold Futures on the CME 2. Example of use a. Japanese investor holds $US10m of corporate bond with 2 years to maturity i. Worried that dollar will fall ii. Does not want to sell bond because yield spread against US Treasuries and other corporates looks high iii. By selling dollars forward, locks in an exchange rate based on current spot rate b. Forward rates quoted by bank based on arbitrage i. Bank which buys dollars forward will hedge against fall in dollar by borrowing dollars, selling dollars spot, and depositing yen c. Forward allows investor to manage exposure to different risks (corporate credit spread, currency exposure) separately 3. Forward contract a. A forward contract is a contract to buy an asset at a pre-specified price at a specified future date b. Have already met one example – the forward interest rate contract i. Spot rate is 4.00% for one year. 4.62% for two ii. One year rate one year forward is 1.04622/1.04 – 1 = 5.26% iii. Can agree now that I will receive £100 in 2 years time in return for paying £100/1.0526 = £95 in one year iv. This is a forward contract, with a maturity of 1 year, on a 2 year zero coupon bond with a forward price or strike price of 95.00.

c. Forward markets for shares i. Investor holds $1000 in a mutual fund indexed to the S&P 500.(S0) ii. Assume dividends of $20 will be paid on the index fund at the end of the year. (D) iii. A forward contract with delivery in one year is available for $1,010. (F0)

iv. The investor hedges by selling or shorting one contract. v. One can interpret the formula F=S(1+rf -d) as follows: vi. If the forward contract is priced correctly, you should be indifferent between: 1. Arranging today to buy the underlying asset at a cost of F one year from now; and, 2. Paying S to buy the asset today, foregoing interest of rf*S on your money (over the next year), but receiving d*S (at the end of the year) as a benefit to holding on to the asset for the year.

d. Forward markets for FX i. There are two risk-free ways to get foreign currency (say, USD) T periods from now: 1. Convert your GBP now at the spot rate 1/S and then invest them at the riskfree rate rus in the US for T periods. If you convert one GBP you will get.

2. OR…You could invest your GBP in the U.K. for one year at the rate ruk and at the same time enter into a forward contract to convert the pounds into dollars T periods from now at a rate F

e. Forward FX Contract i. Since both methods are risk free their payoffs must be the sa...