Distinction Between Interest Rates and Returns

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1 Distinction Between Interest Rates and Returns Rate of Return RET = C + P t+1 P t =i c + g P t C where: i c = = current yield P t g = P t+1 P t P t = capital gain

3 Maturity and the Volatility of Bond Returns Only bond whose return = yield is one with maturity = holding period For bonds with maturity > holding period, i Pfl implying capital loss Longer is maturity, greater is price change associated with interest rate change

4 Maturity and the Volatility of Bond Returns Longer is maturity, more return changes with change in interest rate Bond with high initial interest rate can still have negative return if i Prices and returns more volatile for long-term bonds because have higher interest-rate risk No interest-rate risk for any bond whose maturity equals holding period

5 Distinction Between Real and Nominal Interest Rates Real Interest Rate Interest rate that is adjusted for expected changes in the price level i r = i π e 1. Real interest rate more accurately reflects true cost of borrowing 2. When real rate is low, greater incentives to borrow and less to lend if i = 5% and π e = 3% then: i r = 5% 3% = 2% if i = 8% and π e = 10% then i r = 8% 10% = 2%

7 Thinking about Uncertain Returns need to plan for different return scenarios think of returns being generated by a chance mechanism use historical series of returns to infer probabilities for future returns the investment horizon matters a lot for what these probabilities are potential problem: probabilities for tomorrow's returns might depend on current return, or other variables (e.g. boom, recession) actually, returns not well predictable over short periods of time, but over long periods there is predictability

8 Risks to Bondholders (1) One-Period Nominal Return for a bond with coupon payment C, price P : if maturity is one period, then price is known, nominal return can be foreseen (except perhaps if bankruptcy risk) if maturity is longer than one period, price fluctuations imply that bond is a risky asset! long bond prices fluctuate if one-period interest rate changes (higher short rate! bond prices go down)

9 Risks to Bondholders (2) Real Return on a bond, with expected rate of inflation: `ex post' (with realized inflation), Fisher equation true by definition `ex ante' (with expected inflation), it is hard to test, because expected inflation is not observed but: many countries have indexed bonds, so real returns become observable in countries with hi inflation risk, firms often borrow in foreign currency

10 ³ ² ± Probability Distributions Suppose a random variable, such as a return R, cantaken values, R 1,R 2,..., R N, with probabilities π 1,π 2,..., π N, respectively. We summarize its behavior by Mean E[R] = N n=1 π nr n Measures of Risk Variance var(r) = N n=1 π n(r n E[R]) 2 Standard Deviation σ(r) = var(r)

11 Risk Return Tradeoff mean-standard-deviation diagram shows risk and expected return of possible portfolios every asset (or portfolio) can be represented by one point given a risky and a riskless asset, we can generate any portfolio on a straight line through the two assets points between the two assets correspond to positive portfolio weights. Preferences determine the optimal mix of risky and riskless asset.

12 Several Risky Assets Consider two stocks State 1 2 Probability.5.5 Pepsi return R P Coke return R C 0.2 Expected Returns: E[R P ] = 0 E[R C ] = :1 Standard Deviations StD(R C )= p :5(:1) 2 + :5( :1) 2 ) = :1, StD(R P ) = p :5( :15) 2 + :5(:15) 2 ) = :15! Coke looks more attractive than Pepsi, but: combining the two can give a riskless portfolio (pick w P = :4; w C = :6)!

13 Diversification Bottom Line: by holding several risky assets, risk can be reduced (diversification) expect to see people spreading wealth across many assets how `good' an asset looks depends not only on own risk and return; for example, emerging markets are attractive because of low correlation with US index, puzzle: not a lot of cross-border equity holdings

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