2. Interest Rates

Present Value

PV = CF / (1 + i)ⁿ

The value today of a cash flow received in n years

Yield to maturity

Interest rate that makes the present value of cash flows equal to its value(price) today

Useful to compare debt instruments of different types

Yield to maturity of a simple loan

If n = 1: it is equal to the simple interest rate i = FV - PV/PV *n

If n > 1: i = (FV/PV)^1/n -1

Yield to maturity of a fixed-payment loan

LV = FP / (1 + i) + FP/(1 + i)² + ….. + FP/(1 + i)ⁿ

Yield to maturity of a coupon bond

P = C / (1 + i) + C/(1 + i)² + ….. + C /(1 + i)ⁿ + F/(1 + i)ⁿ

Characteristics of the yield to maturity for coupon bonds

1. YTM = C when P = F

2. YTM > C when P < F

3. P and YTM are negatively correlated

Current yield of a perpetuity

P = C/i_c

Can approximate the behavior of coupon bonds with a long term to maturity

Yield to maturity of a discount bond

If n = 1: FV - PV/PV *n

If n>1: (FV/PV)^1/n -1

Fisher equation

Nominal interest rate equals the (ex-ante) real interest rate plus expected inflation:

i = i_{r +} π^e

When i_r is low, it is more convenient to borrow and less convenient to lend

Rate of return

Payments plus change in value, expressed as a fraction of the purchase price

R = (C + P_t+1 - P_t)/P_t = i_c + g

Characteristics of the rate of return for bonds

• If YTM = R then time until maturity = holding period

• If maturity > holding period and interest rates increase, then P > P at time of sale (capital loss)

• The more distant the maturity, the more impact interest rate change has on price change

• The more distant the maturity, the lower the rate of return caused by an increase in the interest rate

• Even if the initial interest rate is high, a bond’s return can be negative if they rise

Interest rate risk

The riskiness of an asset’s return caused by change in the interest rate

Higher the further away maturity is (long-term bonds particularly exposed)

Non-existent when maturity = holding period

Reinvestment risk

Type of interest rate risk where it is uncertain if proceeds from bond can be reinvested

Present when maturity < holding period
Investors benefit from a rise in the interest rate and viceversa

Duration

Weighted average of the maturities of cash payments

Measures effective maturity (average lifetime)

Σ_{t =1} ⁿ t CP/(1+i)^t / Σ_{t =1} ⁿ CP/(1+i^)t

Characteristics of duration

AEE, the longer the term to maturity, the longer the duration

AEE, the higher the coupon rate, the shorter the duration

AEE, when interest rates rise, duration (of a coupon bond) falls

The duration of a portfolio is the weighted average of the duration of the individual securities, based on the proportion invested in each

The greater the duration, the greater the interest rate risk

Price change given duration formula

%∆ P ≈ -DUR * ∆i/ 1 + i

where %∆ P = (P_t+1 - P_t)/P_t