Chapter 3 The concept of present value (or present discounted value) is based on the common-sense notion that a dollar of cash flow paid to you one year from now is less valuable to you than a dollar paid to you today: Simple Loan Principal – amount of funds given by lender Maturity Date – when it must be paid off If you would have lent out $100 and interest of 10% by the end of year 1 Formula of figuring out how each year would pan out. Working backwards from future amounts to the present. The process of calculating today’s value of dollars received in the future, is called discounting the future. PV – present value CF – cash flow What is the present value of $250 to be paid in two years if the interest rate is 15%? Cash flow = $250 I = 15% N = 2 Years PV = 250 / (1+15%) ^2 PV = 250 /1.3225 PV = $189.04 Year 1 Value – 189.04 Year 2 Value = 250 Present value is extremely useful because it enables us to figure out today’s value of a credit market instrument at a given simple interest rate i by just adding up the present value of all the future cash flows received The Types of Credit Market Instruments Simple Loan – loan that must be paid to lender at maturity date Fixed payment loan - must be repaid by making the same payment every period consisting of part of the principal and interest for a set number of years – instalment loans, car loans etc. $125 every month for 48 months Coupon Bond – pays the owner of the bond a fixed interest payment (coupon payment) every year until the maturity date, when a specified final amount aka face value – Example the coupon bond has a yearly coupon payment of $100 and a face value of $1,000. The coupon rate is then $100/$1,000 = 0.10, or 10%. A coupon bond is identified by three pieces of information. First is the corporation or government agency that issues the bond. Second is the maturity date of the bond. Third is the bond’s coupon rate, the dollar amount of the yearly coupon payment expressed as a percentage of the face value of the bond. Discount Bond – bought below face value and repaid at maturity date. Discount bond does not make any interest payments; it just pays off the face value. YIELD TO MATURITY The interest rate that equates the present value of cash flows received from a debt instrument with its value today In a simple loan If Pete borrows $100 from his sister and next year, she wants $110 back from him, what is the yield to maturity on this loan? We use present value formula backwards to figure out (i) interest PV - $100 – amount that is borrowed CF - $110 – cash flow in a year N – 1 year – number of years 100 = 110 / (1+i) (1+i) *100 = 110 (1+i) = 110/100 (1+i) = 1.10 I = 1.10 – 1 I = 0.10 = 10% for simple loans, the simple interest rate equals the yield to maturity Fixed Payment Loan You decide to purchase a new home and need a $100,000 mortgage. You take out a loan from the bank that has an interest rate of 7%. What is the yearly payment to the bank to pay off the loan in 20 years? LV (Loan Value) = $100,000 I = 7% - interest rate N = 20 years Coupon Bond P – price of coupon bond C – yearly coupon payment F – face value of the bond N = years to maturity date Perpetuity Coupon Bond Fixed income security with no maturity date. This type of bond is often considered a type of equity, rather than debt. Or ic = c/pc What is the yield to maturity on a bond that has a price of $2,000 and pays $100 annually forever? C – yearly payment – 100 Pc – price of perpetuity – 2000 Ic = 100/2000 Ic – 5% DISCOUNT BOND I – F-P/P F = face value of the discount bond P = current price of the discount bond A discount bond such as a one-year U.S. Treasury bill, which pays a face value of $1,000 in one year’s time. If the current purchase price of this bill is $900, then equating this price to the present value of the $1,000 received in one year Present Value - $900 CF – 1000 Need to solve for i $900 = $1000/ (1+i) (1+i) x $900 = $1000 $900 + 900i = $1000 900i = $1000-$900 I – $1000-$900/$900 = 0.111 I = 1.11% Our calculations of the yield to maturity for a variety of bonds reveal the important fact that current bond prices and interest rates are negatively related: When the interest rate rises, the price of the bond falls, and vice versa. The Distinction Between Real and Nominal Interest Rates Nominal – not adjusted for inflation Real – adjusted for inflation, ex ante real interest rate because it is adjusted for expected changes in the price level Nominal Interest Rate Formula - i=ir +πe i – nominal interest rate ir – real interest rate πe – expected rate of inflation Real Interest Rate Formula – ir = i – πe What is the real interest rate if the nominal interest rate is 8% and the expected inflation rate is 10% over the course of a year? i – 8% πe – 10% ir = 8% -10% = -2% In this case it is better for the borrower as they pay 2% less but worse for the loaner as they earn 2% less. When the real interest rate is low, there are greater incentives to borrow and fewer incentives to lend. The Distinction Between Interest Rates and Returns The rate of return is defined as the payments to the owner plus the change in its value, expressed as a fraction of its purchase price. R = return from holding the bond from time t to time t + 1 Pt = price of the bond at time t Pt+1 = price of the bond at time t + 1 C = coupon payment What would the rate of return be on a bond bought for $1,000 and sold one year later for $800? The bond has a face value of $1,000 and a coupon rate of 8%. C = Coupon Payment = 1000 x 8% = $80 Pt+1 = price of the bond one year later = $800 Pt = price of bond today = $1000 R = $80 + ($800-$1000)/$1000 R = -120/1000 = -0.12 R= 12% This can be rewritten as R = C/Pt + Pt+1 -Pt/Pt Then we see the current yield Ic = C/Pt Then we find the rate of capital gain G = Pt+1-Pt/Pt Rate of Capital = R R= ic + g RATE OF CAPITAL GAIN EXAMPLES Calculate the rate of capital gain or loss on a 10-year zero-coupon bond for which the interest rate has increased from 10% to 20%. The bond has a face value of $1,000. G = Pt+1-Pt/Pt Pt+1 = price of the bond one year from now = $1000 / (1+0.20) ^9(9 because it’s one year from now, now is year 10) = $193.81 Pt = price of the bond today = $1000 / (1+0.10) ^10 = $385.84 G = $193.81 - $385.54 / $385.54 G = -0.947 = - 49.7% CALCULATING DURATION Duration is a measure of the sensitivity of the price of a bond or other debt instrument to a change in interest rates. All else being equal, the longer the term to maturity of a bond, the longer its duration. All else being equal, when interest rates rise, the duration of a coupon bond falls. All else being equal, the higher the coupon rate on the bond, the shorter the bond’s duration. A pension fund manager is holding a 10-year 10% coupon bond in the fund’s portfolio, and the interest rate is currently 10%. What loss would the fund be exposed to if the interest rate rises to 11% tomorrow? DUR – Duration – 6.76 Δi – change in interest rate = 0.11-0.10 = 0.01 i = current interest rate = 0.10 %∆P = -6.76 x 0.01 / 1+0.10 %∆P = -0.0615 = -6.15% Now the pension manager has the option to hold a 10-year coupon bond with a coupon rate of 20% instead of 10%. As mentioned earlier, the duration for this 20% coupon bond is 5.98 years when the interest rate is 10%. Find the approximate change in the bond price when the interest rate increases from 10% to 11%. Dur = -5.98 Δi = 0.11-0.10 = 0.01 i = 0.10 %∆P = -5.98 x 0.01/1+0.10 %∆P = -0.054 = -5.4% Maturity and the Volatility of Bond Returns: Interest-Rate Risk Prices and returns for long-term bonds are more volatile than those for shorter-term bonds. Although long-term debt instruments have substantial interest-rate risk, short- term debt instruments do not. Reinvestment Risk However, if an investor’s holding period is longer than the term to maturity of the bond, the investor is exposed to a type of interest-rate risk called reinvestment risk. Reinvestment risk occurs because the proceeds from the short-term bond need to be reinvested at a future interest rate that is uncertain. The yield to maturity, which is the measure most accurately reflecting the interest rate, is the inter- est rate that equates the present value of future cash flows of a debt instrument with its value today The real interest rate is defined as the nominal inter- est rate minus the expected rate of inflation. It is both a better measure of the incentives to borrow and lend and a more accurate indicator of the tight- ness of credit market conditions than the nominal interest rate. The return on a security, which tells you how well you have done by holding this security over a stated period of time, can differ substantially from the interest rate as measured by the yield to maturity. Long-term bond prices have substantial fluctuations when interest rates change and thus bear interest- rate risk. The resulting capital gains and losses can be large, which is why long-term bonds are not considered to be safe assets with a sure return. Bonds whose maturity is shorter than the holding period are also subject to reinvestment risk, which occurs because the proceeds from the short-term bond need to be reinvested at a future interest rate that is uncertain.
