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Nominal and real rates of interest
Cash flows can either be in current (nominal) or constant (real) dollars. If you deposit €100 in a bank account with an interest rate of 5 percent, the balance is €105 by the end of the year. Whether €105 can buy you more goods and services that €100 today depends on the rate of inflation over the year.
Inflation is the rate at which prices as a whole are increasing, whereas nominal interest rate is the rate at which money invested grows. The real interest rate is the rate at which the purchasing power of an investment increases.
The formula for converting nominal interest rate to a real interest rate is:
For small inflation and interest rates the real interest rate is approximately equal to the nominal interest rate minus the inflation rate.
Investment analysis can be done in terms of real or nominal cash flows, but discount rates have to be defined consistently
- Real discount rate for real cash flows
- Nominal discount rate for nominal cash flows
Valuing bonds using present value formulas
A bond is a debt contract that specifies a fixed set of cash flows which the issuer has to pay to the bondholder. The cash flows consist of a coupon (interest) payment until maturity as well as repayment of the par value of the bond at maturity.
The value of a bond is equal to the present value of the future cash flows:
11) Value of bond = PV(cash flows) = PV(coupons) + PV(par value)
Since the coupons are constant over time and received for a fixed time period the present value can be found by applying the annuity formula:
12) PV(coupons) = coupon • annuity factor
- Consider a 10-year US government bond with a par value of $1,000 and a coupon payment of $50. What is the value of the bond if other medium-term US bonds offered a 4% return to investors?
Value of bond = PV(Coupon) + PV(Par value)
= $50 • [1/0.04 - 1/(0.04-1.0410)] + $1,000 • 1/1.0410 = $50 • 8.1109 + $675.56 = $1,081.1
Thus, if other medium-term US bonds offer a 4% return to investors the price of the 10-year government bond with a coupon interest rate of 5% is $1,081.1.
The rate of return on a bond is a mix of the coupon payments and capital gains or losses as the price of the bond changes:
Because bond prices change when the interest rate changes, the rate of return earned on the bond will fluctuate with the interest rate. Thus, the bond is subject to interest rate risk. All bonds are not equally affected by interest rate risk, since it depends on the sensitivity to interest rate fluctuations.
The interest rate required by the market on a bond is called the bond's yield to maturity. Yield to maturity is defined as the discount rate that makes the present value of the bond equal to its price. Moreover, yield to maturity is the return you will receive if you hold the bond until maturity. Note that the yield to maturity is different from the rate of return, which measures the return for holding a bond for a specific time period.
To find the yield to maturity (rate of return) we therefore need to solve for r in the price equation.
- What is the yield to maturity of a 3-year bond with a coupon interest rate of 10% if the current price of the bond is 113.6?
Since yield to maturity is the discount rate that makes the present value of the future cash flows equal to the current price, we need to solve for r in the equation where price equals the present value of cash flows:
PV(Cash flows) = Price on bond
The yield to maturity is the found by solving for r by making use of a spreadsheet, a financial calculator or by hand using a trail and error approach.
Thus, if the current price is equal to 113.6 the bond offers a return of 5 percent if held to maturity.
The yield curve is a plot of the relationship between yield to maturity and the maturity of bonds.
Figure 1: Yield curve
As illustrated in Figure 1 the yield curve is (usually) upward sloping, which means that long-term bonds have higher yields. This happens because long-term bonds are subject to higher interest rate risk, since long-term bond prices are more sensitive to changes to the interest rate.
The yield to maturity required by investors is determined by
1. Interest rate risk
2. Time to maturity
3. Default risk
The default risk (or credit risk) is the risk that the bond issuer may default on its obligations. The default risk can be judged from credit ratings provided by special agencies such as Moody's and Standard and Poor's. Bonds with high credit ratings, reflecting a strong ability to repay, are referred to as investment grade, whereas bonds with a low credit rating are called speculative grade (or junk bonds).
In summary, there exist five important relationships related to a bond's value:
1. The value of a bond is reversely related to changes in the interest rate
2. Market value of a bond will be less than par value if investor's required rate is above the coupon interest rate
3. As maturity approaches the market value of a bond approaches par value
4. Long-term bonds have greater interest rate risk than do short-term bonds
5. Sensitivity of a bond's value to changing interest rates depends not only on the length of time to maturity, but also on the patterns of cash flows provided by the bond
Valuing stocks using present value formulas
The prier of a stock is equal to the present value of all Suture dividends. The intuition behind this insight is that the cash payoff to owners of the stock is equal to cash dividends plus capital gains or losses. Thus, the expected return that an investor expects from a investing in a stock over a set period of time is equal to:
Where Divt and Pt denote the dividend and stock price in year t, respectively. Isolating the current stock price P0 in the expected return formula yields:
The question then becomes "What determines next years stock price PJ". By changing the subscripts next year's price is equal to the discounted value of the sum of dividends and expected price in year 2:
Inserting this into the formula for the current stock price P0 yields:
By recursive substitution the current stock price is equal to the sum of the present value of all future dividends plus the present vdue of the horizon stock price, PH.
The final insight is that as Hi approaches zero,[PH / (1+ r)H] approaches zero. Thus, in the limit the curer stock price, P0, can be expressed as the sum of the present value of all future dividends.
Discounted dividend model
In cases where firms have constant growth in the dividend a special version of the discounted dividend model can be applied. If the dividend grows at a constant rate, g, the present value of the stock can be found by applying the present value formula for perpetuities with constant growth.
Discounted dividend growth model
The discounted dividend growth model is often referred to as the Gordon growth model.
Some firms have both common and preferred shares. Common stockholders are residual claimants on corporate income and assets, whereas preferred shareholders are entitled only to a fixed dividend (with priority over common stockholder). In this case preferred stocks can be valued as a perpetuity paying a constant dividend forever.
The perpetuity formula can also be applied to value firms in general if we assume no growth and that all earnings are paid out to shareholders.
If a firm elects to pay a lower dividend, and reinvest the funds, the share price may increase because future dividends may be higher.
Growth can be derived from applying the return on equity to the percentage of earnings ploughed back into operations:
20) g = return on equity • plough back ratio
Where the plough back ratio is the fraction of earnings retained by the firm. Note that the plough back ratio equals (1 - payout ratio), where the payout ratio is the fraction of earnings paid out as dividends.
The value of growth can be illustrated by dividing the current stock price into a non-growth part and a part related to growth.
Where the growth part is referred to as the present value of growth opportunities (PVGO). Inserting the value of the no growth stock from (22) yields:
Firms in which PVGO is a substantial fraction of the current stock price are referred to as growth stocks, whereas firms in which PVGO is an insignificant fraction of the current stock prices are called income stocks.