FM Exam Application
Cash flow analysis
NPV = present value of inflow - present value of outflow
- The internal rate of return is the interest rate that gives an NPV of 0.
Investment Funds
Setup: the fund is worth {$F_0$} at time 0. For {$k=1, 2, \ldots, n$}, {$c_k\,$} amount of money is invested in the fund at time {$t_k\,$}. The fund is worth {$F_T\, $} at time {$T$}. We want to measure the rate of return on this fund. There are two methods.
- The dollar-weighted rate of interest is the interest {$i$} computed via the following
accumulated value of the initial value of the fund
+ accumulated values of each payment
= the value of the fund at time {$T$};
{$$ F_0(1+i)^T +c_1(1+i)^{T-t_1} +c_2(1+i)^{T-t_2} +c_3(1+i)^{T-t_3} +\cdots+c_n(1+i)^{T-t_n} =F_T\,.$$}
When {$T$} is small, we can approximate a solution by using the binomial approximation
{$$ (1+i)^p \approx 1+pi. $$}
The dollar-weighted rate depends on the amount of money {$c_k\,$} invested and the time {$t_k\,$} at which the investments are made .
- We can also find the time-weighted rate of interest. Here we no longer need to know the {$t_k$} values, but we do need to know the values of the fund {$F_{k}\,$} right before each investment {$c_{k}\,$} is added. The interest {$i$} is computed via the following formula
{$$(1+i)^T = \frac{F_1}{F_0} \cdot \frac{F_2}{F_1+c_1} \cdot \frac{F_3}{F_2+c_2} \cdot \cdots \cdot \frac{F_T}{F_n+c_n}.$$}
The time-weighted method minimizes the roles of the time periods between payments and the amount of payments.
Loans
Two different ways to pay back a loan.
Amortization method
- Part of each payment is used for the interest and the rest goes to reducing the loan principal.
- The interest portion will decrease over time because the principal is reduced over time.
- The present value of the loan equals the present value of the loan payments, which form an annuity.
- To figure out how much of the loan is outstanding, use one of the following two methods:
remaining value of the loan
= future value of the total loan - future values of the payments so far
remaining value of the loan
= present value of payments not yet submitted
Either method should give us the same answer, but it seems like the retrospective method is more often used.
Sinking fund method
- A payment consists of the service portion and the sinking fund portion.
- The service portion is the interest of the loan; it stays constant throughout because the principal also stays constant and isn't paid off until at the very end.
- The other portion goes into a sinking fund that may or may not accrue interest at the same rate as the loan. The future value of the series of payments should equal the value of the loan.
Relationship
- When the two interest rates are the same in the sinking fund method, the payments are the same as in the amortization method.
{$$ \frac{1}{a_{\overline{n}|}} = i + \frac{1}{s_{\overline{n}|}}$$}
Bonds
Setup
- Face amount (par amount) {$F$} vs. redemption amount {$C$} vs. price of bond {$P$}. Usually {$C=F$}. The bond is bought (sold) at par if {$P=C$}; at a premium if {$P>C$}; at a discount if {$P<C$}.
- Coupon rate {$r$} vs. yield rate {$j$} (usually a semiannual effective rate).
Value of a bond
- The initial value is {$P$}, obtained by the following
Price= present value of coupons + present value of redemption value
- The book value of the bond (right after a coupon payment at any time) is obtained by doing the same thing with the remaining coupons.
Think of a bond as the bond issuer trying to pay back a loan to the bond holder. The loan interest rate is the yield rate, the loan payments are the coupons and the redemption payment at the end.
- When a bond is bought at a premium, the price is larger than the redemption amount, so that the coupon rate is higher than the yield rate. So each coupon payment is bigger than the interest earned, which knocks down the bond price a little bit with each coupon payment.
The series of differences forms an amortization of the remium:
amount of premium
= sum of the differences between the coupon payments and interests.
(Take difference to be positive, i.e., coupon - interest).
- When a bond is bought at a discount, the price is lower than the redemption amount, so the coupon rate is lower than the yield rate. Each coupon payment is lower than the interest payment. That leaves a little bit of interest to accumulate in the bond price.
The series of differences form an accumulation of discount:
amount of discount
= sum of the differences between the coupon payments and interests.
(Take interest - coupon.)
- Price {$P_t$} at a fractional {$t$} periods after a payment is determined by two methods:
- Price-plus-accrued of the bond is
{$$ P_0(1+j)^t, $$}
where {$P_0$} is the book price right after the last payment.
{$$ P_0(1+j)^t-t(Fr) $$}
Note this works for {$t=1$} as well.